Calculating Distance Between Two Speakers Producing 180Hz Sound in Phase

[lilkrazyrae]

Two speakers produce a 180Hz sound in phase. If an intensity is located 2.10m from the nearer spearker, find the distance to the further speaker if the speed of sound in the air is 345m/s.

I don't understand exactly what this problem is asking for, and I can't find an equation relating these things. Please help

 

[Tide]

I think the question is asking you to take into account the interference between the two waves in order to determine the intensity of the combined waves at a given point. You know the speed of sound, the frequency and direction of travel for each of the waves along with the relative phase of the sources. You should be able to work out the relative phase at the observation point. :)

 

[lilkrazyrae]

I don't understand our book doesn't say anything about the relative phase; therefore, I don't fully understand what you mean I gathered that the interference between the two waves was the point of the problem but I still don't quite understand it.

 

[Tide]

Actually, looking at your original statement again, there appears to be a word missing where you say "... an intensity is located ..." I think you mean to say there is an intensity node or maximum or something of that sort.

You are dealing with two waves presumably of equal source amplitude propagating in opposite directions. Before we proceed, have you seen anything like $A \cos (kx - \omega t)$ to describe wave propagation?

 

[lilkrazyrae]

I did miss a word and it is intensity minimum and I found an equation similar to that one except it was Asin(kx+wt).

 

[Tide]

Okay. We have one thing left to decide. The A in $A \sin (kx - \omega t)$ represents the amplitude of the wave. I am going to assume (though it's not exactly correct) that the amplitude is a constant for your problem.

Notice that $A \sin (kx - \omega t)$ represents a wave traveling toward the right (toward increasing x) while $A \sin (kx + \omega t)$ travels to the left. Suppose the wave from the nearer speaker is located at x = 0 and is given by $A \sin (kx - \omega t)$. We have to make a slight adjustment to describe the wave from the far speaker (besides just traveling in the opposite direction). Your problem stated that the speakers are in phase so if the second speaker is at x = L then its wave looks like $A \sin (k(x-L) + \omega t)$.

Therefore, the combined wave at any point is

$$A \left( \sin(kx -\omega t) + \sin(k(x-L) + \omega t)\right)$$

Note that $k = \frac {2\pi}{\lambda}$ where $\lambda$ is the wavelength and $\omega = 2 \pi f$ where f is the frequency of the wave. You should also know that $f \lambda = c_s$ the speed of sound.

Can you write the intensity of the wave?

 

[lilkrazyrae]

The only way I see to write the intensity uses mass, and I don't know the mass.

 

[lilkrazyrae]

Ok so do you just solve for x and that is your other distance

 

[Tide]

No, you don't need a mass. Does your textbook tell you how to find the intensity of a wave or the intensity of combined waves?

 

[lilkrazyrae]

Well I have an intensity related to power equation, and I=I(1)+I(2)