- #1

hraghav

- 44

- 5

- Homework Statement
- Two speakers are located a distance 2d apart along the x-axis, where the origin is located at the midpoint between the two speakers. This is shown in the image below. The speakers emit sound with wavelength λ=1.34m. The distance from the origin to either of the speakers is d=6.74m. At distances y≫d, the lines connecting the speakers to the listener can be treated as effectively parallel, similar to two slit diffraction. In such a case, the difference in distance from the speakers to the listening point can be approximated as Δr=2dsinθ, where the angle θ is shown in green in the image above. For the following questions, assume that the listener is at a location

r→=(x,1000m).

a) What is the smallest distance x from the y-axis such that constructive interference occurs?

b) What is the smallest distance x from the y-axis such that destructive interference occurs?

- Relevant Equations
- For constructive interference: Δr=mλ

For destructive interference: Δr=(m+ 0.5)λ

a) For constructive interference: Δr=mλ where m = 1 for smallest distance

Δr= λ where Δr= 2dsinθ given in the question

so 2dsinθ = λ

θ = sin^-1( λ / 2d) where λ = 1.34m and 2d = 2*6.74 = 13.48m

θ = 5.7049 degrees

tan θ = x/1000

1000*tan(5.7049) = x

x = 99.9013 m

b) For destructive interference: Δr=(m+ 0.5)λ where m = 0 for smallest distance

Δr = 0.5 λ but Δr= 2dsinθ given in the question

2dsinθ = 0.5 λ

sinθ = λ / 4d where λ = 1.34m and 4d = 4*6.74 = 26.96m

θ = 2.8489 degrees

tan θ = x/1000

1000*tan(2.8489) = x

x = 49.76 m

Both these answers are not correct. Could someone please look at these and let me know where is the error?

Thank you

Δr= λ where Δr= 2dsinθ given in the question

so 2dsinθ = λ

θ = sin^-1( λ / 2d) where λ = 1.34m and 2d = 2*6.74 = 13.48m

θ = 5.7049 degrees

tan θ = x/1000

1000*tan(5.7049) = x

x = 99.9013 m

b) For destructive interference: Δr=(m+ 0.5)λ where m = 0 for smallest distance

Δr = 0.5 λ but Δr= 2dsinθ given in the question

2dsinθ = 0.5 λ

sinθ = λ / 4d where λ = 1.34m and 4d = 4*6.74 = 26.96m

θ = 2.8489 degrees

tan θ = x/1000

1000*tan(2.8489) = x

x = 49.76 m

Both these answers are not correct. Could someone please look at these and let me know where is the error?

Thank you