Question

A car has two horns having a difference in frequency of \[180Hz\] . The car is approaching a stationary observer with a speed of \[60m{{s}^{-1}}\] . Calculate the difference in frequencies of the notes as heard by the observer. If the velocity of sound is \[330m{{s}^{-1}}\] .

Answer 1

Hint: The Doppler Effect (or the Doppler shift) is the change in frequency of a wave concerning an observer who is moving relative to the wave source. A common example of Doppler shift is the change of pitch/frequency heard when a vehicle sounding a horn approaches and moves away from an observer. Compared to the emitted frequency, the received frequency is higher during the approach, identical at the instant of passing by and lower during the recession.

Formula Used:
\[\Delta f'=\Delta f\times \left( \dfrac{v-{{v}_{O}}}{v-{{v}_{S}}} \right)\]

Complete step by step answer:
From the formula for the apparent frequency heard by the observer in case of Doppler’s Effect (that is when there is relative motion between the source and the observer), we have
\[\Delta f'=\Delta f\times \left( \dfrac{v-{{v}_{O}}}{v-{{v}_{S}}} \right)\] where \[\Delta f\] is the difference in frequencies produced by the source, is the velocity of sound, \[v-{{v}_{O}}\] is the relative velocity of the sound with respect to the observer and \[v-{{v}_{S}}\] is the relative velocity of the sound with respect to the source.
We know that source produces sound frequencies of difference \[180Hz\] , that is, \[\Delta f=180Hz\]
Since the observer is at rest, the relative velocity of the sound with respect to the observer \[(v-{{v}_{O}})=330m{{s}^{-1}}\]
Now, the relative velocity of sound with respect to the source \[(v-{{v}_{S}})=(330-60)m{{s}^{-1}}=270m{{s}^{-1}}\]
Substituting the values in our formula, we get, the difference in frequencies heard by the observer \[\Delta f'=180\times \left( \dfrac{330}{270} \right)Hz=220Hz\]
Hence the observer will hear sound frequencies having a difference of \[220Hz\] .

Note: Most of us only know that Doppler’s Effect in sound is used to find the frequency of the sound heard by the observer when there is relative motion between the source and the observer. But the amazing thing is that the formula for Doppler Effect can also be applied to find the difference in the apparent frequencies heard by the observer when there is a difference in the apparent frequency of the source, as we did in the above question.