Question

Two small identical speakers are connected to the same sinusoidal source, as shown in figure 13. At a distance 40.0 m in front of one speaker, the sound intensity is minimum at two consecutive frequencies 4500 Hz and 7500 Hz. What is the distance X between the speakers?

Press F11 to exit full screen Speaker Speaker 40.0 m Figure 13

Answer 1

In triangle ABC , using Pythagorean theorem

AB = sqrt(AC² + BC²)

AB = sqrt(X² + 40² )

AB = sqrt(X² + 1600 )

path difference is given as

$\Delta$x = AB - BC

$\Delta$x = sqrt(X² + 1600 ) - 40

v = speed of sound = 343 m/s

f= frequency

wavelength is given as

$\lambda$ = v/f

we know that for destructive interference

path difference = (2 n - 1) $\lambda$/2

hence

path difference = (0.5) (2 n - 1) v/f

for f = 4500

path difference = (0.5) (2 n - 1) v/4500                                              eq-1

for f = 7500

path difference = (0.5) (2 (n + 1) - 1) v/7500                                       eq-2

using eq-1 and eq-2

(0.5) (2 n - 1) v/4500 = (0.5) (2 (n + 1) - 1) v/7500

n = 2

using eq-1

path difference = (0.5) (2 n - 1) v/4500     

sqrt(X² + 1600 ) - 40 = (0.5) (2 n - 1) v/4500     

sqrt(X² + 1600 ) - 40 = (0.5) (2 (2) - 1) (343)/4500   

X = 3 m