Find smallest distance x from y-axis for both constructive & destructive interference to occur

[hraghav]

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

 

[TSny]

Your work looks correct to me for the case where the listener is depicted by the green ear in the diagram. Do you know what answers you are supposed to get? Did you consider $x = 0$ as a possible answer?

 

[hraghav]

Your work looks correct to me for the case where the listener is depicted by the green ear in the diagram. Do you know what answers you are supposed to get? Did you consider $x = 0$ as a possible answer?

0 worked for part a ie constructive interference but I still don't have the answer for part b. I can't see the final answers until and unless I don't enter the correct ones.

 

[TSny]

0 worked for part a ie constructive interference but I still don't have the answer for part b. I can't see the final answers until and unless I don't enter the correct ones.

I still don't see any error in your calculation for (b).

 

[hraghav]

I still don't see any error in your calculation for (b).

Sounds good thank you

 

[Steve4Physics]

... but I still don't have the answer for part b. I can't see the final answers until and unless I don't enter the correct ones.

The answer may need to have an appropriate number of significant figures. It depends on the software checking the value.

Four significant figures is excessive. You could try entering the value rounded to 3 sig. figs.