How can the invariance of the wave equation be shown without using tensors?

In summary, the conversation discusses the problem of proving that the wave equation is invariant and the attempts to solve it without using tensors. The suggested approach involves reducing the equation to 1D and using Lorentz transformations to show its invariance. The conversation also touches on the meaning of invariance and the importance of understanding coordinate transformations in solving this problem.
  • #1
loonychune
92
0
The problem is, rather briefly:
Show that the wave equation is INVARIANT
The equation is given as:

[the Laplacian of phi] - 1/(c^2)*[dee^2(phi)/dee(t^2)]

dee being the partial derivative.. phi is a scalar of (x, y, z, t)


Now, i want, and think i should be able, to solve this problem without resorting to tensors.

What I've tried to do is this:
Given a 4-vector, X, X.X = X'.X' would imply that whatever makes up that 4-vector is invariant. So, i have to write the above equation as a 4-vector, apply the Lorentz transformation to get X' and then check to see if X.X = X'.X' ?
i can't actually write that equation as a 4-vector!
Given that the 4-velocity involves manipulating r = (x, y, z, ict) to get u = gamma(u, ic) - i am thinking, what do i do to x, y and z and to r to get the equation ? - this line of thinking seems to have me stumped.
..perhaps using the chain rule to get the laplacian in terms of dee x / dee t(proper time) as it relates to velocity ...
Perhaps i need to look at tensors, and relativity of electrodynamics ? But i am assuming that I'm over complicating things given the detail of the book i got this problem from (it's actually Marion & Thornton: Classical Dynamics chapter 14 problem 1)

I hope I've given enough to warrant a reply, as, even though i probably wouldn't be asked this on the examination (it's more likely to be applications in relativistic kinematics), I'm pretty aggrivated as to why i can't seem to even get close to an answer given the study of the relevant chapter...
 
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  • #2
What does it mean for an equation to be invariant?
 
  • #3
Well, i think it means that under a transformation (Lorentz invariance concerns rotations of x1-x4, so in this case, under a rotation) the equation, relative to its new axis, remains unchanged. So measuring the Xn components yields the same answer in all inertial frames...

That's where i was going with that equation at least, I'm just not able to work out 'its components'...
 
  • #4
not sure that makes sense in saying 'measure the Xn components'... the equation will be the same with x replaced with x' and y, y' z, z' t, t'
 
  • #5
...help :(
 
  • #6
Well, the first thing I would do is to reduce it to the 1D wave equation, to save time, since we can take the lorentz transformation to be only in the x direction. Now, consider the lorentz transformation [tex]\bar{x}=\gamma(x-vt) \hspace{2cm} \bar{t}=\gamma\left(t-\frac{vx}{c^2}\right)[/tex].

Now, the equation is [tex]\frac{\partial^2\phi}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}=0[/tex]

You want to show that this is invariant under the above transformation, so you need to calculate [tex]\frac{\partial \bar{\phi}}{\partial t}[/tex] (and the other derivatives) in terms of the barred coordinates, using the chain rule, and substitute into the equation to show that is invariant.

I should also point out that there is not really any physics needed to do this-- its simply an exercise on evaluating partial derivatives with respect to some transformed coordinates.
 
Last edited:
  • #7
Thanks, i did look roughly along those lines once i found a thread showing why the equation ISN'T invariant under galilean transformations, but found the algebra a bit messy so i spose now i know where I'm going - bob's my uncle.. thanks
 

1. What is the wave equation and why is it important?

The wave equation is a mathematical formula that describes how waves behave and propagate through a medium. It is important because it is used to model and understand many different phenomena such as sound, light, and electromagnetic waves.

2. What is the principle of invariance in the wave equation?

The principle of invariance in the wave equation states that the form of the equation remains the same regardless of any changes in the coordinates or reference frame. This means that the equation will hold true no matter how the observer or the source of the wave is moving.

3. How does the invariance of the wave equation relate to the laws of physics?

The invariance of the wave equation is closely related to the principles of relativity and conservation of energy in physics. It ensures that the laws of physics remain consistent in all reference frames and that energy is conserved regardless of the observer's perspective.

4. Can the invariance of the wave equation be violated?

No, the invariance of the wave equation is a fundamental principle in physics and cannot be violated. If it were to be violated, it would mean that the laws of physics are not consistent and could lead to contradictory results.

5. How is the invariance of the wave equation applied in real-world situations?

The invariance of the wave equation is used in many fields, such as acoustics, optics, and electromagnetics, to model and predict the behavior of waves. It is also utilized in the development of technologies such as radar, sonar, and medical imaging devices.

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