{
    "componentChunkName": "component---src-templates-markdown-doc-tsx",
    "path": "/2025.1.3/knowledge_base/boundary_conditions/absorbing_boundaries",
    "result": {"data":{"mdx":{"id":"8a6cfe7d-00ec-5844-832f-e17c83b6ac16","frontmatter":{"title":"Absorbing Boundaries","images":[{"publicURL":"/static/87b961a0676d801f78071332e7544de5/absorbing_layers.png","childImageSharp":{"gatsbyImageData":{"layout":"constrained","backgroundColor":"#f8f8f8","images":{"fallback":{"src":"/static/87b961a0676d801f78071332e7544de5/c7240/absorbing_layers.png","srcSet":"/static/87b961a0676d801f78071332e7544de5/de3a1/absorbing_layers.png 150w,\n/static/87b961a0676d801f78071332e7544de5/30cdc/absorbing_layers.png 300w,\n/static/87b961a0676d801f78071332e7544de5/c7240/absorbing_layers.png 600w","sizes":"(min-width: 600px) 600px, 100vw"},"sources":[{"srcSet":"/static/87b961a0676d801f78071332e7544de5/c65bc/absorbing_layers.webp 150w,\n/static/87b961a0676d801f78071332e7544de5/078c3/absorbing_layers.webp 300w,\n/static/87b961a0676d801f78071332e7544de5/6d09e/absorbing_layers.webp 600w","type":"image/webp","sizes":"(min-width: 600px) 600px, 100vw"}]},"width":600,"height":600}}}]},"body":"var _excluded = [\"components\"];\nfunction _extends() { _extends = Object.assign || function (target) { for (var i = 1; i < arguments.length; i++) { var source = arguments[i]; for (var key in source) { if (Object.prototype.hasOwnProperty.call(source, key)) { target[key] = source[key]; } } } return target; }; return _extends.apply(this, arguments); }\nfunction _objectWithoutProperties(source, excluded) { if (source == null) return {}; var target = _objectWithoutPropertiesLoose(source, excluded); var key, i; if (Object.getOwnPropertySymbols) { var sourceSymbolKeys = Object.getOwnPropertySymbols(source); for (i = 0; i < sourceSymbolKeys.length; i++) { key = sourceSymbolKeys[i]; if (excluded.indexOf(key) >= 0) continue; if (!Object.prototype.propertyIsEnumerable.call(source, key)) continue; target[key] = source[key]; } } return target; }\nfunction _objectWithoutPropertiesLoose(source, excluded) { if (source == null) return {}; var target = {}; var sourceKeys = Object.keys(source); var key, i; for (i = 0; i < sourceKeys.length; i++) { key = sourceKeys[i]; if (excluded.indexOf(key) >= 0) continue; target[key] = source[key]; } return target; }\n/* @jsxRuntime classic */\n/* @jsx mdx */\n\nvar _frontmatter = {\n  \"title\": \"Absorbing Boundaries\",\n  \"images\": [\"../images/absorbing_layers.png\"]\n};\nvar layoutProps = {\n  _frontmatter: _frontmatter\n};\nvar MDXLayout = \"wrapper\";\nreturn function MDXContent(_ref) {\n  var components = _ref.components,\n    props = _objectWithoutProperties(_ref, _excluded);\n  return mdx(MDXLayout, _extends({}, layoutProps, props, {\n    components: components,\n    mdxType: \"MDXLayout\"\n  }), mdx(\"h1\", null, \"Absorbing Boundaries\"), mdx(\"p\", null, \"In many situations the computational domain needs to be truncated, which requires absorbing boundaries to avoid artificial reflections.\"), mdx(\"h2\", null, \"First-order boundary condition\"), mdx(\"h3\", null, \"Acoustic media\"), mdx(\"p\", null, \"In acoustic we apply the \", mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"#references\"\n  }, \"Sommmerfeld radiation condition\"), \":\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mfrac\", {\n    parentName: \"mrow\"\n  }, mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"1\"), mdx(\"mi\", {\n    parentName: \"mfrac\"\n  }, \"\\u03C1\")), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2207\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03D5\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u22C5\"), mdx(\"mover\", {\n    parentName: \"mrow\",\n    \"accent\": \"true\"\n  }, mdx(\"mi\", {\n    parentName: \"mover\"\n  }, \"n\"), mdx(\"mo\", {\n    parentName: \"mover\"\n  }, \"\\u20D7\")), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mfrac\", {\n    parentName: \"mrow\"\n  }, mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"1\"), mdx(\"mrow\", {\n    parentName: \"mfrac\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C1\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"p\")))), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03D5\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mspace\", {\n    parentName: \"mrow\",\n    \"width\": \"1em\"\n  }), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"on\\xA0\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u0393\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\frac{1}{\\\\rho}\\\\nabla \\\\phi \\\\cdot \\\\vec{n} = - \\\\frac{1}{\\\\rho\\\\, v_p} \\\\partial_{t}\\\\phi,\\\\quad\\\\text{on~}\\\\Gamma,\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.20188em\",\n      \"verticalAlign\": \"-0.8804400000000001em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen nulldelimiter\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mfrac\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.32144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.314em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C1\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.23em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"frac-line\",\n    \"style\": {\n      \"borderBottomWidth\": \"0.04em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.677em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"1\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8804400000000001em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2207\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03D5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u22C5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.714em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord accent\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"n\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"accent-body\",\n    \"style\": {\n      \"left\": \"-0.2355em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"overlay\",\n    \"style\": {\n      \"height\": \"0.714em\",\n      \"width\": \"0.471em\"\n    }\n  }, mdx(\"svg\", {\n    parentName: \"span\",\n    \"width\": \"0.471em\",\n    \"height\": \"0.714em\",\n    \"style\": {\n      \"width\": \"0.471em\"\n    },\n    \"viewBox\": \"0 0 471 714\",\n    \"preserveAspectRatio\": \"xMinYMin\"\n  }, mdx(\"path\", {\n    parentName: \"svg\",\n    \"d\": \"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"\n  }))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.293548em\",\n      \"verticalAlign\": \"-0.972108em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen nulldelimiter\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mfrac\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.32144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.314em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C1\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15139200000000003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"p\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.23em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"frac-line\",\n    \"style\": {\n      \"borderBottomWidth\": \"0.04em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.677em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"1\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.972108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\"))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03D5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"1em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"on\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord nobreak\"\n  }, \"\\xA0\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u0393\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"where we assumed either a pressure or displacement potential formulation for \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03D5\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\phi\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.8888799999999999em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03D5\"))))), \", see \", mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"../parameterizations\"\n  }, \"here\"), \" for a general parameterization of the acoustic wave equation.\"), mdx(\"h3\", null, \"Elastic media\"), mdx(\"p\", null, \"The generalization of this condition to elastic medium is usually called the\\n\", mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"#references\"\n  }, \"Clayton-Engquist\"), \" condition, which relates\\nthe normal traction on the boundary to the normal and tangential components of the velocity field.\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"C\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \":\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B5\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u22C5\"), mdx(\"mover\", {\n    parentName: \"mrow\",\n    \"accent\": \"true\"\n  }, mdx(\"mi\", {\n    parentName: \"mover\"\n  }, \"n\"), mdx(\"mo\", {\n    parentName: \"mover\"\n  }, \"\\u20D7\")), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"p\")), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C1\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"u\"), mdx(\"mover\", {\n    parentName: \"msub\",\n    \"accent\": \"true\"\n  }, mdx(\"mi\", {\n    parentName: \"mover\"\n  }, \"n\"), mdx(\"mo\", {\n    parentName: \"mover\"\n  }, \"\\u20D7\"))), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"+\"), mdx(\"munder\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"munder\"\n  }, \"\\u2211\"), mdx(\"mi\", {\n    parentName: \"munder\"\n  }, \"i\")), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"s\")), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C1\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"u\"), mdx(\"msub\", {\n    parentName: \"msub\"\n  }, mdx(\"mover\", {\n    parentName: \"msub\",\n    \"accent\": \"true\"\n  }, mdx(\"mi\", {\n    parentName: \"mover\"\n  }, \"\\u03C4\"), mdx(\"mo\", {\n    parentName: \"mover\"\n  }, \"\\u20D7\")), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"i\"))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mspace\", {\n    parentName: \"mrow\",\n    \"width\": \"1em\"\n  }), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"on\\xA0\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u0393\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"C : \\\\varepsilon(u) \\\\cdot \\\\vec{n} =\\nv_p\\\\,\\\\rho\\\\,\\\\partial_t u_{\\\\vec{n}}\\n+ \\\\sum_i v_s\\\\,\\\\rho\\\\,\\\\partial_t u_{\\\\vec{\\\\tau}_i},\\\\quad\\\\text{on~}\\\\Gamma,\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.07153em\"\n    }\n  }, \"C\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \":\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03B5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u22C5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.714em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord accent\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"n\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"accent-body\",\n    \"style\": {\n      \"left\": \"-0.2355em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"overlay\",\n    \"style\": {\n      \"height\": \"0.714em\",\n      \"width\": \"0.471em\"\n    }\n  }, mdx(\"svg\", {\n    parentName: \"span\",\n    \"width\": \"0.471em\",\n    \"height\": \"0.714em\",\n    \"style\": {\n      \"width\": \"0.471em\"\n    },\n    \"viewBox\": \"0 0 471 714\",\n    \"preserveAspectRatio\": \"xMinYMin\"\n  }, mdx(\"path\", {\n    parentName: \"svg\",\n    \"d\": \"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"\n  }))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.980548em\",\n      \"verticalAlign\": \"-0.286108em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15139200000000003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"p\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C1\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3448em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5450000000000004em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord accent mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.714em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"n\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.714em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"accent-body\",\n    \"style\": {\n      \"left\": \"-0.2355em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"overlay mtight\",\n    \"style\": {\n      \"height\": \"0.714em\",\n      \"width\": \"0.471em\"\n    }\n  }, mdx(\"svg\", {\n    parentName: \"span\",\n    \"width\": \"0.471em\",\n    \"height\": \"0.714em\",\n    \"style\": {\n      \"width\": \"0.471em\"\n    },\n    \"viewBox\": \"0 0 471 714\",\n    \"preserveAspectRatio\": \"xMinYMin\"\n  }, mdx(\"path\", {\n    parentName: \"svg\",\n    \"d\": \"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"\n  }))))))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15499999999999997em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"+\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.327674em\",\n      \"verticalAlign\": \"-1.277669em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-limits\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.0500050000000003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.872331em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"i\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.050005em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\"\n  }, \"\\u2211\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.277669em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.151392em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"s\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C1\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.34480000000000005em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.545em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord accent mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.714em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.1132em\"\n    }\n  }, \"\\u03C4\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.714em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"accent-body\",\n    \"style\": {\n      \"left\": \"-0.20772em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"overlay mtight\",\n    \"style\": {\n      \"height\": \"0.714em\",\n      \"width\": \"0.471em\"\n    }\n  }, mdx(\"svg\", {\n    parentName: \"span\",\n    \"width\": \"0.471em\",\n    \"height\": \"0.714em\",\n    \"style\": {\n      \"width\": \"0.471em\"\n    },\n    \"viewBox\": \"0 0 471 714\",\n    \"preserveAspectRatio\": \"xMinYMin\"\n  }, mdx(\"path\", {\n    parentName: \"svg\",\n    \"d\": \"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"\n  }))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3280857142857143em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.357em\",\n      \"marginRight\": \"0.07142857142857144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.5em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size3 size1 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"i\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.143em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.25509999999999994em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"1em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"on\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord nobreak\"\n  }, \"\\xA0\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u0393\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"where \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"u\"), mdx(\"mover\", {\n    parentName: \"msub\",\n    \"accent\": \"true\"\n  }, mdx(\"mi\", {\n    parentName: \"mover\"\n  }, \"n\"), mdx(\"mo\", {\n    parentName: \"mover\"\n  }, \"\\u20D7\")))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\partial_t u_{\\\\vec{n}}\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.84944em\",\n      \"verticalAlign\": \"-0.15499999999999997em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3448em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5450000000000004em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord accent mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.714em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"n\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.714em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"accent-body\",\n    \"style\": {\n      \"left\": \"-0.2355em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"overlay mtight\",\n    \"style\": {\n      \"height\": \"0.714em\",\n      \"width\": \"0.471em\"\n    }\n  }, mdx(\"svg\", {\n    parentName: \"span\",\n    \"width\": \"0.471em\",\n    \"height\": \"0.714em\",\n    \"style\": {\n      \"width\": \"0.471em\"\n    },\n    \"viewBox\": \"0 0 471 714\",\n    \"preserveAspectRatio\": \"xMinYMin\"\n  }, mdx(\"path\", {\n    parentName: \"svg\",\n    \"d\": \"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"\n  }))))))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15499999999999997em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \" denotes the normal component of the velocity field and \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"u\"), mdx(\"msub\", {\n    parentName: \"msub\"\n  }, mdx(\"mover\", {\n    parentName: \"msub\",\n    \"accent\": \"true\"\n  }, mdx(\"mi\", {\n    parentName: \"mover\"\n  }, \"\\u03C4\"), mdx(\"mo\", {\n    parentName: \"mover\"\n  }, \"\\u20D7\")), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"i\")))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\partial_t u_{\\\\vec{\\\\tau}_i}\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.9495399999999998em\",\n      \"verticalAlign\": \"-0.25509999999999994em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.34480000000000005em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.545em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord accent mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.714em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.1132em\"\n    }\n  }, \"\\u03C4\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.714em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.714em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"accent-body\",\n    \"style\": {\n      \"left\": \"-0.20772em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"overlay mtight\",\n    \"style\": {\n      \"height\": \"0.714em\",\n      \"width\": \"0.471em\"\n    }\n  }, mdx(\"svg\", {\n    parentName: \"span\",\n    \"width\": \"0.471em\",\n    \"height\": \"0.714em\",\n    \"style\": {\n      \"width\": \"0.471em\"\n    },\n    \"viewBox\": \"0 0 471 714\",\n    \"preserveAspectRatio\": \"xMinYMin\"\n  }, mdx(\"path\", {\n    parentName: \"svg\",\n    \"d\": \"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"\n  }))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3280857142857143em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.357em\",\n      \"marginRight\": \"0.07142857142857144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.5em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size3 size1 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"i\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.143em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.25509999999999994em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \" the tangential component(s) of the velocity, respectively.\"), mdx(\"p\", null, \"Note that in both media, the terms appearing on the left-hand-side are the same\\nas for the\\nnatural boundary conditions, and thus vanish in the weak form.\"), mdx(\"h2\", null, \"Sponge layers\"), mdx(\"p\", null, \"The boundary conditions above both work fairly well for waves with normal incident angle, but they are imperfect, in general, and will still allow for small reflections. Because these artificial reflections can have similar magnitudes as reflections from heterogeneities of the medium, an improved version of absorbing boundaries of often necessary.\"), mdx(\"p\", null, \"Following the work by \", mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"#references\"\n  }, \"Kosloff & Kosloff\"), \",\\nSalvus can introduce additional sponge layers close to the boundaries in which the wavefield is tapered.\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"2\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B3\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"+\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"\\u03B3\"), mdx(\"mn\", {\n    parentName: \"msup\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"2\\\\gamma\\\\,\\\\partial_t{u} + \\\\gamma^2 u,\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.8888799999999999em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"2\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u03B3\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"+\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.0585479999999998em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u03B3\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"where the taper \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B3\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\gamma\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.625em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u03B3\"))))), \" is defined as a function of distance \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"d\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.69444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"))))), \" from the boundary as follows\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B3\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"{\"), mdx(\"mtable\", {\n    parentName: \"mrow\",\n    \"rowspacing\": \"0.15999999999999992em\",\n    \"columnalign\": \"center left\",\n    \"columnspacing\": \"1em\"\n  }, mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"false\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B1\"), mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"1\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"msup\"\n  }, mdx(\"mi\", {\n    parentName: \"mo\"\n  }, \"sin\"), mdx(\"mo\", {\n    parentName: \"mo\"\n  }, \"\\u2061\")), mdx(\"mn\", {\n    parentName: \"msup\"\n  }, \"2\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mfrac\", {\n    parentName: \"mrow\"\n  }, mdx(\"mrow\", {\n    parentName: \"mfrac\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C0\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\")), mdx(\"mrow\", {\n    parentName: \"mfrac\"\n  }, mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"2\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"l\"))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\"))))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"false\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2264\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2264\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"l\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"))))), mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"false\"\n  }, mdx(\"mn\", {\n    parentName: \"mstyle\"\n  }, \"0\"))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"false\"\n  }, mdx(\"mtext\", {\n    parentName: \"mstyle\"\n  }, \"else.\"))))))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\gamma(d) = \\\\left\\\\{\\\\begin{array}{cl}\\n\\\\alpha \\\\left(1 - \\\\sin^2(\\\\frac{\\\\pi\\\\, d}{2\\\\,l})\\\\right) & 0 \\\\leq d \\\\leq l,\\\\\\\\\\n0 & \\\\text{else.}\\n\\\\end{array}\\n\\\\right.\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u03B3\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.4401080000000004em\",\n      \"verticalAlign\": \"-0.9700540000000004em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"minner\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \"{\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mtable\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"arraycolsep\",\n    \"style\": {\n      \"width\": \"0.5em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-c\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.470054em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.589946em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, \"\\u03B1\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"minner\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size1\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"1\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, \"sin\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.871868em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.12076em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen nulldelimiter\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mfrac\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8801079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.6550000000000002em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace mtight\",\n    \"style\": {\n      \"marginRight\": \"0.19516666666666668em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.01968em\"\n    }\n  }, \"l\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.23em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"frac-line\",\n    \"style\": {\n      \"borderBottomWidth\": \"0.04em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.394em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"\\u03C0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace mtight\",\n    \"style\": {\n      \"marginRight\": \"0.19516666666666668em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"d\"))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.345em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size1\"\n  }, \")\"))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.3899459999999997em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.9700540000000004em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"arraycolsep\",\n    \"style\": {\n      \"width\": \"0.5em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"arraycolsep\",\n    \"style\": {\n      \"width\": \"0.5em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-l\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.470054em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.589946em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"\\u2264\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"\\u2264\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.01968em\"\n    }\n  }, \"l\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.3899459999999997em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"else.\"))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.9700540000000004em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"arraycolsep\",\n    \"style\": {\n      \"width\": \"0.5em\"\n    }\n  }))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  }))))))), mdx(\"p\", null, \"The sponge layer is parameterized by two parameters, the amplitude \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B1\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\alpha\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, \"\\u03B1\"))))), \" at the boundary and the width of the layer in meters.\"), mdx(props.images.Img1, null), mdx(Message, {\n    color: \"violet\",\n    header: \"Practical Advice\",\n    content: mdx(\"div\", null, mdx(\"p\", null, \"Empirical tests have shown that setting the taper amplitude equal to the dominant frequency (in Hz), and using a layer width of 3-7 wavelengths (depending on how close the sources and receivers are to the absorbing boundary) work usually well.\")),\n    mdxType: \"Message\"\n  }), mdx(\"h2\", null, \"Why not PMLs?\"), mdx(\"p\", null, \"Despite their success in many applications, deriving a stable formulation of Perfectly Matched Layers (PML) for general coupled acoustic-elastic media on arbitrarily shaped domains using continuous Galerkin spectral-elements is an open research question.\"), mdx(\"p\", null, \"That being said, implementing PMLs within Salvus is on the wish list, although not imminently planned.\"), mdx(\"h2\", null, \"Further reading\"), mdx(\"p\", null, \"More information on the practical use of absorbing boundaries can be found on\\nthe following pages:\"), mdx(\"ul\", null, mdx(\"li\", {\n    parentName: \"ul\"\n  }, \"the \", mdx(\"a\", {\n    parentName: \"li\",\n    \"href\": \"../../../references/python_api/\"\n  }, \"simple config interface\"), \" in Salvus\", mdx(\"em\", {\n    parentName: \"li\"\n  }, \"Flow\"), \",\"), mdx(\"li\", {\n    parentName: \"ul\"\n  }, \"the low-level description in the \", mdx(\"a\", {\n    parentName: \"li\",\n    \"href\": \"../../../references/salvus_compute_api/\"\n  }, \"SalvusCompute API\"), \",\"), mdx(\"li\", {\n    parentName: \"ul\"\n  }, \"a \", mdx(\"a\", {\n    parentName: \"li\",\n    \"href\": \"../../../examples/tutorials/advanced_interface/marmousi/tutorial\"\n  }, \"tutorial\"), \" using the Dirichlet condition and absorbing boundaries on different side sets.\")), mdx(\"h3\", null, \"References\"), mdx(\"p\", null, \"Sommerfeld, A.: Die Greensche Funktion der Schwingungsgleichung. Jahresbericht der Deutschen Mathematiker-Vereinigung, 21(1), pp. 309-353, 1912.\"), mdx(\"p\", null, \"Clayton, R. and Engquist, B.: Absorbing boundary conditions for acoustic and elastic wave equations: Bulletin of the Seismological Society of America, 67 (6), pp. 1529-1540, 1977. \", mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"https://pubs.geoscienceworld.org/bssa/article-lookup/67/6/1529\"\n  }, \"doi\")), mdx(\"p\", null, \"Kosloff, R. and Kosloff, D.: Absorbing Boundaries for Wave Propagation Problems. Journal of Computational Physics 63(2), pp. 363-376, 1986. \", mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"https://doi.org/10.1016/0021-9991(86)90199-3\"\n  }, \"doi: 10.1016/0021-9991(86)90199-3\")));\n}\n;\nMDXContent.isMDXComponent = true;"},"site":{"siteMetadata":{"salvusDocVersions":{"current":"2026.5.0"}}}},"pageContext":{"id":"8a6cfe7d-00ec-5844-832f-e17c83b6ac16"}},
    "staticQueryHashes": ["1756726491","1865182279","3419370438","3597190305","4112489441","519097329"]}