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This page lists all supported\\ninterpretations, how to translate saved fields to physically meaningful\\nquantities (pressure, particle velocity, displacement), and the special role of\\nthe coupled case.\"),mdx(\"p\",null,\"In the elastic wave equation, all outputs are given in their canonical SI\\nunits. This is in contrast to the acoustic case, where the units depend on the\\nchosen interpretation of the scalar field, typically some potential. 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mathnormal\"},\"\\u03D5\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\"]\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},\"=\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"Pa\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"s\"))))),\". To get\\npressure and particle velocity, use:\"),mdx(\"table\",null,mdx(\"thead\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"thead\"},mdx(\"th\",{parentName:\"tr\",\"align\":\"center\"},\"Desired output\"),mdx(\"th\",{parentName:\"tr\",\"align\":\"center\"},\"Expression\"),mdx(\"th\",{parentName:\"tr\",\"align\":\"center\"},\"Field to save\"),mdx(\"th\",{parentName:\"tr\",\"align\":\"center\"},\"Further operations\"))),mdx(\"tbody\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":\"center\"},\"Pressure (Pa)\"),mdx(\"td\",{parentName:\"tr\",\"align\":\"center\"},mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\",\"mathvariant\":\"normal\"},\"\\u2202\"),mdx(\"mi\",{parentName:\"msub\"},\"t\")),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03D5\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\partial_t \\\\phi\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"marginRight\":\"0.05556em\"}},\"\\u2202\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.2805559999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.05556em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"t\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03D5\")))))),mdx(\"td\",{parentName:\"tr\",\"align\":\"center\"},mdx(\"inlineCode\",{parentName:\"td\"},\"[\\\"phi_t\\\"]\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"center\"},\"None\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":\"center\"},\"Particle velocity (\",mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"m\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2009\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msup\"},\"s\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\")))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{m}\\\\,\\\\text{s}^{-1}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8141079999999999em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"m\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"s\")),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8141079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"\\u2212\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"1\"))))))))))))),\")\"),mdx(\"td\",{parentName:\"tr\",\"align\":\"center\"},mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msup\"},\"\\u03C1\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\"))),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},\"\\u2207\"),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03D5\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\rho^{-1} \\\\nabla \\\\phi\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.008548em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03C1\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8141079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"\\u2212\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"1\"))))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"\\u2207\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03D5\")))))),mdx(\"td\",{parentName:\"tr\",\"align\":\"center\"},mdx(\"inlineCode\",{parentName:\"td\"},\"[\\\"gradient-of-phi\\\"]\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"center\"},\"Multiply by inverse density\")))),mdx(\"h3\",null,\"Interpretation B: Divergence of Force Density Source\"),mdx(\"p\",null,\"Assume \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"f\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"f\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10764em\"}},\"f\"))))),\" carries divergence of force density units\\n\",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"[\"),mdx(\"mi\",{parentName:\"mrow\"},\"f\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"]\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mtext\",{parentName:\"mrow\"},\"N\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2009\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msup\"},\"m\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"4\")))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"[f]=\\\\text{N}\\\\,\\\\text{m}^{-4}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"[\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10764em\"}},\"f\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\"]\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},\"=\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8141079999999999em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"N\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"m\")),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8141079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"\\u2212\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"4\"))))))))))))),\", we find\\n\",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"[\"),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03D5\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"]\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msup\"},\"kg\"),mdx(\"mn\",{parentName:\"msup\"},\"2\")),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2009\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msup\"},\"m\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"4\"))),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2009\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msup\"},\"s\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"2\")))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"[\\\\phi]=\\\\text{kg}^2\\\\,\\\\text{m}^{-4}\\\\,\\\\text{s}^{-2}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"[\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord 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text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"kg\")),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8984479999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.1473400000000002em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"2\")))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"m\")),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8141079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"\\u2212\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord 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Denote the coupled-form\\nsource term by \",mdx(\"inlineCode\",{parentName:\"p\"},\"f_c(t)\"),\" (dimensionless) and the time variable by \",mdx(\"inlineCode\",{parentName:\"p\"},\"t\"),\".\"),mdx(\"p\",null,\"Given one of the following physical source descriptions in a \",mdx(\"em\",{parentName:\"p\"},\"pure acoustic\"),\"\\nsetting:\"),mdx(\"table\",null,mdx(\"thead\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"thead\"},mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"Physical description\"),mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"Symbol & units\"),mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"Conversion to dimensionless \",mdx(\"inlineCode\",{parentName:\"th\"},\"f_c(t)\"),\" (Interpretation C)\"),mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"Notes\"))),mdx(\"tbody\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Volume injection rate\"),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"q(t)\"),\" \",\"[s^-1]\"),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"f_c(t) = - \\u222B q(t) dt\"),\" (constant chosen so f_c\\u21920 outside source)\"),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Time integral; remove DC after integration\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Pressure history (localized)\"),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"p(t)\"),\" \",\"[Pa]\"),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"f_c(t) = - p(t) / (\\u03C1 c^2)\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Uses local acoustic impedance (\",mdx(\"inlineCode\",{parentName:\"td\"},\"\\u03C1 c^2\"),\") scaling\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Divergence of force density\"),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"g(t)\"),\" \",\"[N m^-4]\"),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"f_c(t) = - (1/\\u03C1) \\u222B\\u222B g(t) dt^2\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Two time integrals; detrend afterwards\")))),mdx(\"p\",null,\"Each physical description corresponds to a way of injecting \",mdx(\"em\",{parentName:\"p\"},\"work\"),\" or\\n\",mdx(\"em\",{parentName:\"p\"},\"mass/volume change\"),\" into the acoustic field. In the coupled Interpretation C,\\nthe governing equation has a dimensionless right-hand side, so any dimensional\\nsource must be rescaled to produce the correctly scaled potential \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03D5\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\phi\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03D5\"))))),\" with\\nunits \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"kg\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},\"/\"),mdx(\"mtext\",{parentName:\"mrow\"},\"m\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{kg}/\\\\text{m}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"kg\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"/\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"m\")))))),\". A volume injection rate \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"q\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"t\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"q(t)\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\"}},\"q\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"(\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"t\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\")\"))))),\" is (up to constants)\\nthe time derivative of a dimensionless monopole strength-hence we integrate\\nonce to obtain its cumulative effect. A localized pressure history \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"p\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"t\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"p(t)\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"p\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"(\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"t\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\")\"))))),\"\\nenters as an imposed normal stress; dividing by the bulk modulus proxy\\n\",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03C1\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msup\"},\"c\"),mdx(\"mn\",{parentName:\"msup\"},\"2\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\rho\\nc^2\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.008548em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03C1\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"c\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8141079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"2\")))))))))))),\" yields a dimensionless amplitude consistent with \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"p\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"\\u03D5\"),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"mrow\"},\"t\"),mdx(\"mi\",{parentName:\"mrow\"},\"t\")))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"p = -\\\\phi_{tt}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.625em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"p\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},\"=\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"\\u2212\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03D5\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.2805559999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"t\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"t\"))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\".\\nFinally, a divergence of force density \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"g\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"t\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"g(t)\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\"}},\"g\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"(\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"t\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\")\"))))),\" is (in the pure-acoustic\\ninterpretations) the second time derivative of a monopole-like loading;\\nrecovering the dimensionless \",mdx(\"inlineCode\",{parentName:\"p\"},\"f_c\"),\" therefore requires two time integrations\\n(and detrending to remove integration constants). This mapping ensures that\\nafter substitution into the coupled equation, \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03D5\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\phi\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03D5\"))))),\" and its derivatives\\nreproduce the originally intended physical pressure / traction and\\nvolume-change effects.\"),mdx(\"p\",null,\"Conversely, once you have a dimensionless \",mdx(\"inlineCode\",{parentName:\"p\"},\"f_c(t)\"),\" (coupled interpretation):\"),mdx(\"table\",null,mdx(\"thead\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"thead\"},mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"Quantity (pure acoustic meaning)\"),mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"Recover from \",mdx(\"inlineCode\",{parentName:\"th\"},\"f_c(t)\")),mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"Comment\"))),mdx(\"tbody\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Volume injection rate \",mdx(\"inlineCode\",{parentName:\"td\"},\"q(t)\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"q(t) = - d f_c / dt\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Central differences recommended\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Local pressure \",mdx(\"inlineCode\",{parentName:\"td\"},\"p(t)\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"p(t) = - \\u03C1 c^2 f_c(t)\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Near-source approximation\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Divergence force density \",mdx(\"inlineCode\",{parentName:\"td\"},\"g(t)\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"g(t) = - \\u03C1 d^2 f_c / dt^2\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Central differences recommended\")))),mdx(\"p\",null,\"Implementation tips:\"),mdx(\"ol\",null,mdx(\"li\",{parentName:\"ol\"},\"Always window the source time function so its value and its first derivative\\nvanish near simulation start to avoid long-period artifacts.\"),mdx(\"li\",{parentName:\"ol\"},\"Numerical differentiation/integration should use the native solver time step\\nto avoid interpolation phase errors (or heavily supersampled signals).\"),mdx(\"li\",{parentName:\"ol\"},\"After integration, remove any residual linear trend (detrending) to enforce\\nzero mean; this prevents persistent offsets in \",mdx(\"inlineCode\",{parentName:\"li\"},\"\\u03C6\"),\".\")),mdx(\"p\",null,\"Design guidance:\"),mdx(\"ul\",null,mdx(\"li\",{parentName:\"ul\"},\"Use Interpretation A if your physical source is naturally specified as a\\nvolume injection rate and you mainly care about pressure time series.\"),mdx(\"li\",{parentName:\"ul\"},\"Use Interpretation B if you have (or conceptually think in terms of) a known\\ndivergence of force density field.\"),mdx(\"li\",{parentName:\"ul\"},\"Use Interpretation C if you will ever couple to elasticity, or if you prefer\\npressure via a second derivative and a straightforward formula for normal\\ndisplacement at interfaces.\"),mdx(\"li\",{parentName:\"ul\"},\"Converting previously archived A/B datasets to the coupled view can be done\\nvia the conversion tables by reconstructing \",mdx(\"inlineCode\",{parentName:\"li\"},\"f_c(t)\"),\" and recomputing derived\\nfields.\")),mdx(\"h2\",null,\"Elastic simulations\"),mdx(\"p\",null,\"In the elastic case, Salvus solves the general elastic wave equation. For the\\nanalysis of units involved in the source term specifically, let's limit\\nourselves to purely elastic linear media. The strong form of this equation is:\"),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03C1\"),mdx(\"mtext\",{parentName:\"mrow\"},\"\\u2009\"),mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"normal\"},\"\\u2202\"),mdx(\"mi\",{parentName:\"msubsup\"},\"t\"),mdx(\"mn\",{parentName:\"msubsup\"},\"2\")),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},\"u\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},\"\\u2207\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22C5\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},\"C\"),mdx(\"mo\",{parentName:\"mrow\"},\":\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},\"\\u2207\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},\"u\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"+\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"script\"},\"f\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},\".\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\rho \\\\, \\\\partial _t^2 \\\\mathbf{u} = \\\\nabla \\\\cdot (\\\\mathbf{C} : \\\\nabla\\n  \\\\mathbf{u}) + \\\\mathbf{\\\\mathcal{f}}.\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.1111079999999998em\",\"verticalAlign\":\"-0.247em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03C1\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"marginRight\":\"0.05556em\"}},\"\\u2202\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t 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text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"stf\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"(\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"t\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\")\"))))),\" of these\\nsources should be interpreted, we start with the formulation of this point\\nsource \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"f\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"f\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10764em\"}},\"f\"))))),\" as injected into the strong form:\"),mdx(\"p\",null,\"$$ f(\\\\mathbf{x}, t) = \\\\text{stf}(t) \\\\, \\\\mathbf{F} \\\\,\\\\delta(\\\\mathbf{x}) $$\"),mdx(\"p\",null,\"In this definition, the vector \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},\"F\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"[\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\mathbf{F} = [\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.68611em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},\"F\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},\"=\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"[\"))))),mdx(\"inlineCode\",{parentName:\"p\"},\"fx\"),\", \",mdx(\"inlineCode\",{parentName:\"p\"},\"fy\"),\", \",mdx(\"inlineCode\",{parentName:\"p\"},\"fz\"),mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"]\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"]\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\"]\"))))),\" allows one\\nto orient (and scale) the source in any direction. The symbol \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03B4\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\delta\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03785em\"}},\"\\u03B4\"))))),\"\\nindicates the (multidimensional) Dirac delta function.\"),mdx(\"p\",null,\"The units of the Dirac delta function should be interpreted to collapse to\\ndimensionless if integrated over its argument. In this case, we would integrate\\nover three spatial coordinates, giving it the units \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msup\"},\"m\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"3\")))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{m}^{-3}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8141079999999999em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"m\")),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8141079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"\\u2212\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"3\"))))))))))))),\". If now we\\ninterpret the orientation vector \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},\"F\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\mathbf{F}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.68611em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},\"F\")))))),\" as dimensionless, the units of\\n\",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"stf\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"t\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{stf}(t)\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"stf\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"(\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"t\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\")\"))))),\" are found in the following way:\"),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"[\"),mdx(\"mtext\",{parentName:\"mrow\"},\"N\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22C5\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msup\"},\"m\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"3\"))),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"]\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mtext\",{parentName:\"mrow\"},\"stf\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"t\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22C5\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"[\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"]\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22C5\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"[\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msup\"},\"m\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"3\"))),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"]\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"[\\\\text{N} \\\\cdot \\\\text{m}^{-3}] = \\\\text{stf}(t) \\\\cdot [ 1 ] \\\\cdot\\n  [\\\\text{m}^{-3}]\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"[\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"N\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},\"\\u22C5\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.1141079999999999em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"m\")),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.864108em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"\\u2212\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"3\"))))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\"]\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},\"=\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"stf\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"(\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"t\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\")\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},\"\\u22C5\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"[\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"1\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\"]\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},\"\\u22C5\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.1141079999999999em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"[\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"m\")),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.864108em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"\\u2212\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"3\"))))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\"]\")))))),mdx(\"p\",null,\"Yielding that \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"stf\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"t\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{stf}(t)\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"stf\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"(\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"t\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\")\"))))),\" should be interpreted as Newtons when\\n\",mdx(\"span\",{parentName:\"p\",\"className\":\"math 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math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"[\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msup\"},\"\\u03B4\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mn\",{parentName:\"mrow\"},\"2\"),mdx(\"mi\",{parentName:\"mrow\"},\"D\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\"))),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},\"x\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"]\"),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msup\"},\"m\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"2\")))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"[\\\\delta^{(2D)}(\\\\mathbf{x})] = \\\\text{m}^{-2}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.138em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"[\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord 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= \\\\text{m}^{-3}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.174108em\",\"verticalAlign\":\"-0.286108em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"[\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"marginRight\":\"0.05556em\"}},\"\\u2202\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t 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mtight\"},\"i\"))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.286108em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\")\"))))),\", which\\nhas no counterpart in the symmetric moment tensor) represents rotational force\\ncouples that produce a net torque but no net moment. This is useful for\\nmodeling certain types of dislocations, CLVD (compensated linear vector dipole)\\ndecompositions, or other non-double-couple source mechanisms.\"),mdx(\"h3\",null,\"Summary: STF units for elastic point sources\"),mdx(\"p\",null,\"Salvus supports five source mechanisms: \",mdx(\"inlineCode\",{parentName:\"p\"},\"Scalar\"),\", \",mdx(\"inlineCode\",{parentName:\"p\"},\"ScalarGradient\"),\", \",mdx(\"inlineCode\",{parentName:\"p\"},\"Vector\"),\",\\n\",mdx(\"inlineCode\",{parentName:\"p\"},\"MomentTensor\"),\", and \",mdx(\"inlineCode\",{parentName:\"p\"},\"VectorGradient\"),\". Of these, only \",mdx(\"inlineCode\",{parentName:\"p\"},\"Vector\"),\", \",mdx(\"inlineCode\",{parentName:\"p\"},\"MomentTensor\"),\",\\nand \",mdx(\"inlineCode\",{parentName:\"p\"},\"VectorGradient\"),\" are valid in elastic elements (the solver enforces this\\nconsistency check). The \",mdx(\"inlineCode\",{parentName:\"p\"},\"Scalar\"),\" and \",mdx(\"inlineCode\",{parentName:\"p\"},\"ScalarGradient\"),\" mechanisms are\\nacoustic-only and their unit interpretations depend on which acoustic\\nconvention is chosen (see the acoustic section above). The table below collects\\nthe STF units for all three elastic point source types, in both 2-D and 3-D:\"),mdx(\"table\",null,mdx(\"thead\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"thead\"},mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"Source type\"),mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"STF units (3-D)\"),mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"STF units (2-D)\"),mdx(\"th\",{parentName:\"tr\",\"align\":\"left\"},\"Physical interpretation\"))),mdx(\"tbody\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"VectorPoint\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"N\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{N}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"N\"))))))),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"N\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22C5\"),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msup\"},\"m\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\")))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{N} \\\\cdot \\\\text{m}^{-1}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord 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text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"N\"))))))),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Seismic moment (symmetric dipole)\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"inlineCode\",{parentName:\"td\"},\"VectorGradientPoint\")),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"N\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22C5\"),mdx(\"mtext\",{parentName:\"mrow\"},\"m\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{N} \\\\cdot \\\\text{m}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"N\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},\"\\u22C5\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"m\"))))))),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"N\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{N}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"N\"))))))),mdx(\"td\",{parentName:\"tr\",\"align\":\"left\"},\"Generalized moment (non-symmetric dipole)\")))),mdx(\"p\",null,\"Salvus does not enforce a particular split between the STF and the spatial\\nweights. Only their product \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"stf\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"t\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22C5\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"w\"),mdx(\"mi\",{parentName:\"msub\"},\"k\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{stf}(t) \\\\cdot w_k\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"stf\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"(\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"t\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\")\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},\"\\u22C5\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.58056em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02691em\"}},\"w\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02691em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.03148em\"}},\"k\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\" is physically meaningful,\\nand it must carry the units listed in the table. The derivations above assume\\ndimensionless weights for clarity, so that the STF alone carries the full\\nphysical amplitude. In practice, many users specify the weights in physical\\nunits (e.g., \",mdx(\"inlineCode\",{parentName:\"p\"},\"mxx\"),\" in \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\"},\"N\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22C5\"),mdx(\"mtext\",{parentName:\"mrow\"},\"m\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\text{N} \\\\cdot \\\\text{m}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"N\")),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},\"\\u22C5\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},\"m\")))))),\" for a 3-D moment tensor), in\\nwhich case the STF is correspondingly dimensionless. Either convention is valid\\nas long as it is applied consistently.\"),mdx(\"blockquote\",null,mdx(\"p\",{parentName:\"blockquote\"},mdx(\"strong\",{parentName:\"p\"},\"Note on interpretation dependence\"),\": unlike the acoustic case, where the\\nunits of the STF depend on which interpretation (A, B, or C) of the scalar\\npotential \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03D5\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\phi\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03D5\"))))),\" is chosen, the elastic case has a \",mdx(\"strong\",{parentName:\"p\"},\"unique\"),\" state-variable\\nconvention: \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},\"u\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\mathbf{u}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.44444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},\"u\")))))),\" is displacement in meters. 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