"Light cone coordinates and gauge"의 두 판 사이의 차이

2011년 10월 5일 (수) 10:08 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

http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

encyclopedia

http://en.wikipedia.org/wiki/Light_cone_gauge
http://en.wikipedia.org/wiki/
http://www.scholarpedia.org/
Princeton companion to mathematics(Companion_to_Mathematics.pdf)

books

[[4909919|]]

articles

question and answers(Math Overflow)

blogs

구글 블로그 검색
- http://blogsearch.google.com/blogsearch?q=
- http://blogsearch.google.com/blogsearch?q=

experts on the field

http://arxiv.org/

@@ 2번째 줄: / 2번째 줄: @@
 *  light cone gague<br>
+<h5>1차원에서의 일반해</h5>
+* <math>\frac{\partial^2 Y}{\partial t^2}=v^2\frac{\partial^2 Y}{\partial x^2}</math> 또는 <math>\mu\frac{\partial^2 Y}{\partial t^2}=T\frac{\partial^2 Y}{\partial x^2}</math> (<math>v=\sqrt{\frac{T}{\mu}}</math>)
+*  일반해는 <math>Y=f(x+vt)+g(x-vt)</math>로 주어진다<br>
+*  f는 왼쪽, g는 오른쪽으로 이동하는 파동이며, Y는 그 중첩으로 주어진다<br>
+(증명)
+<math>u=x+at</math>, <math>v=x-at</math>라 두자.
+그러면 <math>Y=f(u)+g(v)</math>로 쓸 수 있다.
+<math>\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)</math>
+ <math>W(u,v)=\frac{\partial Y}{\partial t}=af'(u)-ag'(v)</math>.
+<math>\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))</math>
+<math>\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)</math>
+<math>Z(u,v)=\frac{\partial Y}{\partial x}=f'(u)+g'(v)</math>
+<math>\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)</math>
+따라서
+<math>\frac{\partial^2 Y}{\partial t^2}=a^2\frac{\partial^2 Y}{\partial x^2}=a^2(f''(u)+g''(v))</math>■
@@ 21번째 줄: / 59번째 줄: @@
 <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
+* http://en.wikipedia.org/wiki/Light_cone_gauge
 * http://en.wikipedia.org/wiki/
 * http://www.scholarpedia.org/