Two sources of sound located at A and B. A and B are equidistant (at equal dista

Question

Two sources of sound located at A and B. A and B are equidistant (at equal distance) from the center (o) of the system. A is located 1.5 m to the left of O and B is locate 1.5 m to the right of o. O is the center of a circle with radius R=10 m. There are three points located on this circle P2 at 0 degrees, P1 at 90 degrees, P3 at 180 degrees where the sound is measured.

Two waves are emitted from a (s1) and from b (s2) with the same amplitude (sm=.75m), wavelength = 2 m and frequency. We can write the experssion for the sound waves as:
s1 (x,t) = 0.75m cos(kx-wt) s2 (x,t) = 0.75 cos(kx-wt+phaseshift)

Two sound waves are initially in phase when they are emitted from A and B, which means that there are no phaseshift between two waves. phaseshift is equal to 0rad

1. What is the phase difference between the two waves when they arrive at the following positions:
a at p1
b at p2
c at p3

2 What is the amplitude of the resultant wave, due to the combination of the two waves, at the following positions

d. at p1
e at p2
e at p3

Explanation / Answer

s1 (x,t) = 0.75 cos(kx-wt) s2 (x,t) = 0.75 cos(kx-wt+) where = phaseshift

Amplitude, sm=.75m, wavelength , = 2 m

velocity of sound in air v= 343.2 m/s at 20 degree

wave number k = 2/ = 2*3.14/2 = 3.14 m-1

angular frequeny w = 2*frequencey= 2c/ = 2*3.14*343.2/2 = 1077.65 s-1

let ta1 be the time taken by the wave s1 to reach at P1 , ta1= distance/ v = (102+1.52)/ 343.2 = 0.0295 s

let ta2 be the time taken by the wave s1 to reach at P2 , ta2= distance/ v = (10+1.5)/ 343.2 = 0.0335 s

let ta3 be the time taken by the wave s1 to reach at P3 , ta3= distance/ v = (10-1.5)/ 343.2 = 0.0248 s

let tb1 be the time taken by the wave s2 to reach at P1 , tb1= distance/ v = (102+1.52)/ 343.2 = 0.0295 s

let tb2 be the time taken by the wave s2 to reach at P2 , tb2= distance/ v = (10-1.5)/ 343.2 = 0.0248 s

let tb3 be the time taken by the wave s2 to reach at P3 , ta3= distance/ v = (10+1.5)/ 343.2 = 0.0335 s

The total phase of sound produced from point A (s1) at P1 ,a1= kx-wta1

= 3.14* (102+1.52) - 1077.65*0.0295 = -0.0394

The total phase of sound produced from point A (s1) at P2 ,a2= kx-wta2

= 3.14* (10+1.5) - 1077.65*0.0335 = -2.1466

The total phase of sound produced from point A (s1) at P3 ,a3= kx-wta3

= 3.14* (10-1.5) - 1077.65*0.0248 = -0.0357

The total phase of sound produced from point B (s2) at P1 ,b1= kx-wtb1

= 3.14* (102+1.52) - 1077.65*0.0295 = -0.0394

The total phase of sound produced from point B (s2) at P2 ,b2= kx-wtb2

= 3.14* (10-1.5) - 1077.65*0.0248 = -0.0357

The total phase of sound produced from point B (s2) at P3 ,b3= kx-wtb3

= 3.14* (10+1.5) - 1077.65*0.0335 = -2.1466

1) a) phase difference between waves at P1 = a1-b1 = -0.0394 - (-0.0394) = 0

b) phase difference between waves at P2 = a2-b2 = -2.1466 - (-0.0357) = -2.1109

c) phase difference between waves at P3 = a3-b3 = -0.0357 - (-2.1466) = 2.1109

2) When two waves ( E1 = E01 Cos (kx - t) and E2 = E02 sin (k(x+ x) - t) = E02 sin (kx - t + )   ) interfere, resultant is = E02 = E012 + E022 + 2E01E02 cos , where is the phase difference between them

a) amplitude at p1 = 0.752+0.752+2*0.75*0.75 * Cos 0 =2.25 m

b) amplitude at p2 = 0.752+0.752+2*0.75*0.75 * Cos (-2.1109) =2.249 m

c) amplitude at p3 = 0.752+0.752+2*0.75*0.75 * Cos (2.1109) =2.249 m

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