(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

1) Show that the electromagnetic wave equation,

d^2(phi)/dx^2 + d^2(phi)/dy^2 + d^2(phi)/dz^2 –(1/c^2)( d^2(phi)/dt^2) = 0

is not invariant under Galilean transformation.

Note: here d is a partial differential operator.

2. Relevant equations

3. The attempt at a solution

I have the solution but I couldn’t understand one particular step. The solution is as follows:

The equation will be invariant if it retains the same form when expressed in terms of the new variables x’,y’,z’,t’. From Galilean transformation we have,

dx’/dx=1, dx’/dt=-v, dt’/dt=dy’/dy=dz’/dz=1, dx’/dy= dx’/dz= dy’/dx= dt’/dx=0

From chain rule and using the above results we have,

d(phi)/dx= [d(phi)/dx’][dx’/dx] + [d(phi)/dy’][dy’/dx] + [d(phi)/dz’][dz’/dx] + d(phi)/dt’][dt’/dx] = d(phi)/dx’

And,

d^2(phi)/dx^2= d^2(phi)/dx’^2

Similarly,

d^2(phi)/dy^2= d^2(phi)/dy’^2 &

d^2(phi)/dz^2= d^2(phi)/dz’^2

Moreover,

d(phi)/dt= -v[d(phi)/dx’] + d(phi)/dt’

Differentiating the above equation with respect to t ,

d^2(phi)/dt^2 = d^2(phi)/dt’^2 -2v[d^2(phi)/dx’dt’] + v^2[d^2(phi)/dx’^2]

This is where I have a doubt. I differentiated in the following way:

d^2(phi)/dt^2= -v[d^2(phi)/dx’dt] - [d(phi)/dx’][dv/dt] + [d^2(phi)/dt’^2]

= -v[(d^2(phi)/dx’^2)(dx’/dt)] –[d(phi)/dx’][dv/dt] + [d^2(phi)/dt’^2]

= (v^2)[ d^2(phi)/dx’^2] –[d(phi)/dx’][dv/dt] + [d^2(phi)/dt’^2]

I am able derive 2 of the terms but how to derive the third term -2v[d^2(phi)/dx’dt’] from –[d(phi)/dx’][dv/dt]. Could somebody please help me with this derivation?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Galilean transformation

**Physics Forums | Science Articles, Homework Help, Discussion**