Question

Sound with a frequency 710 Hz from a distant source passes through a doorway 1.17 m wide in a sound-absorbing wall.

(a) Find the number of the diffraction minima at listening positions along a line parallel to the wall.


(b) Find the angular directions of the diffraction minima at listening positions along a line parallel to the wall. (Enter your answers from smallest to largest starting with the first answer blank. Enter NONE in any remaining answer blanks.)

Answer 1

Its given that,

frequency of sound wave = f = 710 Hz ;

width of doorway = w = 1.17 m

(a)In the given case, consider door way as the single slit diffraction.

We know that the consition for max. of interference is

d sin (theta) = k eq(1)

where d will be d = w/2 = 1.17/2 = 0.585

we know that = v/f where v= 343 m/s is speed of sound in air at 20 deg C and f is the freq.

= 343/710 = 0.483 meters

Putting the value of and d in eq (1) we get

(0.585) sin (theta) = k (0.483) < for theta = 90 deg we have

k = 0.585/0.483 = 1.211,

K has to be an integer so k =1. There will be two directions for maxima that is at k = 0 and k = 1. And there will be only a single direction for minima at k = 0

(can be proved if u put any other value u will get invalid results for the vaue of theta)

(b)Therefore the angular direction for minima is

Theta = sin -1 [ (k + 1/2) /d] = sin -1 [ (0+0.5) ( 0.483) / (0.585) ] = 24.38 degrees

Hence the angular direction of the minima is Theta = +/-24.38 deg.