Question

Two sound speakers are driven in phase by an audio amplifier at frequency $600Hz$. The speed of sound is $340m/s$. The speakers are on the $y$-axis, one at $y=+1.0m$ and the other at $y=-1.0m$. A listener begins at $y=0$ and walks along a line parallel to the $y$-axis at a very large distance $x$ away. At ${\frac{\theta }{2}}^o$ angle she will first hear a maximum (after $0^{\circ }$) sound intensity. Find $\theta$ :

Answer 1

Path difference between sound from the two sources at a point of the line along which the listener travels =$\frac{yd}{x}$(where y is the y-coordinate of the point, d is the separation between the two sources).

Hence to obtain a maxima, $\frac{yd}{x}=\lambda =\frac{c}{f}$

$⟹\tan \theta =\frac{y}{x}=\frac{c}{fd}$

$\tan \theta =\frac{340}{600\times 2}$

Using approximation, $\tan \theta \approx \theta$

$⟹\theta \approx {16.5}^{\circ }$