Question

A man marries a great Wagnerian soprano but alas, he discovers that he cannot stand Wagnerian opera. To save his eardrum, the unhappy man decides to silence his lark-like wife for good. His plan is to tie her to the front of the car and send the car and the soprano speeding towards a brick wall. The soprano, however, is shrewd as she studied physics in her student days at the music conservatory. She realized that the brick wall has a resonant frequency of 600 Hz. That is, the wall will fall down if a continuous sound wave of 600 Hz hits it and saves her to sing again. The car is heading towards the wall at a velocity of $30{\text{m}}{{\text{s}}^{ - 1}}$. Find the frequency at which the soprano must sing so that the wall crumbles. (Given $v = 330{\text{m}}{{\text{s}}^{ - 1}}$ ).
A) 545.45 Hz
B) 550.2 Hz
C) 560 Hz
D) 565.3 Hz

Answer 1

Hint:Here, the soprano is the source of the sound wave and the brick wall is the observer. The soprano is moving towards the brick wall. This situation invokes the concept of Doppler effect which deals with change in frequency due to the motion of the source, observer or both.

Formula used:
The frequency of a moving source of sound is given by, ${f_s} = {f_o}\left( {\dfrac{{v + {v_s}}}{v}} \right)$ where ${f_o}$ is the frequency heard by the observer if the source was stationary, $v$ is the velocity of the sound wave and ${v_s}$ is the velocity of the source.

Complete step by step answer.
Step 1: List the parameters known from the question.
A soprano is tied to a car so that she crashes into a brick wall. The soprano sings at some frequency to match the resonant frequency of the brick wall so that the wall falls down.
In this scenario the soprano is the source moving towards the stationary observer being, the brick ball. Doppler effect refers to the change in frequency due to the motion of the source, the observer or both.
The velocity of the car is the velocity of the soprano and is given to be ${v_s} = 30{\text{m}}{{\text{s}}^{ - 1}}$.
The resonant frequency of the wall is given to be ${f_w} = 600{\text{Hz}}$ .
The speed of the sound wave is given to be $v = 330{\text{m}}{{\text{s}}^{ - 1}}$ .
Step 2: Express the relation of the frequency of the sound wave produced by the soprano and make appropriate substitutions.
The frequency of the sound wave produced by the soprano is given by, ${f_s} = {f_w}\left( {\dfrac{{v + {v_s}}}{v}} \right)$ -----(1).
where ${f_w}$ is the resonant frequency of the wall, $v$ is the velocity of the sound wave and ${v_s}$ is the velocity of the soprano.
Substituting values for ${v_s} = - 30{\text{m}}{{\text{s}}^{ - 1}}$, ${f_w} = 600{\text{Hz}}$ and $v = 330{\text{m}}{{\text{s}}^{ - 1}}$ in equation (1) we get, ${f_s} = 600 \times \left( {\dfrac{{330 - 30}}{{330}}} \right) = 545.45{\text{Hz}}$
Thus the frequency at which the soprano must sing is ${f_s} = 545.45{\text{Hz}}$ .

Hence the correct option is A.

Note: The source being, the soprano is moving towards her observer. So the velocity of the soprano is taken to be negative i.e. ${v_s} = - 30{\text{m}}{{\text{s}}^{ - 1}}$ while substituting in equation (1). The resonant frequency of the brick wall refers to the frequency of the sound heard by the wall if the soprano was not moving. If the soprano was not moving, she would have to sing at the resonant frequency of 600 Hz to knock down the brick wall. The motion of the soprano led her to change her frequency of singing.