Lorentz transf. of a spherical wave in Euclidean space

In summary, a moving spherical standing wave still solves the wave equation, when boosted in the following way.
  • #1
Rumo
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This thread is not about the lorentz invariance of the wave equation: [tex] \frac{1}{c^2}\frac{\partial^2\Phi}{\partial t^2}-\Delta \Phi = 0[/tex]

It is about an interesting feature of a standing spherical wave:
[tex] A\frac{\sin(kr)}{r}\cos(wt) [/tex]

It still solves the wave equation above, when it is boosted in the following way:
[tex] z' = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}(z-vt) [/tex]
and
[tex] t' = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}(t-\frac{vz}{c^2}) [/tex]
This means that a transformation, which looks like the lorentz transformation, is needed for a moving standing spherical wave to still solve the wave equation. It is important to notice, that this takes place in an Euclidean space!

Source: http://arxiv.org/abs/1408.6195

I would like to know, what you think about this. Including the paper above. The paper was written for educational purposes, because it shows, that the lorentz transformation can arise in an Euclidean space.
 
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  • #2
I calculated this with Mathematica. I would have uploaded the notebook, but I can't.
Hence I made a screenshot. You can calculate this easily with any program of your choice.
Just replace c with w/k.
 

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  • #3
I only don't understand the first sentence in #1. It's of course all about the Lorentz transformation, which is a symmetry of the wave equation.

[OT for the admins] Why can't one upload an nb (Mathematica notebook) file? Wouldn't this be very nice for those among us, who have Mathematica at hand? I never thought to upload a Mathematica notebook so far, but it's a nice idea, isn't it?
 
  • #4
vanhees71 said:
I only don't understand the first sentence in #1. It's of course all about the Lorentz transformation, which is a symmetry of the wave equation.

Thanks for your reply!
Isn't the lorentz transformation transforming the spatial and time coordinates? Hence, I would use the chain rule to show the invariance. The Φ would not be important at all, just a solution to the wave equation. It would describe how Φ would look like from the point of view of a boosted observer!?

But in this case the spatial and time coordinates are not transformed.

[tex] \Phi = A\frac{\sin(k\sqrt{x^2+y^2+(\gamma(z-vt))^2})}{\sqrt{x^2+y^2+(\gamma(z-vt))^2}}\cos(w\gamma(t-\frac{vz}{c^2}) [/tex]
is a solution of the wave equation. v is the velocity of the 'standing' spherical wave, relative to the e.g. ideal gas.
 
  • #5
Both space and time are transformed. The spherical standing wave carries 0 momentum in the old frame and a momentum in z-direction in the new one.
 
  • #6
vanhees71 said:
Both space and time are transformed. The spherical standing wave carries 0 momentum in the old frame and a momentum in z-direction in the new one.

But in this case, it is not a coordinate transformation!
It is a moving spherical standing wave.
I still use the same coordinates to differentiate.
x, y, z, t. not x, y, z', t'!
 

Related to Lorentz transf. of a spherical wave in Euclidean space

What is a spherical wave in Euclidean space?

A spherical wave in Euclidean space is a type of electromagnetic wave that propagates outward in all directions from a central point. It is characterized by a spherical wavefront and has a decreasing amplitude as it travels further from the source.

What is the Lorentz transformation of a spherical wave?

The Lorentz transformation of a spherical wave is a mathematical formula that describes how the wave's properties, such as its frequency and wavelength, change when observed from different reference frames moving at different velocities. It takes into account the effects of time dilation and length contraction predicted by Einstein's theory of special relativity.

Why is the Lorentz transformation important for understanding spherical waves?

The Lorentz transformation is important for understanding spherical waves because it allows us to accurately describe and predict the behavior of these waves in different reference frames. This is crucial in many areas of physics, such as optics and electromagnetism, where the effects of special relativity must be taken into account.

What are the key equations involved in the Lorentz transformation of a spherical wave?

The key equations involved in the Lorentz transformation of a spherical wave are the Lorentz transformations for time and space, given by t' = γ(t − vx/c²) and x' = γ(x − vt), where γ is the Lorentz factor, t and x are the time and position in the original frame, and t' and x' are the time and position in the transformed frame.

How does the Lorentz transformation of a spherical wave differ from the transformation of a plane wave?

The Lorentz transformation of a spherical wave differs from that of a plane wave in that the spherical wavefronts are no longer perpendicular to the direction of motion in the transformed frame. This is due to the fact that a spherical wave expands in all directions, while a plane wave propagates in a single direction. Additionally, the frequency and wavelength of a spherical wave are not constant in the transformed frame, unlike a plane wave which maintains its frequency and wavelength in all reference frames.

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