Problem 21.64 You are standing 2.5 directly in front of one of the two loudspeak

Question

Problem 21.64 You are standing 2.5 directly in front of one of the two loudspeakers shown in the figure. They are 3.0 apart and both are playing a 686 tone in phase. (Figure 1) Part A As you begin to walk directly away from the speaker, at what distances from the speaker do you hear a minimum sound intensity? The room temperature is 20 . Express your answer numerically using two significant figures. If there is more than one answer, enter your answers in ascending order separated by commas. Part B A tuner first tunes the A string very precisely by matching it to a 440 Hz tuning fork. She then strikes the A and E strings simultaneously and listens for beats between the harmonics. What beat frequency indicates that the E string is properly tuned? Part C The tuner starts with the tension in the E string a little low, then tightens it. What is the frequency of the E string when she hears four beats per second?

Explanation / Answer

Speed of sound, v = 331.3 + 0.6 * T = 343.3 m/s

frequency, f = 686 Hz
Wavelength , = v/f = 0.5

For a minimum sound intensity, the path difference between the two sounds must be an odd multiple of /2
x = (2.5^2 + 3^2) - 2.5 = 1.4051

The nearest odd multiple of /2 = 5/2 = 1.25 and next is 3/2 = 0.75 and /2 = 0.25

Let x be the distance between the straight back speaker and the person...

So,

For x = 5/2

x = 1.25 = (x^2 + 3^3) - x

So, x = 2.975 m

For x = 3/2

x = 0.75 = (x^2 + 3^3) - x

So, x =5.625 m

For x = /2

x = 0.25 = (x^2 + 3^3) - x

So, x =17.875 m

So answer : 2.975 m, 5.625 m, 17.875 m

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