Set 56 Problem number 19

Problem

Assume that sound travels at 340 m/s.

A sound source is initially 40 meters from a detector, moving toward the detector at 14 m/s.  The source emits a sound at a constant frequency of 330 Hz.  

• How long is it between the instant the microphone receives the first pulse and the last?
• How many pulses are emitted as the source approaches the detector?
• What sound frequency is therefore received by the detector?

Solution

The source requires 40 meters /( 14 m/s) = 2.857 seconds to reach the detector; sound requires 40 m / (340 m/s) = .1176 seconds to reach the detector. 

The first pulse emitted at the 40 meter distance is not detected until .1176 seconds later, while the last pulse emitted is at the detector and is detected immediately.  Thus all the pulses are detected in .1176 seconds less than the time required to emit them.  Since the pulses are emitted in the 2.857 seconds required for the source to reach the detector, the time required for detection is

• time required for detector to receive pulses = 2.857 sec - .1176 sec = 2.739 sec.

During the 2.857 seconds the number of pulses emitted is 2.857 sec * 330 cycles/sec = 942.8 pulses.  Thus 942.8 pulses are received in 2.739 seconds, and the observed frequency is

• observed frequency = 942.8 pulses / 2.739 seconds = 344.2 Hz.

Generalized Solution

If a source moving toward an observer at velocity vSource emits sound at frequency f, and if sound travels at velocity vSound, then in any time interval `dt the source will move distance

• distance moved by source = vSource * `dt

and will emit f * `dt pulses.

The first pulse will reach the position at which the last pulse is emitted in the time required for sound to travel the distance vSource * `dt moved by the source.  The first pulse will therefore reach the position at which the last pulse is received after a delay of

• delay in receiving first pulse:  `dtPulse = vSource * `dt / vSound.

Thus the f * `dt pulses will be received in time interval `dt - `dtPulse, and the observed frequency must be

• observed frequency:  f ' = # pulses received / time required for reception = f `dt / (`dt - `dtPulse)

= f `dt / (`dt - vSource * `dt / vSound) = f / (1 - vSource/vSound).