{
    "componentChunkName": "component---src-templates-markdown-doc-tsx",
    "path": "/2025.1.3/knowledge_base/boundary_conditions/homogeneous_dirichlet",
    "result": {"data":{"mdx":{"id":"86566994-b728-56e1-8a75-32deb5a52cfa","frontmatter":{"title":"","images":null},"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 = {};\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, \"Homogeneous Dirichlet\"), mdx(\"p\", null, \"Homogeneous Dirichlet conditions force the wavefield to be zero at the specified boundary.\"), mdx(\"p\", null, \"In the scalar acoustic case, this simply gives\"), 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  }, \"\\u03D5\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), 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  }, \"\\\\phi = 0,\\\\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\": \"0.8888799999999999em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03D5\"), 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\": \"0.8777699999999999em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), 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, \"and in elastic media, we obtain\"), 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  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), 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(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \".\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"u = 0,\\\\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\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\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\": \"0.8777699999999999em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), 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\": \"mord\"\n  }, \".\")))))), mdx(\"p\", null, \"It is possible to fix only certain components of the displacement field \", 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  }, \"u\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"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\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"))))), \" in the elastic case.\"), mdx(Message, {\n    color: \"violet\",\n    header: \"Practical Advice\",\n    content: mdx(\"div\", null, mdx(\"p\", null, \"For coupled simulations of waves propagating along a fluid-solid interface, the homogeneous Dirichlet condition is the physically correct condition to describe the \\\"free-surface condition\\\" in water.\")),\n    mdxType: \"Message\"\n  }), 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/models_and_topo/marmousi/tutorial\"\n  }, \"tutorial\"), \" using the Dirichlet condition and absorbing boundaries on different side sets.\")));\n}\n;\nMDXContent.isMDXComponent = true;"},"site":{"siteMetadata":{"salvusDocVersions":{"current":"2026.5.0"}}}},"pageContext":{"id":"86566994-b728-56e1-8a75-32deb5a52cfa"}},
    "staticQueryHashes": ["1756726491","1865182279","3419370438","3597190305","4112489441","519097329"]}