Problem 14.2.15

Solution

The main idea of the method is that the phase of a plane wave is a Lorentz invariant. It follows that the frequency and the wavenumber transform during transitions between inertial reference frames according to the same Lorentz transformation formulas as time and coordinate .

The solution algorithm for both problems is the same:

1. Transition from the laboratory frame () to the rest frame of the moving object ().
2. Analysis of the process (reflection or refraction) in frame , where the boundary is stationary.
3. Inverse transition from to the laboratory frame for the modified wave.

Solution to problem 14.2.5 (Reflection from a mirror)

Let the incident wave travel along the -axis. Its frequency is , and the wavenumber is . The mirror moves towards the wave with a velocity . The projection of the mirror's velocity in frame is .

Step 1. Transition to the mirror frame .
In the rest frame of the mirror , the frequency of the incident wave is determined by the formula of the relativistic Doppler effect for an approaching source (which directly follows from the Lorentz transformations):

Step 2. Reflection in frame .
In frame , the mirror is stationary. Upon normal reflection from a stationary ideal mirror, the wave frequency is conserved, and the direction of propagation is reversed.
Reflected frequency: .
Reflected wavenumber (taking into account the direction along the axis): .

Step 3. Return to the laboratory frame .
Frame (the mirror) moves relative to with a velocity .
We apply the inverse Lorentz transformation for the frequency of the reflected wave:


Substituting from Step 1:

The change in wave frequency :

Solution to problem 14.2.6 (Wave inside a dielectric)

Step 1. Transition to the dielectric frame .
Similarly to the first problem, in the rest frame of the dielectric , the frequency of the incident wave is:

Step 2. Entering the dielectric in frame .
In frame , the boundary of the dielectric is stationary. When passing through a stationary boundary between media, the wave frequency does not change:

The wave continues to propagate in the same direction (along ). The phase velocity in the dielectric is , so the wavenumber inside is:

Step 3. Return to the laboratory frame .
The velocity of frame relative to is still .
We apply the inverse Lorentz transformation for the frequency of the transmitted wave:


Substituting the expression for :

Let us find the difference in frequencies of the plane wave outside () and inside () the dielectric: