Question

A musician uses an open flute of length $50{\text{ cm}}$ to produce second harmonic sound waves. A person runs towards the musician from another end of a hall at a speed of $10{\text{ }}km/hr$ . If the wave speed is $330{\text{ }}m/s$ , the frequency heard by the running person shall be close to:
A. 753
B. 500
C. 333
D. 666

Answer 1

Hint:-Frequency of the sound produced by flute is given by $n\dfrac{v}{{2l}}$ where $n$ is the harmonic number, $v$ is the speed of the wave and $l$ is the length of the flute.
Use the Doppler effect to find the apparent change in the frequency of sound due to the relative motion between the source of the sound (musician) and the observer (the person).

Complete step-by-step solution:-
Let us first know about harmonic
A harmonic number is basically an increasing series of sound components that can be heard above the audible fundamental frequency.
Frequency of the sound produced by flute is given by $n\dfrac{v}{{2l}}$ where $n$ is the harmonic number, $v$ is the speed of the wave and $l$ is the length of the flute.
So, as given in the question that the flute produces second harmonic sound waves, so $n = 2$
Also, the wave speed $v = 330m/s$ and length of the flute $l = 50cm = 0.5m$
Therefore, Frequency of the sound produced by flute
$f = \dfrac{{2 \times 330}}{{2 \times 0.5}} = 660Hz$
As given in the question velocity of observer, ${v_o} = 10km/hr = \dfrac{{25}}{9}m/s$
Now, we use the Doppler effect to find the apparent change in the frequency of sound heard by the observer due to the relative motion between the source (musician) and the observer (the person).
As the person runs towards the musician, so the frequency detected by observer, $f' = \left[ {\dfrac{{v + {v_o}}}{v}} \right]$
On substituting the values we have
$f' = \left[ {\dfrac{{\dfrac{{25}}{9} + 330}}{{330}}} \right]660 = 335.56 \times 2 \approx 666Hz$
Hence, option D is correct.

Note:- As we increase the frequency harmonics that can be heard above the fundamental frequency will cover up the harmonic spectrum of the sound. Harmonics can be difficult to differentiate among themselves as single components, nevertheless, they are there.
The Doppler effect is used to measure the velocity of detected objects in some types of radar. A beam of sound is fired from the radar at a moving target like a car or plane. For instance, a police can use a radar to detect speed of a motorist as it approaches or recedes from the radar source.