Wave equation and Galilean Transformation

In summary, the conversation discusses the Galilean invariance of the classical wave equation and the use of coordinate frames to show its transformation. The correct equations are derived for both space and time, and the need to use the Lorentz transforms is mentioned. It is also mentioned that this exercise is relevant for the special relativity exam.
  • #1
Advent
30
0
Hi! I was reading some notes on relativity (Special relativity) (http://teoria-de-la-relatividad.blogspot.com/2009/03/3-la-fisica-es-parada-de-cabeza.html) and it says that the classical wave equation is not Galilean Invariant. I tried to show it by myself, but I think there is some point that I'm missing.

Given two coordinate frames, if [tex] x=\bar{x}+vt [/tex] the classical wave equation transforms to:

[tex]\frac{\partial^2 \phi}{\partial \bar{x}^2} + \frac{\partial^2 \phi}{\partial \bar{y}^2} + \frac{\partial^2 \phi}{\partial \bar{z}^2} - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial \bar{t}^2} + \frac{1}{c^2} \left( 2v \frac{\partial^2 \phi}{\partial \bar{x} \partial \bar{t}}- v^2 \frac{\partial^2 \phi}{\partial \bar{x}^2} \right) = 0[/tex]

But i can really get that answer. Supose I want to compute

[tex] \frac{\partial \phi}{\partial \bar{x}} = \frac{\partial \phi}{\partial x} \frac{\partial x }{\partial \bar{x}} + \frac{\partial \phi}{\partial y } \frac{\partial y }{\partial \bar{x}} + \frac{\partial \phi }{\partial z } \frac{\partial z }{\partial \bar{x}} + \frac{\partial \phi }{\partial t } \frac{\partial t }{\partial \bar{x}} [/tex]

And

[tex] t = \bar{t} = \frac{x-\bar{x}}{v} [/tex]

should be my hints to get to that result?

Thanks!
 
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  • #2
Advent said:
And

[tex] t = \bar{t} = \frac{x-\bar{x}}{v} [/tex]

No, this is incorrect. You need to use

[tex] t = \bar{t} [/tex]

and

[tex] x=\bar{x}+v \bar{t} [/tex]
 
  • #3
So the right thing is [tex]\frac{\partial t}{\partial \bar{x}} = 0[/tex] ? and every partial derivative involving time except [tex]\frac{\partial t}{\partial \bar{t}} = \frac{\partial \bar{t}{\partial t}[/tex] which are one
 
  • #4
Advent said:
So the right thing is [tex]\frac{\partial t}{\partial \bar{x}} = 0[/tex]

Yes, of course
 
  • #5
Ok, thank you! Let's see if I can do it now
 
  • #6
Ok, I'm almost done, because now I have problems with the indices.

I started by [tex]x=\bar{x} + vt[/tex] where [tex]v[/tex] is a constant. Now the wave equation [tex]\nabla^2 \phi = \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t}[/tex].

So i start writing the wave equation in the "\bar" coordinates. The Laplacian is the same [tex]\nabla_{bar}^2 \phi = \nabla^2 \phi[/tex] because

[tex]\frac{\partial \phi}{\partial \bar{x}} = \frac{\partial \phi}{\partial x} \frac{\partial x}{\partial \bar{x}} + \frac{\partial \phi}{\partial t} \frac{\partial t}{\partial \bar{x}} = \frac{\partial \phi}{\partial x}[/tex]

For time:

[tex]\frac{\partial \phi}{\partial \bar{t}} = \frac{\partial \phi}{\partial t} \frac{\partial t}{\partial \bar{t}} + \frac{\partial \phi}{\partial x} \frac{\partial x}{\partial \bar{t}} = \frac{\partial \phi}{\partial t} + v \frac{\partial \phi}{\partial x}[/tex]

[tex]\frac{\partial^2 \phi}{\partial \bar{t}^2} = \frac{\partial }{\partial \bar{t}} \frac{\partial \phi}{\partial \bar{t}} = \frac{\partial \phi^2}{\partial t ^2} + 2v \frac{\partial ^2 \phi}{\partial t \partial x} + v^2 \frac{\partial ^2 \phi}{\partial x^2}[/tex]

And the wave equation reads what I said in the first post but without bars. It should be a very silly thing, but I can't see it... Thanks for your time.
 
  • #7
Advent said:
And the wave equation reads what I said in the first post but without bars. It should be a very silly thing, but I can't see it... Thanks for your time.

What happens if you start by calculating the partial derivatives wrt (x,t) instead of (x_bar,t)?
 
  • #8
I was thinking that should be that... in fact, a very very silly thing.

PS: Btw the equation you have calculating with "bar" instead with normal is not the one in my first post. Actually is calculating with normal instead of bars.

Absolutely clear now, thanks again!
 
  • #9
Advent said:
I was thinking that should be that... in fact, a very very silly thing.

PS: Btw the equation you have calculating with "bar" instead with normal is not the one in my first post. Actually is calculating with normal instead of bars.

Absolutely clear now, thanks again!

You are welcome, glad that I could help.
Now, if you do the same exercise by replacing the Galilean transforms with the Lorentz ones, you should get the famous invariance of the wave equation.
 
  • #10
I know, in fact i should do it for my speclal relativity exam, but there is some little work and theory before Lorentz Transformations.
 

1. What is the wave equation?

The wave equation is a mathematical equation that describes the behavior of waves. It is a second-order partial differential equation that relates the wave's amplitude to its position and time, and is used to model various phenomena such as sound, light, and water waves.

2. How is the wave equation derived?

The wave equation is derived from the fundamental laws of physics, such as Newton's laws of motion and the principles of conservation of mass and energy. It can also be derived using mathematical techniques such as Fourier analysis and separation of variables.

3. What is the importance of the wave equation in science?

The wave equation is important in many areas of science and engineering. It is used to understand and predict the behavior of waves, which are fundamental to the functioning of many natural and man-made systems. This equation has applications in fields such as acoustics, optics, electromagnetics, and seismology.

4. What is the Galilean transformation?

The Galilean transformation is a mathematical transformation that relates the coordinates of an event in one inertial frame of reference to the coordinates of the same event in another inertial frame of reference. It was first described by Galileo Galilei in the 17th century and is used to study the motion of objects in classical mechanics.

5. How is the wave equation related to the Galilean transformation?

The wave equation and the Galilean transformation are related through the concept of relative velocity. The wave equation is used to describe the propagation of waves in different reference frames, and the Galilean transformation allows us to transform between these frames. This is important in understanding how waves behave in different situations and how their properties may change depending on the observer's perspective.

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