Question

Two speakers, A and B, are at the same point on an x-axis and each emits sound with a wavelength of 0.25 m. Speaker B's phase constant is 260 degrees larger than speaker A's phase constant and each produces an amplitude of 10 Pa. What amplitude occurs along the x-axis in front of these speakers (Pa)? b. What is the minimum distance you can move speaker A to achieve constructive interference along the x-axis? Give a positive answer regardless of direction.

Answer 1

given two speakers A dn B are on the same point on the x axis
phase constant of A = 0
phase constant pf B = 260 deg
hence

wave emitted by speaker A can be written as
ya = A*sin(kx - wt)
for speaker B
yb = A*sin(kx - wt + 260 deg)
A = 10 Pa

hence
from super imposition
ya + yb = 10*sin(kx - wt) + 10*sin(kx - wt + 260 deg) = 10[sin(kx - wt) + sin(kx - wt)*cos(260) + cos(kx - wt)*sin(260)]
ya + yb = 10[sin(kx - wt)*0.8263518 - 0.9848*cos(kx - wt)]
consider
[sin(kx - wt)*0.8263518 - 0.9848*cos(kx - wt)] = A[sin(kx - wt)*cos(phi) - cos(kx - wt)sin(phi)]
comparing
Acos(phi) = 0.8263518
Asin(phi) = 0.9848
A = 1.285574048
hence
resultant amplitude is A*10 = 12.85574048 Pa

b. for constructive interference, phase difference has to be 0 deg or 180 deg or 360 deg
hence smallest phase shift would be 260 - 180 = 80 deg
   now, phase shift of 180 deg means lambda/2 distance
   80 dg = lambda*80/360 = 0.25*80/360 = 0.055555555 m
   hence speaker A should be moved by 5.5 cm to get constructive interference on x axis