Chapter 24, Problem 67GP

To determine

The angle at which the whistle is not possible to hear.

Answer to Problem 67GP

  $31°$

Explanation of Solution

Given data:

Distance, $\text{d}=0.88\text{m}$

Frequency, $\text{f}=750\text{Hz}$

The speed of the sound, $v=343\text{m/s}$

Formula used:

  $d\sin \theta =m\lambda$ and $\lambda =\frac{v}{\text{f}}$

Where,

  $d$ is the distance.

  $\lambda$ is wavelength.

  $\text{f}$ is the frequency.

  $v$ is the speed.

Calculation:

The wavelength of the sound: $\lambda =\frac{v}{\text{f}}$

  $\lambda =\frac{v}{\text{f}}=\frac{343\text{m/s}}{750\text{Hz}}=0.457\text{m}$

The angles of minima: $d\sin \theta =m\lambda ,m=1,2,3,\mathrm{...},$

  $\left(0.88\right)\sin {\theta }_1=\left(1\right)\left(0.457\text{m}\right)$

  $\therefore {\theta }_1=31°$

Similarly put $\text{m}=\text{2,}$

  $\therefore {\theta }_2=1.04°$

Hence, there is no ${\theta }_2$ .

Conclusion: The whistle would not be heard at angles of $31°$ on both sides of the normal.