Two speakers are driven by the same amplifier and emit sinusoidal waves in phase. The speakers...

Question:

Two speakers are driven by the same amplifier and emit sinusoidal waves in phase. The speakers are on the x axis a distance d = 2.50 m apart. The frequency of the sound waves produced by the loudspeakers is 200 Hz. For a point on the x axis between the two speakers, at what values of x will (a) destructive interference occur, and b) constructive interference occur? (c) Can destructive or constructive interference occur at a point on the x axis to the right of both speakers or to the left of both speakers? Explain. (Assume the temperature is 86 F (30 C).)

Interference of Sound:

When two traveling waves which exist in the same medium will interfere with each other, the intensity of sound rises as some points and falls at some other points. If the two sound wave superpose in the same phase, the interference is said to be constructive interference, and destructive interference if they are in opposite phase.

Answer and Explanation:

speed of sound at a temperature t is given by

{eq}v_t=331+0.61t=331+0.61 \times 30 =349.3\ m/s {/eq}

for constructive interference , path difference= integer multiple of wavelength. That is when

{eq}\lambda=\dfrac{v}{f}=\dfrac{349.3}{200}=1.745\ m \\ (d-x)-x=n \lambda \\ \Rightarrow x=\dfrac{d- n \lambda}{2} \\ for\ n=0 \\ x=\dfrac{d}{2}=\dfrac{2.5}{2}=1.25\ m \\ for\ n=1 \\ x=\dfrac{d- \lambda}{2} \\ x=\dfrac{2.5- 1.745}{2} =0.376\ m {/eq}

For destructive interference

{eq}(d-x)-x=n \dfrac{ \lambda }{2} \\ \Rightarrow x=\dfrac{d- n \lambda}{4} \\ for\ n=0 \\ x=\dfrac{d}{4}=\dfrac{2.5}{4}=0.625\ m \\ for\ n=1 \\ x=\dfrac{d- \lambda}{4} \\ x=\dfrac{2.5- 1.745}{4} =0.188\ m {/eq}

c)

All the points between the two speakers can be the points of constructive or destructive interference.


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Interference Patterns of Sound Waves

from MTEL Physics (11): Practice & Study Guide

Chapter 16 / Lesson 5
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