Problem 3

A small source emits a pure tone of frequency

f_s

in still air, where sound travels at speed

c

. It moves uniformly straight toward a stationary listener at speed

v_s<c

; neglect reflections and wind. (a) During one emission period, determine the spacing between successive wavefronts in front of the source. (b) Find the frequency

f_o

heard by the listener. (c) Find the limits of

f_o

v_s\to0

and

v_s\to c^-

, explaining why an approaching source raises the frequency.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{f_o=f_s\frac{c}{c-v_s};\quad f_o(0)=f_s,\quad f_o\to\infty\ (v_s\to c^-)}

problems.proof.analysis

Approach shortens the interval between crests reaching the listener, so heard frequency rises. The formula applies to a subsonic source in still air as assumed.

problems.proof.pitfall. In still air, wave speed at the listener is

c

, not

c+v_s

; source motion is already accounted for in the wavelength.