91Ó°ÊÓ

Two speakers spaced a distance \(1.5 \mathrm{m}\) apart emit coherent sound waves at a frequency of \(680 \mathrm{Hz}\) in all directions. The waves start out in phase with each other. A listener walks in a circle of radius greater than one meter centered on the midpoint of the two speakers. At how many points does the listener observe destructive interference? The listener and the speakers are all in the same horizontal plane and the speed of sound is \(340 \mathrm{m} / \mathrm{s} .\) [Hint: Start with a diagram; then determine the maximum path difference between the two waves at points on the circle. I Experiments like this must be done in a special room so that reflections are negligible.

Step by step solution

01

Draw and analyze a diagram

To visualize the scenario, draw a diagram with two speakers 1.5 meters apart and a listener walking around the circle centered at the midpoint between the speakers. Label the speakers as A and B, the midpoint as C, and a point on the circle where the listener is standing as P. Also, label the distances AC and BC.

02

Calculate the wavelength of the sound waves

Since we are given the speed of sound (\(v=340 \mathrm{m} / \mathrm{s}\)) and the frequency of the sound waves (\(f=680 \mathrm{Hz}\)), we can calculate the wavelength of the sound waves using the formula: \(\lambda = \frac{v}{f}\). In this case, \(\lambda = \frac{340}{680} = 0.5 \mathrm{m}\).

03

Determine the path difference

The path difference between the sound waves at point P can be represented as \(|AP - BP|\). Notice that \(\bigtriangleup ACP\) and \(\bigtriangleup BCP\) are right triangles, and \(d = AC = BC = \frac{1.5}{2}=0.75 \mathrm{m}\).

04

Calculate the maximum path difference for destructive interference

For destructive interference to occur, the path difference must be equal to half a wavelength or an odd multiple of it. The maximum path difference occurs when the listener is at the furthest point from one of the speakers, such as the point on the circle diametrically opposite to speaker A or B. With this in mind, the maximum path difference that still causes destructive interference is \(2d = 1.5 \mathrm{m}\)

05

Identify the number of points of destructive interference

Since the maximum path difference that still causes destructive interference is exactly equal to \(3(\frac{1}{2}\lambda)\), the listener will experience destructive interference at 3 points around the circle (1 point per multiple of \(\frac{1}{2} \lambda\)): once when the path difference is \(\frac{1}{2} \lambda\), once when the path difference is \(\lambda\), and once when the path difference is \(\frac{3}{2} \lambda\). Therefore, the listener experiences destructive interference at 3 points on the circle.

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