# Phase constant and reflection of waves

1. Mar 25, 2014

### NATURE.M

1. The problem statement, all variables and given/known data
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:
$ψ_{1}(x, t)$ = Asin(($ω_{1}/v$)x - ($ω_{1}$)t + θ)
$ψ_{2}(x, t)$ = Asin(($ω_{2}/v$)x - ($ω_{2}$)t + θ), would θ = 0 (the phase constant) in both these cases ?

And the reflected waves:
$ψ_{1}(x, t)$ = Asin(($ω_{1}/v$)x + ($ω_{1}$)t + θ)
$ψ_{2}(x, t)$ = Asin(($ω_{2}/v$)x + ($ω_{2}$)t + θ), would θ = 0 in these cases as well?

2. Mar 26, 2014

### Simon Bridge

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: Mar 26, 2014
3. Mar 26, 2014

### NATURE.M

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. Mar 26, 2014

### Simon Bridge

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?