# Superposition of sound waves

Two speakers are driven by the same oscillator whose frequency is f. They are located a distance d from each other on a vertical pole. A man walks straight toward the lower speaker in a direction perpendicular to the pole, as shown in Figure P14.10.

(Note the answers must be in all variables)

(a) How many times will he hear a minimum in sound intensity? (Use f, d, and v as necessary.)

b) How far is he from the pole at these moments? Let v represent the speed of sound, and assume that the ground does not reflect sound. (Use f, d, v, and n as necessary where n = 1, 2, ..., nmax.)

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#### Solution Summary

Two speakers are producing same sound. The positions at which the sound intensities are minimum are determined.

Physics Phenomenon of beating: sound of a siren, beat frequency of two sounds

See attached file for formulas.

Temporal interference results from the superposition of two waves having slightly different frequencies. When the two waves are observed at a point in space, there is a temporal (time) alternation between constructive and destructive interference. If, for example, two tuning forks of slightly different frequencies are struck, one hears a sound of periodically varying amplitude. This phenomenon is called beating, and the number of amplitude maxima one hears per second gives the beat frequency. (a) Let the time-dependent variations of the displacement due to two sound waves of slightly different frequencies, at a point where kx = π /2, be

and

where ω1 > ω2. Use the principle of superposition to show that the resulting displacement is

where and . (b) The derivation in (a) shows that the resultant displacement is a cosine function whose angular frequency is ω and whose time varying amplitude is the absolute value of the quantity in the brackets. Show that the number of beats per second (periodically varying amplitude) is . (c) A siren emitting a sound of 1000 Hz moves away from you towards the face of a cliff at a speed of 10 m / s. Take the speed of sound in air as 330 m / s. You hear the direct sound of the siren and the sound reflected from the cliff. What is the beat frequency of these two sounds? Is it perceptible (it must be less than 20 Hz to be perceived)?

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