Question

A source of the sound of frequency \[256\,Hz\] is moving towards a wall with a velocity of \[5m/s\]. The number of beats per second heard by a stationary observer between the source and the wall is (Velocity of sound \[ = 330\,m/s\]).
A. 8
B. 3
C. 5
D. 2

Answer 1

Hint: In physics, the term frequency refers to the number of waves that pass a fixed point in unit time. It also describes the number of cycles or climates experienced during one unit of time by a body in periodic motion.

Formula used:
The expression of number of beats is given as,
\[\left( {\dfrac{v}{{v - {v_s}}}} \right)n - \left( {\dfrac{v}{{v + {v_s}}}} \right)n\]
Where v is the velocity of the observer, $v_s$ is the velocity of sound, and n is the source frequency.

Complete step by step solution:
We are given that
Sound frequency \[ = 256\,Hz\]
Velocity \[ = 5m/s\]
Velocity of sound \[ = 330\,m/s\]
In this question, the observer will hear two sounds, one directly from the source and the other from the reflected image of the sound.

Now, if the sound reaches the observer after being reflected from a stationary surface and the medium is also stationary, the image of the source in the reflecting surface will become the source of the reflected sound then,
The number of beats \[ = \left( {\dfrac{v}{{v - {v_s}}}} \right)n - \left( {\dfrac{v}{{v + {v_s}}}} \right)n \\ \]
Now substitute all the given values in the formula, we get
The number of beats \[ = \dfrac{{2nv{v_s}}}{{{v^2} - v_s^2}} \\ \]
The number of beats \[ = \dfrac{{2 \times 256 \times 330 \times 5}}{{335 \times 325}} \\ \]
The number of beats \[ = \dfrac{{844,800}}{{108,875}} \\ \]
The number of beats \[ = 7.75\]
Therefore, the number of beats per second heard by a stationary observer between the source and the wall is \[8\] approximate.

Hence, option A is correct.

Note: The formulas differ for each case, that is when the observer moves away from the source, the velocity of the observer should be subtracted; when the observer moves towards the source, the velocity of the observer should be added. The units of the parameters must be considered.