Question

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

c) Determine the three lowest frequencies that give complete destructive interference in point P.

Answer 1

given, speaker A is at origin, B is at (3,0)
everything is in xy plane, measures in meters
Given, speakers are the same
consider point P(1,4)

a. the time graphs are given for f = 500 Hz

b.
speed of sound, v = 350 m/s
B is phase pi/4 after A
now phase of 2*pi corresponds to wavelenth lambda
hence pi/4 corresponds to lambda/8
so speaker A leads speaker B wavelength by lamdba/8 [where lamdba is wavelength of both the speakers as bot are same speakers at same frequency]
so, for constructive interference,
path difference should be 0 at point P
now actual difference in PA and PB
PA - PB = sqrt(1 + 16) - sqrt((1 - 3)^2 + (4 - 0)^2) = sqrt(17) - sqrt(20)
PA - PB = -0.34903032938 m

so for constructive interference
0.34903032938 = lambda/8 + n*lambda
now, lambda*f = v
hence ,
f = v[1 + 8n] /0.34903032938*8

for three lowest frequencies
n = 0, 1, 2
f = 350[1 + 8n] /0.34903032938*8 = 125.34727(1 + 8n)
f0 = 125.34727 Hz
f1 = 1128.1254 Hz
f2 = 2130.9036 Hz

c.for destructive interfernece
0.34903032938 = lambda/8 + n*lambda/2 [ n = 1, 3, 5, n != 0 as at n = 1, path difference shall become 0 which is true for constructive interference and not destructive interference]
f = 350(1 + 4*n)/0.34903032938*8
f = 125.34727305( 1 + 4n)
three lowest frequencies for destructive interference at point P are
f1 = 626.736365 Hz
f2 = 1629.514549 Hz
f3 = 2632.29273407 Hz