Electromagnetic wave equation not invariant under galilean trans.

[bfusco]

Homework Statement

Prove that the electromagnetic wave equation: 
(d^2ψ)/(dx^2) + (d^2ψ)/dy^2) + (d^2ψ)/(dz^2) − (1/c^2) * [(d^2ψ)/(dt^2)]= 0 is NOT invariant under Galilean transformation. (i.e., the equation does NOT have the same form for a moving observer moving at speed of, say, v in the x direction).

*note: all the "d"s in this equation are the partial derivatives.

Homework Equations

galilean transformation:
t'=t, x'=x-vt, y'=y, z'=z

The Attempt at a Solution

-from what i understand to solve this involves the chain rule for partial derivatives.
-looking at the equation i can't help but jump to something with the gradient of ∇ψ, but i don't know where to go from there.

 

[mfb]

The y,z,t components in the equation just stay the same, you can ignore them. You just have to change the x-component.
Hint: It will get an additional t-component afterwards.

 

[bfusco]

d/dx(dψ/dx' * dx'/dt + dψ/dt' * dt'/d(something)) ?

 

[bfusco]

d/dx(dψ/dx' * dx'/dt + dψ/dt' * dt'/d(something)) ?

wait is it d/dx(dψ/dx' * dx'/dx + dψ/dt' * dt'/dx), which equals d/dx(dψ/dx' * 1 + dψ/dt' * 0)?

 

[Nickie]

I would like to clarify that the statement "prove that the electromagnetic wave equation is not invariant under Galilean transformation" is not entirely accurate. The fact is that the electromagnetic wave equation is not invariant under Galilean transformation in the traditional sense, as the equation does not have the same form for a moving observer. However, this does not mean that the equation is not valid or useful in describing electromagnetic phenomena.

To understand this better, let's first define what is meant by Galilean transformation. Galilean transformation is a mathematical concept that describes the transformation of coordinates and velocities between two inertial frames of reference that are moving at a constant velocity relative to each other. In other words, it is a transformation that takes into account the relative motion between two frames of reference.

Now, let's consider the electromagnetic wave equation. This equation describes the behavior of electromagnetic waves in a vacuum, and it is a fundamental equation in electromagnetics. It is derived from Maxwell's equations, which are the laws that govern the behavior of electromagnetic fields. The electromagnetic wave equation shows that the electric and magnetic fields in an electromagnetic wave are perpendicular to each other and to the direction of propagation of the wave.

When we apply Galilean transformation to the electromagnetic wave equation, we are essentially changing the frame of reference from which we are observing the wave. This means that the coordinates and velocities in the equation will be transformed according to the Galilean transformation rules. However, the fundamental properties of the wave, such as the perpendicularity of the electric and magnetic fields, remain the same.

In other words, while the form of the equation may change, the underlying physical phenomenon described by the equation remains the same. This means that the electromagnetic wave equation is still valid and useful in describing electromagnetic waves, even when we apply Galilean transformation.

In conclusion, while it is true that the electromagnetic wave equation is not invariant under Galilean transformation, this does not invalidate the equation or its use in describing electromagnetic phenomena. As scientists, it is important to understand the limitations and assumptions of mathematical concepts and equations, but also to recognize their usefulness and applicability in describing the natural world.