Chapter 37, Problem 11P

To determine

The angles measured from the perpendicular bisector of the line joining the speakers where the distant observer would hear maximum and minimum intensity.

Answer to Problem 11P

The angles measured from the perpendicular bisector of the line joining the speakers where the distant observer would hear maximum and minimum intensity are $\underset{\_}{0°,29.1°\text{and}\text{76}\text{.3}°}$ and $\underset{\_}{14.1°\text{and}\text{46}\text{.8}°}$ respectively.

Explanation of Solution

Write the expression to find the wavelength.

    $\lambda =\frac{v}{f}$                                                                                                                         (I)

Here, $\lambda$ is the wavelength, $v$ is the velocity if the sound and $f$ is the frequency.

Write the expression for location of maxima.

    $d\sin \theta =m\lambda$                                                                                                             (II)

Here, $d$ is the distance between the speakers, $\theta$ is the angle and $m$ is the order.

Write the expression for location of minima.

    $d\sin \theta =\left(m+\frac{1}{2}\right)\lambda$                                                                                                (III)

Conclusion:

Substitute $340\text{m}/\text{s}$ for $v$ and $2000\text{Hz}$ for $f$ in equation (I).

    $\begin{array}{c}\lambda =\frac{340\text{m}/\text{s}}{2000\text{Hz}}\\ =0.170\text{m}\end{array}$

Write the expression to find the angle of maxima from equation (II).

    $\theta ={\sin }^{-1}\left(\frac{m\lambda }{d}\right)$                                                                                                      (IV)

Write the expression to find the angle of minima from equation (III).

    $\theta ={\sin }^{-1}\left(\left(m+\frac{1}{2}\right)\frac{\lambda }{d}\right)$                                                                                             (V)

Substitute $0$ for $m$, $0.170\text{m}$ for $\lambda$ and $0.350\text{m}$ for $d$ in equation (IV).

    $\begin{array}{c}\theta ={\sin }^{-1}\left(\frac{\left(0\right)\left(0.170\text{m}\right)}{0.350\text{m}}\right)\\ =0°\end{array}$

Substitute $1$ for $m$, $0.170\text{m}$ for $\lambda$ and $0.350\text{m}$ for $d$ in equation (IV).

    $\begin{array}{c}\theta ={\sin }^{-1}\left(\frac{\left(1\right)\left(0.170\text{m}\right)}{0.350\text{m}}\right)\\ =29.1°\end{array}$

Substitute $2$ for $m$, $0.170\text{m}$ for $\lambda$ and $0.350\text{m}$ for $d$ in equation (IV).

    $\begin{array}{c}\theta ={\sin }^{-1}\left(\frac{\left(2\right)\left(0.170\text{m}\right)}{0.350\text{m}}\right)\\ =76.3°\end{array}$

Higher orders have no solution.

Substitute $0$ for $m$, $0.170\text{m}$ for $\lambda$ and $0.350\text{m}$ for $d$ in equation (V).

    $\begin{array}{c}\theta ={\sin }^{-1}\left(\frac{1}{2}\frac{\left(0.170\text{m}\right)}{0.350\text{m}}\right)\\ =14.1°\end{array}$

Substitute $1$ for $m$, $0.170\text{m}$ for $\lambda$ and $0.350\text{m}$ for $d$ in equation (V).

    $\begin{array}{c}\theta ={\sin }^{-1}\left(\frac{3}{2}\frac{\left(0.170\text{m}\right)}{0.350\text{m}}\right)\\ =46.8°\end{array}$

Higher orders have no solution.

Therefore, the angles measured from the perpendicular bisector of the line joining the speakers where the distant observer would hear maximum and minimum intensity are $\underset{\_}{0°,29.1°\text{and}\text{76}\text{.3}°}$ and $\underset{\_}{14.1°\text{and}\text{46}\text{.8}°}$ respectively.