The figure shows two point sources S 1 and S 2 that emit sound of wavelength ? =

Question

The figure shows two point sources S1 and S2 that emit sound of wavelength ? = 1.6 m. The emissions are isotropic and in phase, and the separation between the sources is d = 15.0 m. At any point P on the x axis, the wave from S1 and the wave from S2 interfere. Start with Pvery far away (x = infin). As you then move P in along the x axis toward the origin, (a) does the phase difference between the waves increase or decrease? At what distance x do the waves have a phase difference of (b) 0.50?, (c) 1.00?, and (d ) 1.50??

Explanation / Answer

Ans.

call d1 the distance from S1 to the point and d2 the distance from S2

the difference in distance, d2 - d1 = Sqrt[15+x^2] -x

as x gets large, this difference approaches zero, so that when x-> infinity, the waves are in phase and interfere constructively

a) since the phase difference is zero at x-> infinity, the phase difference will increase as you first move toward S1, but will oscillate

b, c,d)

the difference in distance is Sqrt[15^2+x^2]-x

this will equal 0.5L = 0.8m when

Sqrt[15^2+x^2]-x = 0.8

or Sqrt[15^2+x^2]=0.8+x

square both sides

15^2+x^2 = 0.64+1.6x +x^2

x=140.225 m Ans. of (b)

for a phase difference of 1.6m, follow the same procedure

d2-d1 = Sqrt[225+x^2] - x =1.6

225+X^2 = x^2+3.2x+2.56

x = 69.51m Ans. of (c)

similarly for a phase difference of 2.4 m:

225+x^2 = x^2+4.8x+5.76

x=45.675 m Ans. of (d)

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