A siren emits sound of frequency while receding at constant speed from a stationary…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Variational and asymptotic methods

Mathematical Methods for Physics, Ch Series, limits and controlled approximation

Problem

A siren emits sound of frequency

f_0 = 750\,\mathrm{Hz}

while receding at constant speed

v

from a stationary listener, in air where the speed of sound is

c_{\mathrm{s}} = 343\,\mathrm{m\,s^{-1}}

. The classical Doppler formula for a receding source gives

f_{\mathrm{obs}} = f_0\left(1 + \frac{v}{c_{\mathrm{s}}}\right)^{-1}.

((a)) Expand

f_{\mathrm{obs}}

in powers of

v/c_{\mathrm{s}}

up to and including the term of order

(v/c_{\mathrm{s}})^3

, writing the discarded term in

O

notation. ((b)) For

v = 30.0\,\mathrm{m\,s^{-1}}

, evaluate the fractional frequency shift

(f_0 - f_{\mathrm{obs}})/f_0

and the observed frequency

f_{\mathrm{obs}}

, both to three significant figures.

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