Solving Billy and Speakers: 16m, 0.8m, 3000 Hz

[axgalloway]

Homework Statement

Two lound speakers are each located 16m from a wall and are 0.8m apart. They play a 3000 Hz sound, both speakers are in phase. Bill is located at the wall at the m=0 maximum of sound interference. If he walks along the wall, how far does he have to walk to hear m=1 maximum? (The speed of sound is 343 m/s.)

Homework Equations

2pi * (x2-x1) / lambda = 2mpi
vsound = lambda*f

The Attempt at a Solution

343 m/s = lambda*3000Hz
lambda= .1143 meters

Beyond that, I do not know how to solve it.
Thank you.

 

[LowlyPion]

Sounds like you have a little geometry problem. Don't you just need to find the point away from the mid-line whose distance from each speaker is different by half a wavelength?

 

[nlightner]

I would first clarify the problem by asking for additional information such as the direction of the sound waves and the orientation of the speakers. This will help determine the specific equation to use in solving the problem.

Assuming that the sound waves are traveling in a straight line and the speakers are oriented towards the wall, we can use the equation for constructive interference to solve for the distance that Bill needs to walk to reach the m=1 maximum.

Constructive interference occurs when the path difference between the two sound waves is equal to an integer multiple of the wavelength (λ). In this case, we can use the following equation:

2π * (x2-x1) / λ = 2π * m

Where:
x2 = distance from the second speaker to the wall (16m)
x1 = distance from the first speaker to the wall (16m+0.8m = 16.8m)
λ = wavelength (0.1143m)
m = order of maximum interference (1)

By substituting the given values, we can solve for x2:

2π * (x2-16.8m) / 0.1143m = 2π * 1
x2-16.8m = 0.1143m
x2 = 16.9143m

Therefore, Bill needs to walk 0.9143m along the wall to reach the m=1 maximum of sound interference. This can also be confirmed by using the equation for the speed of sound to calculate the time it takes for the sound wave to travel the additional distance of 0.9143m:

Time = Distance / Velocity
Time = 0.9143m / 343m/s = 0.0027s

Since the frequency of the sound is 3000Hz, the time for one cycle is 1/3000s = 0.000333s. Therefore, in 0.0027s, the sound wave will complete 8 cycles, which corresponds to the m=1 maximum of interference.