Phase constant and reflection of waves

[NATURE.M]

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:
$ψ_{1}(x, t)$ = 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?
I'm really confused about this??

 

[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 $k=\omega/v$.

 

[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?

 

[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?