Question

Two speakers spaced a distance 1.5 m apart emit coherent sound waves at a frequency of 680 Hz in all directions. The waves start out in phase with each other. A listener walks in a circle of radius greater than 1 m centered on the midpoint of the two speakers. At how many points does the listener observe destructive interference? The listener and the speakers are all in the same horizontal plane and the speed of sound is 340 m/s. Experiments like this must be done in a special room so that reflections are negligible.

Answer 1

Speed of the sound wave v = 340 m/s

Frequency of the sound wave f = 680 Hz

Distance between the speakers d = 1.5 m

 

The wavelength of the sound wave is

λ = v/f

   = (340 m/s) / 680 Hz

   = 0.5 m

 

For destructive interference, the path difference between the two sound waves is odd multiple of half wavelength. Path difference that causes destructive interference is

 

⇒ nλ/2  where n = 1, 3, 5, …..

⇒ 0.25m, 0.75m, 1.25m,

 

The number of occurrence of destructive interference in one quadrant of the circle is 3. The listener when moves in one complete circle, the number of occurrence of destructive interference are 4(quadrants in a circle) × 3 = 12. 

 

At 12 points the listener finds the destructive interference in a circle.

At 12 points the listener finds the destructive interference in a circle.