Question

Two loudspeakers 42.0 m apart and facing each other emit identical 115 Hz sinusoidal sound waves in a room where the sound speed is 345 m/s. Susan is walking along a line between the speakers. As she walks, she finds herself moving through loud and quiet spots. If Susan stands 19.5 m from one speaker, is she standing at a quiet spot or a loud spot?

Answer 1

Answer:

She will be standing at a loud spot.

Explanation:

In order to define if Susan is standing at a quiet spot or a loud spot, we need to know first the difference in the path of the sound that reach to Susan from one speaker and the one from the other speaker.

If we call d₁ to the distance from speaker A, that we know is equal to 19.5m, the distance d₂ to the other speaker will be as follows:

d₂ = 42.0 m - 19.5m = 22.5 m

So, the difference in path for both speakers is just:

d = d₂-d₁ = 22.5 m -19.5 m = 3.0 m

Now, we need to relate this distance with the wavelength of the sound, as we know for a constructive interference, the difference in the paths must be equal to an even multiple of the semi-wavelength, as follows:

d = (2n)*(λ/2)

In order to get the value of λ, we know that at any wave, there exists a fixed relationship between the speed, the frequency and the wavelength:

v = λ*f

If v = 345 m/s and f= 115 1/sec, we can easily solve for λ:

λ = v/f = 345 m/s / 115 (1/s) = 3.0 m

As the difference between the distances from both speakers to Susan equals exactly to one wavelength, this means that both arrive in the same phase, which means that there will be a constructive interference, i.e., it will be a loud spot.