Learning Goal:

To practice Problem$-$Solving Strategy $16\mathrm{.}2$ Doppler effect.

The sound source of a ship$$s sonar system operates at a frequency of $22\mathrm{.}0$ kHzkHz $.$ The speed of sound in water $($assumed to be at a uniform $20$C$$C$)$ is $1482$ m$/$sm$/$s $.$ What is the difference in frequency between the directly radiated waves and the waves reflected from a whale traveling straight toward the ship at $4\mathrm{.}95$ m$/$sm$/$s $?$ Assume that the ship is at rest in the water.Problem$-$Solving Strategy $16\mathrm{.}2$: Doppler effect

IDENTIFY the relevant concepts:

The Doppler effect is relevant whenever the source of waves, the wave detector $($listener$),$ or both are in motion.

SET UP the problem using the following steps:

Establish a coordinate system. Define the positive direction to be the direction from listener to source, and make sure that you know the signs of all the relevant velocities. Also, the velocities must all be measured relative to the medium in which the sound is traveling.

Use consistent notation to identify the various quantities: subscripts $"$S$"$ for source and $"$L$"$ for listener.

Determine which unknown quantities are your target variables.

EXECUTE the solution as follows:

Relate the frequencies at the source and the listener, the sound speed, and the velocities of the source and the listener using

fL$=$v$+$vLv$+$vSfSfL$=$v$+$vLv$+$vSfS

If the source is moving, you can find the wavelength measured by the listener using

in$$front$=$v$$vSfSin$$front$=$v$$vSfS or behind$=$v$+$vSfSbehind$=$v$+$vSfS

When a wave is reflected from a surface, either stationary or moving, the analysis can be carried out in two steps. In the first, the surface plays the role of listener; the frequency with which the wave crests arrive at the surface is fLfL$.$ Then think of the surface as a new source, emitting waves with this same frequency fLfL$.$ Finally, determine what frequency is heard by a listener detecting this new wave.

EVALUATE your answer:

Ask whether your final result makes sense. If the source and listener are moving toward each other, fL$>$fSfL$>$fS; if they are moving apart, fL