Amplitude in plane x=y from two speakers placed on x and y axes.

[Saraphim]

Homework Statement

Two speakers at (x,y,z)=-L,0,0 and (x,y,z)=(0,-L,0)

Find the amplitude at all positions in the plane x=y

Homework Equations

The waves are given by:

$$\tilde{f}_x(\overline{r},t)=\frac{A}{r_x} e^{i(kr_x-\omega t)}$$

$$\tilde{f}_y(\overline{r},t)=\frac{A}{r_y} e^{i(kr_y-\omega t+\delta)}$$

And the amplitude A is real.

The Attempt at a Solution

I'm unsure how to proceed here, at least with finding the amplitude in the plane itself. I'm thinking that the amplitude for all points must be given by the real part of

$$\tilde{f}(\overline{r},t)=\tilde{f}_x+\tilde{f}_y=Ae^{-i \omega t}\left( \frac{1}{r_x} e^{i kr_x} + \frac{1}{r_y} e^{i(kr_y+\delta)}\right)$$

But how do I go from here and to only the x=y plane? Do I just set $$r_x=r_y$$? How does this carry any information about z? I think I just need a nudge in the right direction.

 

[Saraphim]

To elaborate a bit, what is stumping me is that both functions completely discard any information about z, and if I take, for instance, $$\tilde{f}_x((x,y,z),t)$$ for any set values of (x,y), then the result doesn't depend at all of z! This would make the wave propagate as a cylinder with infinite z-length. Am I going insane?

 

[Saraphim]

Wow, okay, that was utter nonsense. I've now managed to confuse myself to the point where I don't know what I'm doing.