Question

Two identical loudspeakers are driven in phase by a common oscillator at

800 Hz

and face each other at a distance of

1.25 m .

Locate the points along the line joining the two speakers where relative minima of sound pressure amplitude would be expected. (Use

v = 343 m / s .

)

Accepted Answer

Let two loudspeakers $(1)$ and $(2)$ be separated by a distance $1.25 m,$ and observer moves between the two speakers along the line joining two as shown in figure

Two identical waves moving in opposite directions constitute a standing wave. We must find the nodes.
The wavelength is $λ = \frac{v}{f} = \frac{343 m / s}{800 Hz} = 0.429 m$

The two waves moving in opposite directions along the line between the two speakers will add to produce a standing wave with this distance between nodes.

Distance, node to node $= λ / 2 = 0.214 m$

Because the speakers vibrate in phase, air compressions from each will simultaneously reach the point halfway between the speakers, to produce anti-node of pressure here. A node of pressure will be located at this distance on either side of the midpoint.

Distance, node to anti-node $= λ / 4 = 0.107 m$

Therefore, nodes of sound pressure will appear at these distances from either speaker:
$[1 / 2 (1.25 m) + 0.107 m = 0.732 m$ and $1 / 2 (1.25 m) - 0.107 m = 0.518 m]$

The standing wave contains a chain of equally spaced nodes at distances from either of the speakers

$0.732 m + 0.214 m = 0.946 m$
$0.947 m + 0.214 m = 1.16 m$

Also at $0.518 m - 0.214 m = 0.303 m$

$0.303 m - 0.214 m = 0.089 m$

The standing wave exists only along the line segment between the speakers. No nodes or anti-nodes appear at distances greater than $1.25 m$ or less than $0,$ because waves add to give standing wave only if they are travelling in opposite directions and not in the same direction. Thus, the distances from either of the speakers to the nodes of pressure between the speakers are $0.089 m, 0.303 m, 0.518 m, 0.732 m, 0.946 m$ and $1.16 m .$

B M Sharma Solutions for Chapter: Superposition and Standing Waves, Exercise 1: ILLUSTRATIONS

B M Sharma Physics Solutions for Exercise - B M Sharma Solutions for Chapter: Superposition and Standing Waves, Exercise 1: ILLUSTRATIONS

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