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

AI Thread Summary
The discussion focuses on calculating the amplitude in the plane where x equals y, influenced by two speakers located at (-L, 0, 0) and (0, -L, 0). The speaker wave equations are provided, and the user is attempting to combine these to find the resultant amplitude. There is confusion regarding how to incorporate the z-dimension and whether to set r_x equal to r_y for simplification. The user expresses frustration over the lack of z-dependence in the wave functions, leading to uncertainty about the propagation characteristics. Overall, the thread highlights the complexities of wave interference and the challenges of visualizing amplitude in a three-dimensional context.
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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.
 
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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?
 
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.
 
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