Question

You are standing 2.50m directly in front of one of the two loudspeakers shown in the figure. They are 3.00m apart and both are playing a 686Hz tone in phase.

As you begin to walk directly away from the speaker, at what distances from the speaker do you hear a minimum sound intensity? The room temperature is 20?C.

Enter your answers numerically in increasing order separated by commas.

Answer 1

Destructive interference occurs when the difference in distance from the in phase speakers is

d = ?/2*(2n -1) where n = 1,2,3...

If the room temp is 20oC then v = 343m/s so ? = 343/686 = 0.500m

So the difference in distances d for minimum sound

will be 0.500/2*(2n -1) = 0.250, 0.750, 1.25, 1.75, 2.25, 2.75, 3.25, 3.75, 4.25,4.75....

So we must find the distance
such that the hypotenuse of the triangle minus the side = 0.250, 0.750, 1.25, 1.75, 2.25, 2.75, ,....

so sqrt(3^2 + (2.5+x)^2) - (2.5+x) = 0.250, 0.750, 1.25, 1.75, 2.25, 2.75, .....

Therefore 9 + 6.25 + 5x + x^2 = (0.25 + 2.5 + x)^2 = (2.75 + x)^2 = 7.5625 + 5.5x + x^2

So 0.5x = 15.25 - 7.5625 = 7.6875

So x1 = 7.6875/0.5 = 15.375m

repeating for d = 0.750

Therefore 9 + 6.25 + 5x + x^2 = (0.75 + 2.5 + x)^2 = (3.25 + x)^2 = 10.5625 + 6.5x + x^2

So 1.5x = 15.25 - 10.5625 = 4.6875

So x2 = 4.6875/1.5 = 3.125m

repeating for d = 1.250

Therefore 9 + 6.25 + 5x + x^2 = (1.25 + 2.5 + x)^2 = (3.75 + x)^2 = 14.0625 + 7.5x + x^2

So 2.5x = 15.25 - 14.0625 = 1.1875

So x2 = 1.1875/2.5 = 0.475m

repeating for d = 1.750

Therefore 9 + 6.25 + 5x + x^2 = (1.75 + 2.5 + x)^2 = (4.25 + x)^2 = 18.0625 + 9.5x + x^2

So 1.5x = 15.25 - 18.0625 = -2.8125 This is not feasible as you would be closer to the speaker than you started

So x = 0.475, 3.125, 15.375

and the distances from the speaker is 2.5 +x = 2.975, 5.625, 17.875