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

v_s\to0:\ f_o\to f_s;\qquad v_s\to c^-:\ f_o\to\infty

problems.proof.analysis

A stationary source gives wavelength $c/f_s$ and no Doppler shift. As source speed approaches sound speed, the classical model compresses the forward spacing toward zero.