{"id":2768,"date":"2022-03-29T17:58:59","date_gmt":"2022-03-29T08:58:59","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2768"},"modified":"2025-02-03T12:17:00","modified_gmt":"2025-02-03T03:17:00","slug":"%e5%8f%82%e8%80%83%ef%bc%9a%e6%b3%a2%e5%8b%95%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ab%e3%81%a4%e3%81%84%e3%81%a6","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e7%9c%9f%e7%a9%ba%e4%b8%ad%e3%81%ae%e3%83%9e%e3%82%af%e3%82%b9%e3%82%a6%e3%82%a7%e3%83%ab%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%a8%e9%9b%bb%e7%a3%81%e6%b3%a2\/%e5%8f%82%e8%80%83%ef%bc%9a%e6%b3%a2%e5%8b%95%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ab%e3%81%a4%e3%81%84%e3%81%a6\/","title":{"rendered":"\u53c2\u8003\uff1a\u6ce2\u52d5\u65b9\u7a0b\u5f0f\u306b\u3064\u3044\u3066"},"content":{"rendered":"

\u3044\u304d\u306a\u308a \\(\\displaystyle \\nabla^2 f -\\frac{1}{v^2} \\frac{\\partial^2 f}{\\partial t^2} = 0\\) \u3092\u898b\u305b\u3089\u308c\u3066\uff0c\u3053\u308c\u306f\u901f\u3055 \\(v\\) \u3067\u4f1d\u308f\u308b\u6ce2\u3092\u8868\u3057\u3066\u3044\u308b\u6ce2\u52d5\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3060\u3068\u8a00\u308f\u308c\u3066\u3082\u56f0\u308b\u3060\u308d\u3046\u304b\u3089\uff0c\u5c11\u3057\u89e3\u8aac\u3002<\/p>\n

1\u6b21\u5143\u6ce2\u52d5\u65b9\u7a0b\u5f0f<\/h3>\n

\u307e\u305a\uff0c\u7c21\u5358\u306e\u305f\u3081\u306b \\(f\\) \u304c1\u6b21\u5143\uff0c\\(x\\) \u65b9\u5411\u306e\u307f\u306e\u7a7a\u9593\u4f9d\u5b58\u6027\u3092\u6301\u3064\u3068\u3059\u308b\u3068<\/p>\n

$$\\nabla^2 f = \\left(\\frac{\\partial^2}{\\partial x^2} +\\frac{\\partial^2}{\\partial y^2} +\u00a0\\frac{\\partial^2}{\\partial z^2} \\right) f \\Rightarrow \\frac{\\partial^2}{\\partial x^2} f$$<\/p>\n

\u3068\u306a\u308b\u3002\u3053\u306e1\u6b21\u5143\u6ce2\u52d5\u65b9\u7a0b\u5f0f<\/p>\n

$$ \\left( \\frac{\\partial^2}{\\partial x^2} -\\frac{1}{v^2} \\frac{\\partial^2 }{\\partial t^2} \\right) f = 0$$\u3092\u4f8b\u306b\uff0c\u3053\u306e\u6ce2\u52d5\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u304c\u4efb\u610f\u306e1\u5909\u6570\u95a2\u6570 \\(F(u), G(v)\\) \u3092\u4f7f\u3063\u3066
$$f(x, t) = F(x-vt) + G(x+vt)$$\u3068\u66f8\u3051\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n

\u5909\u6570\u5909\u63db<\/h3>\n

\u5909\u6570\u5909\u63db \\((x, t) \\rightarrow (u, w)\\) \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n

\\begin{eqnarray}
u &\\equiv& x -v t\\\\
w &\\equiv& x + vt
\\end{eqnarray}<\/p>\n

\u3053\u306e\u9006\u5909\u63db\u306f<\/p>\n

\\begin{eqnarray}
x &=& \\frac{1}{2}(u + w)\\\\
t &=& \\frac{1}{2v}(w -u)
\\end{eqnarray}<\/p>\n

\u5909\u6570\u5909\u63db\u306b\u95a2\u3059\u308b\u504f\u5fae\u5206\u306e\u898f\u5247\u304b\u3089<\/p>\n

<\/span>\\begin{eqnarray}
\\frac{\\partial}{\\partial u} &=& \\frac{\\partial x}{\\partial u}\\frac{\\partial}{\\partial x} + \\frac{\\partial t}{\\partial u}\\frac{\\partial}{\\partial t}\\\\
&=& \\frac{1}{2} \\frac{\\partial}{\\partial x} -\\frac{1}{2v} \\frac{\\partial}{\\partial t}\\\\
\\therefore\\ 2 \\frac{\\partial}{\\partial u} &=& \\frac{\\partial}{\\partial x}-\\frac{1}{v} \\frac{\\partial}{\\partial t}
\\end{eqnarray}<\/p>\n

\u540c\u69d8\u306b\u3057\u3066\u3000
$$ \\ 2 \\frac{\\partial}{\\partial w} = \\frac{\\partial}{\\partial x}+\\frac{1}{v} \\frac{\\partial}{\\partial t}$$<\/p>\n

\u3053\u308c\u3092\u4f7f\u3046\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\\left( \\frac{\\partial^2}{\\partial x^2} -\\frac{1}{v^2} \\frac{\\partial^2 }{\\partial t^2} \\right) f
\n&=& \\left( \\frac{\\partial}{\\partial x} -\\frac{1}{v} \\frac{\\partial }{\\partial t} \\right)\\left( \\frac{\\partial}{\\partial x} +\\frac{1}{v} \\frac{\\partial }{\\partial t} \\right) f \\\\
&=& 4 \\frac{\\partial}{\\partial u}\\frac{\\partial}{\\partial w} f = 0
\\end{eqnarray}<\/p>\n

$$\\therefore\\ \\ \\frac{\\partial}{\\partial u}\\left(\\frac{\\partial}{\\partial w} f\\right) = 0$$
\u3053\u308c\u3088\u308a\uff0c\\(\\displaystyle \\frac{\\partial}{\\partial w} f\\) \u306f \\(u\\)\u00a0 \u306b\u306f\u4f9d\u5b58\u305b\u305a\uff0c \\(w\\) \u306e\u307f\u306b\u4f9d\u5b58\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\uff0c<\/p>\n

$$\\frac{\\partial}{\\partial w} f = G'(w)$$\u3068\u304a\u3051\u308b\u3002\u3053\u308c\u3092 \\(w\\) \u3067\u7a4d\u5206\u3057\u3066<\/p>\n

$$f = G(w) + F(u)$$<\/p>\n

\u3053\u3053\u3067 \\(F(u)\\) \u306f \\(w\\) \u306b\u4f9d\u5b58\u3057\u306a\u3044\u7a4d\u5206\u300c\u5b9a\u6570\u300d\u3067\u3042\u308b\u3002\u3068\u3044\u3046\u3053\u3068\u3067<\/p>\n

$$f = F(u) + G(w) = F(x -vt) + G(x + vt)$$<\/p>\n

\u6ce2\u52d5\u306e\u30b0\u30e9\u30d5\u4f8b<\/h3>\n

1\u6b21\u5143\u6ce2\u52d5\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u3046\u3061\uff0c\\(F(x -vt)\\) \u306f \\(+x\\) \u65b9\u5411\u306b\u9032\u3080\u6ce2\uff0c\\(G(x + vt)\\) \u306f \\(-x\\) \u65b9\u5411\u306b\u9032\u3080\u6ce2\u3092\u8868\u3059\u3002<\/p>\n

\u4f8b\u3068\u3057\u3066 \\(\\sin(x -vt)\\) \u3092 \\(v = 1\\)\u00a0 \u3068\u3057\u3066 Matplotlib \u3067\u30b0\u30e9\u30d5\u306b\u3057\u305f\u4f8b\u304c\u4ee5\u4e0b\u3067\u3042\u308b\u3002<\/p>\n

\"\"<\/p>\n