How Does Phase Accumulation Affect a Wave's Return Journey?

[sapz]

Homework Statement

I hope this question is better suited to this forum...

(See picture)
We have 3 areas in which a wave can move.

The wave Y1 starts at area 1 and goes towards Border 1, some part of it is passed to Area 2.
That part goes towards Border 2, and some part of it is reflected back into Area 2.
That part moves towards Border 1, and some of it passes to Area 1.

I'm interested in that last part that returned to Area 1, which is Y4.
What is its Phase? It would seem that the wave accumulated phase when it was in Area 2, so should it be 2D * k2? (wave number times the distance in that area)?

If the original wave was $Y_1(x,t) = Ae^{i(wt-k_1x)}$, would Y4 be $Y_4(x,t) = Be^{i(wt+k_1x+\phi)}$, where $\phi = 2Dk_2$?
Or should it be $\phi = -2Dk_2$?

(A and B are some amplitudes we can relate through reflectivity and transmittance coefficients)

 

[haruspex]

The extra path length will create phase lag. Think of it this way, it has the same phase as the waves that were reflected from the first boundary at 2D/v in the past.
What about a possible 180 degree phase shift at the reflection?

 

[sapz]

The 180 degrees shift at reflection would be expressed by a minus sign in the reflection coefficient.
But what about the shift lag from the extra path? I am not sure I understand

 

[haruspex]

The 180 degrees shift at reflection would be expressed by a minus sign in the reflection coefficient.

Not necessarily. It depends on which medium has the higher index.

But what about the shift lag from the extra path? I am not sure I understand

A wave reflected at the second boundary has had to travel an extra 2D compared to a wave reflected at the first boundary. That takes time 2D/v. If the frequency at the source is ω, the phase at the source advances ω2D/v in that time. So the phase of a wave reflected at the second boundary will be ω2D/v behind a wave it meets that was reflected at the first boundary.

 

[paranormal]

I can provide a response to this content. The phenomenon described here is known as phase accumulation of a wave. It occurs when a wave travels through different mediums or areas with varying properties, causing a change in its phase. In this case, the wave Y1 starts at area 1 and travels towards Border 1, where some of it is transmitted to Area 2. As it passes through Area 2, it accumulates phase due to the change in medium properties, and then some of it is reflected back into Area 2. When it reaches Border 1 again, some of it is transmitted back to Area 1. This returning wave, Y4, will have a different phase compared to the original wave Y1 due to the phase accumulation in Area 2.

The phase of a wave is determined by the wave number (k), which is related to the wavelength (λ) and the frequency (ω) of the wave. In this case, the phase of Y4 can be calculated using the formula Φ = 2π/λ * D * k2, where D is the distance traveled in Area 2 and k2 is the wave number in Area 2. This formula assumes that the wave travels in a straight line and does not encounter any obstacles or other factors that may affect its phase.

Based on the given information, Y4 can be written as Y_4(x,t) = Be^{i(wt+k_1x+\phi)}, where B is the amplitude of the reflected wave and ϕ is the phase accumulated in Area 2. The value of ϕ can be either positive or negative, depending on the direction of the phase shift in Area 2. If the phase shift is in the same direction as the original wave, then ϕ = 2Dk_2. However, if the phase shift is in the opposite direction, then ϕ = -2Dk_2.

In summary, phase accumulation of a wave occurs when a wave travels through different mediums or areas with varying properties, resulting in a change in its phase. The phase of the returning wave Y4 can be calculated using the formula Φ = 2π/λ * D * k2, where D is the distance traveled in Area 2 and k2 is the wave number in Area 2. The value of ϕ in the formula can be either positive or negative, depending on the direction of the phase shift in