Question

One speaker A is located in origin and another speaker B is located at point (3.0),
the units in the xy plane measure in meters and the speakers are the same. At a point P = (1,4) is one
microphone placed. The speakers are connected to a tone generator that generates sinus tones
(flat waves) so that speaker B lies ?/4 after A in phase and the sound speed is 350m/s.

a) Draw time graphs for both speakers' oscillations when the frequency is 500 Hz.
b) Determine the three lowest frequencies that give complete constructive interference in the point
c) Determine the three lowest frequencies that give complete destructive interference in point P.

Answer 1

Part-(a)

Frequency 500 Hz, Time period T = 1/500 = 2 ms

Phase of wave produced by Speaker B lagging behind pahse of wave produced by speaker A.

Phase difference between these waves is /4 .

------------------------------------------------------

Part-(b)

For destructive interference , phase difference of waves from speaker-A and speaker-B when they reach mike should be (2n+1) , where n = 0, 1, 2, 3 etc.

Since speaker-B is laggibg behind in phase by (/4 ) , the wave from speaker-B when it reaches mike shoud satisfy the following condition for destructive interference

-(/4 ) + = (2n+1) , where n = 0,1,2,3 .. etc

where is phase difference due to path difference between the waves when they reaching mike from their respective starting position.

Hence = [ 2n + (5/4)]

Equivalent path difference [ 2n + (5/4) ] ( / 2 ) = [ n + (5/8) ]

where is wavelength of sound wave

From figure we see the path difference for the waves starting from speaker-A and speaker-B when they reach mike position is m = 0.35 m

Hence condition for destructive interference is given by

[ n + (5/8) ] = 0.35

when n = 0,   = 0.35 / 0.625 = 0.56 m , frequency = velocity / wavelength = 350/0.56 = 625 Hz

when n = 1,   = 0.35 / 1.625 = 0.215 m , frequency = 350/0.215 = 1628 Hz

when n = 2,   = 0.35 /2.625 = 0.133 m , frequency = 350/0.133 = 2632 Hz

-----------------------------------------------------

Part-(c)

For constructive interference , phase difference between two waves should be 2n , where n = 0, 1, 2.. etc

Since speaker-B is laggibg behind in phase by (/4 ) , the wave from speaker-B when it reaches mike shoud satisfy the following condition for constructive interference

-(/4 ) + = 2n , where n = 0,1,2,3 .. etc

Hence = 2n + (/4 )

Equivalent path difference [ 2n ] ( / 2 ) = [ n + (1/8)]

Hence condition for constructive interference is given by

[ n + (1/8) ] = 0.35

when n = 0,   = 0.35 / 0.125 = 2.8 m , frequency = 350/2.8 = 125 Hz

when n = 1,   = 0.35 / 1.125 = 0.311 m , frequency = 350/0.311 = 1125 Hz

when n = 2,   = 0.35 /2.125 = 0.165 m , frequency = 350/0.165 = 2121 Hz