"Light cone coordinates and gauge"의 두 판 사이의 차이
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2012년 11월 2일 (금) 02:05 판
introduction
- light cone gague
1차원에서의 일반해
- \(\frac{\partial^2 Y}{\partial t^2}=v^2\frac{\partial^2 Y}{\partial x^2}\) 또는 \(\mu\frac{\partial^2 Y}{\partial t^2}=T\frac{\partial^2 Y}{\partial x^2}\) (\(v=\sqrt{\frac{T}{\mu}}\))
- 일반해는 \(Y=f(x+vt)+g(x-vt)\)로 주어진다
- f는 왼쪽, g는 오른쪽으로 이동하는 파동이며, Y는 그 중첩으로 주어진다
(증명)
\(u=x+at\), \(v=x-at\)라 두자.
그러면 \(Y=f(u)+g(v)\)로 쓸 수 있다.
\(\frac{\partial Y}{\partial t}=\frac{\partial Y}{\partial u}\frac{\partial u}{\partial t} +\frac{\partial Y}{\partial v}\frac{\partial v}{\partial t}=f'(u)a+g'(v)(-a)=af'(u)-ag'(v)\)
\(W(u,v)=\frac{\partial Y}{\partial t}=af'(u)-ag'(v)\).
\(\frac{\partial^2 Y}{\partial t^2}=\frac{\partial W}{\partial t}=\frac{\partial W}{\partial u}\frac{\partial u}{\partial t} +\frac{\partial W}{\partial v}\frac{\partial v}{\partial t}=af''(u)a-ag''(v)(-a)=a^2(f''(u)+g''(v))\)
\(\frac{\partial Y}{\partial x}=\frac{\partial Y}{\partial u}\frac{\partial u}{\partial x} +\frac{\partial Y}{\partial v}\frac{\partial v}{\partial x}=f'(u)+g'(v)\)
\(Z(u,v)=\frac{\partial Y}{\partial x}=f'(u)+g'(v)\)
\(\frac{\partial^2 Y}{\partial x^2}=\frac{\partial Z}{\partial x}=\frac{\partial Z}{\partial u}\frac{\partial u}{\partial x} +\frac{\partial Z}{\partial v}\frac{\partial v}{\partial x}=f''(u)+g''(v)\)
따라서
\(\frac{\partial^2 Y}{\partial t^2}=a^2\frac{\partial^2 Y}{\partial x^2}=a^2(f''(u)+g''(v))\)■
history
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articles
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- http://www.ams.org/mathscinet
- [1]http://www.zentralblatt-math.org/zmath/en/
- [2]http://arxiv.org/
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