Question

A car moves with a speed of 54 km h⁻¹ towards a cliff. The horn of the car emits sound of frequency 400 Hz at a speed of 335 m s⁻¹. (a) Find the wavelength of the sound emitted by the horn in front of the car. (b) Find the wavelength of the wave reflected from the cliff. (c) What frequency does a person sitting in the car hear for the reflected sound wave? (d) How many beats does he hear in 10 seconds between the sound coming directly from the horn and that coming after the reflection?

Answer 1

Given:
Velocity of car \[v_{car}\] = 54 kmh⁻¹ = \[54 \times \frac{5}{18} = 15   {\text { ms }}^{- 1}\]

Frequency of the car f = 400 Hz
Velocity of sound in air \[v_{air}\]= 335 ms⁻¹
Wavelength in front of the car \[\lambda\]=?
(a) Net velocity in front of the car \[v\] =\[v_{car}  -  v_{air}\]= 335\[-\]15 = 320 m/s

\[\text { As }  v = f\lambda, \] 

\[ \therefore   \lambda = \frac{v}{f}\]

\[ \Rightarrow \lambda = \frac{320}{400} = 80  \text { cm }\]

(b) The frequency \[\left( f_1 \right)\] heard near the cliff is given by :

\[f_1  = \frac{v_{air}}{v_{air} + v_{car}} \times  f_0 \] 

\[ \Rightarrow  f_1  = \frac{335}{335 + 5} \times 400\] 

\[ \Rightarrow  f_1  = \frac{335 \times 400}{320} \text{ Hz }\] 

\[ \Rightarrow  f_1  = 418 . 75  \text { Hz }\]

As we know,

\[v = f\lambda\] 

\[\text { Wavelength  reflected  from  the  cliff  is }  \] \[\lambda = \frac{v}{f_1} = \frac{335}{418 . 75} = 80  \text { cm }\]

(c) Here,

\[v_0\]= 15 `\text { ms}^\(- 1)`

Frequency of the reflected sound wave \[\left( f_2 \right)\]heard by the person sitting in the car :

\[f_2  = \frac{v + v_0}{v} \times  f_1 \] 

\[ \Rightarrow  f_2  = \frac{335 + 15}{335} \times \frac{335}{320} \times 400\] 

\[ \Rightarrow  f_2  = 437 \text{ Hz }\]

(d) He will not hear any beat in 10 seconds because the difference of frequencies is greater than 10 (persistence of sound for the human ear is 1/10 of a second).