Answer
(a) $257Hz; 255Hz$
(b) $2Hz$
(c) $0.670m$
(d) $2.0Hz$
Work Step by Step
(a) When the observer is moving towards the speaker, the frequency is given as
$f_1=(1+\frac{u}{v})f$
We plug in the known values to obtain:
$f_1=(1+\frac{1.35m/s}{342m/s})(256Hz)$
$\implies f_1=257Hz$
If the observer is moving away from the sound speaker, then the frequency can be determined as
$f_2=(1-\frac{u}{v})f$
We plug in the known values to obtain:
$f_2=(1-\frac{1.35m/s}{343m/s})(256Hz)$
$f_2=255hz$
(b) The beat frequency heard by the observer can be determine as
$f^{\prime}=f_1-f_2$
We plug in the known values to obtain:
$f^{\prime}=257Hz-255HZ$
$\implies f^{\prime}=2Hz$
(c) We know that
$d=\frac{1}{2}(\frac{v}{f})$
$\implies d=\frac{1}{2}(\frac{343m/s}{256Hz})$
$d=0.670m$
(d) We know that
$t=\frac{d}{u}$
$t=\frac{0.670m}{1.35 m/s}$
$t=0.5s$
and $\Delta f=\frac{1}{t}$
$\Delta f=\frac{1}{0.5s}=2.0Hz$
Thus, the observer hears the maximum loudness from the speaker 2 times per second and this is equal to the beat frequency.
