Phase constant and reflection of waves

In summary, the waves have a differing frequency. The reflected waves will have the same phase constant θ if the end of the string is not fixed.
  • #1
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Homework Statement


If two waves are created at x = 0 and t = 0 (and are in phase to begin with), and are then sent traveling along a string in the positive x direction, and they get reflected at the end of the string, there will be a similar pair of waves traveling in the negative x-direction. The two waves have a differing frequency.
Now my question is whether there a phase angle associated with these wave (namely the reflected waves).
Like for the waves traveling in the positive direction:
[itex]ψ_{1}(x, t)[/itex] = Asin(([itex]ω_{1}/v[/itex])x - ([itex]ω_{1}[/itex])t + θ)
[itex]ψ_{2}(x, t)[/itex] = Asin(([itex]ω_{2}/v[/itex])x - ([itex]ω_{2}[/itex])t + θ), would θ = 0 (the phase constant) in both these cases ?

And the reflected waves:
[itex]ψ_{1}(x, t)[/itex] = Asin(([itex]ω_{1}/v[/itex])x + ([itex]ω_{1}[/itex])t + θ)
[itex]ψ_{2}(x, t)[/itex] = Asin(([itex]ω_{2}/v[/itex])x + ([itex]ω_{2}[/itex])t + θ), would θ = 0 in these cases as well?
I'm really confused about this??
 
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  • #2
So you are pretty sure of the initial (incident) wave, and are unsure about the reflected wave...
[at some time t> 2L/v ...]

the incident wave travels in the +x direction at speed v:
##y_{i}(x,t)=A\sin k(x-vt)##

the reflected wave travels in the -x direction at speed v:
##y_{r}(x,t)=B\sin [k(x+vt)+\phi]##

(where B and \phi are the unknowns ... though we expect |B|=|A| from your description.)

we require that ##y_{r}(L,t)=-y_{i}(L,t)## ... if the wave inverts on reflection.
The time and space derivatives also have a similar relationship.
So you can solve the simultaneous equations.Off your description - both incident waves will have the same initial phase ##\theta##.
The value of the initial phase depends on when you started your stopwatch ... it is usually convenient to set it to zero or pi/2 unless you have some reason to believe it is something different.

You should be aware that the phase of the wave at x and t is given by the entire argument of the sine function. The \theta in there is the phase at (x,t)=(0,0). You'll also find it easier to represent the waves in terms of wave numbers ##k=\omega/v##.
 
Last edited:
  • #3
So I should just compute θ for the reflected waves by setting (x, t) = (0, 0) ?
Assuming the end of the string is not fixed, then the reflections are not 180 deg out of phase. But will there still be a phase constant θ present?
 
  • #4
Why not try it and see?
But why not just follow the suggestion you were given?
i.e. relate the reflected wave to the incident one and use the boundary conditions?
 
  • #5


I can provide the following response to this content:

Yes, there is a phase angle associated with these reflected waves. In general, the phase angle (θ) represents the initial phase of a wave at a given point in space and time. In the case of the reflected waves, the initial phase is determined by the boundary conditions at the end of the string where the waves are reflected.

In the equations provided, the phase constant (θ) is represented as the term within the parentheses after the negative sign. It is worth noting that the phase constant can take on any value, depending on the boundary conditions and the specific properties of the medium in which the waves are traveling. Therefore, θ may not necessarily be equal to 0 in both cases.

Furthermore, the frequency (ω) of the reflected waves may differ from the frequency of the incident waves, depending on the properties of the medium and the boundary conditions. This can result in a change in the wavelength and amplitude of the reflected waves.

In summary, the phase constant and the frequency of the reflected waves are determined by the boundary conditions and the properties of the medium. They may not necessarily be the same as the incident waves and can vary depending on the specific situation.
 

1. What is the phase constant of a wave?

The phase constant of a wave is a measure of the position of a wave in its cycle at a specific point in space and time. It is represented by the symbol φ and is measured in radians.

2. How is the phase constant related to the wavelength of a wave?

The phase constant is directly proportional to the wavelength of a wave. This means that as the wavelength increases, the phase constant also increases.

3. What is the difference between phase constant and phase shift?

The phase constant and phase shift are both measures of the position of a wave in its cycle. However, the phase constant is a fixed value for a specific wave, while the phase shift can vary depending on the reference point chosen.

4. How does phase constant affect the reflection of a wave?

The phase constant plays an important role in determining how a wave reflects off of a surface. If the phase constant of the reflected wave is the same as the incident wave, they will interfere constructively, resulting in a strong reflection. If they are out of phase, they will interfere destructively, resulting in a weaker reflection.

5. Can the phase constant of a wave change as it reflects off of a surface?

No, the phase constant of a wave remains constant as it reflects off of a surface. The only thing that changes is the phase shift, which depends on the angle of incidence and the properties of the surface.

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