A river flows with speed 5 m/s. A swimmer in the river carries a tuning fork of frequency 110 Hz, always keeping it above water. The swimmer moves with velocity 6 m/s relative to water in a direction perpendicular to the river flow. An observer with an identical tuning fork stands on the o

Question

A river flows with speed 5 m/s. A swimmer in the river carries a tuning fork of frequency 110 Hz, always keeping it above water. The swimmer moves with velocity 6 m/s relative to water in a direction perpendicular to the river flow. An observer with an identical tuning fork stands on the opposite bank. Find the beat frequency (in Hz) heard by the observer due to the sound emitted by the swimmer the moment the swimmer starts swimming. Speed of sound in air = 336 m/s.

Accepted Answer

2 Hz. At the moment the swimmer starts, he is directly across from the observer. His velocity relative to ground: 6 m/s perpendicular to bank (toward other bank) + 5 m/s along bank (river drift). The observer is directly across, so the line between them is perpendicular to the bank. The component of swimmer's velocity along this line = 6 m/s (toward observer). The river drift (5 m/s) is perpendicular to this line. Using Doppler: f_obs = f_source * v_sound / (v_sound - v_source_toward_observer) = 110 * 336 / (336 - 6) = 110 * 336/330 = 112 Hz. Beat frequency = 112 - 110 = 2 Hz.

Solution

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