- #1
Nerrad
- 20
- 0
Homework Statement
(a) Light waves satisfy the wave equation ##u_{tt}-c^2u_{xx}## where ##c## is the speed of light.
Consider change of coordinates $$x'=x-Vt$$ $$t'=t$$
where V is a constant. Use the chain rule to show that ##u_x=u_{x'}## and ##u_{tt}=-Vu_{x'}+u_{t'}##
Find ##u_{xx},u_{tt},## and hence ##u_{tt}-c^2u_{xx}##, in terms of derivatives with respect to ##x'## and ##t'##.
Deduce that if ##u## satisfies the wave equation in ##x,t## coordinates, then it does not satisfy the same equation in the ##x',t'## coordinates.
The Attempt at a Solution
So I've worked out that $$u_{xx}=u_{x'x'}$$ and $$u_{tt}=u_{t't'}+v^2(u_{x'x'})-2v(u_{x'xt})$$ so technically ##u_{tt}-c^2u_{xx}## expressed in terms of derivatives with respect to ##x'## and ##t'## would just be $$u_{t't'}+v^2(u_{x'x'})-2v(u_{x'xt})-c^2(u_{x'x'})=0$$ right?
But how do I do the bit where question says "Deduce that if ##u## satisfies the wave equation in ##x,t## coordinates, then it does not satisfy the same equation in the ##x',t'## coordinates." I don't know where to start with this?Would this be done conceptually or mathematically?