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

In summary: Please ignore all of the above, I'll start againIn summary, the speakers are located at (-L,0,0) and (0,-L,0) and the objective is to find the amplitude at all positions in the x=y plane. The waves are given by two equations and the amplitude A is real. The attempt at a solution involves finding the amplitude at all points in the x=y plane by setting r_x=r_y and using the real part of the equation. However, the issue is that the equations do not take into account the z-coordinate, leading to confusion and the need to start again.
  • #1
Saraphim
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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:

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

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

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

[tex]\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)[/tex]

But how do I go from here and to only the x=y plane? Do I just set [tex]r_x=r_y[/tex]? How does this carry any information about z? I think I just need a nudge in the right direction.
 
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  • #2
To elaborate a bit, what is stumping me is that both functions completely discard any information about z, and if I take, for instance, [tex]\tilde{f}_x((x,y,z),t)[/tex] 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?
 
  • #3
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.
 

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

1. What is amplitude in plane x=y from two speakers placed on x and y axes?

Amplitude refers to the maximum displacement or distance from the equilibrium position of a wave. In the case of sound waves, it is the maximum displacement of air particles caused by the vibrations of the speakers.

2. How does the arrangement of two speakers on the x and y axes affect the amplitude in plane x=y?

The arrangement of two speakers on the x and y axes allows for sound waves to travel in both the x and y directions, resulting in an interference pattern that can increase or decrease the amplitude at various points in the plane x=y.

3. Can the amplitude in plane x=y be controlled by adjusting the placement of the speakers?

Yes, the amplitude in plane x=y can be controlled by adjusting the distance between the speakers and the angle at which the sound waves are emitted. This will change the interference pattern and therefore affect the amplitude at different points in the plane.

4. What is the significance of amplitude in plane x=y from two speakers in practical applications?

The amplitude in plane x=y can be used to create a more immersive and realistic sound experience in audio systems, as well as to improve the overall sound quality by reducing interference and unwanted echoes.

5. Are there any limitations to using two speakers on the x and y axes to produce amplitude in plane x=y?

Yes, this method may not work as effectively in larger or outdoor spaces where sound waves can reflect off of surfaces and cause interference. Additionally, the placement of the speakers needs to be carefully considered to achieve the desired amplitude pattern.

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