Young's Double Slit Experiment (Diffraction)

[jones268]

A lecturer is demonstrating two-slit interference with sound waves. Two speakers are used, 1.9 m apart. The sound frequency is 1220Hz and the speed of sound is 343 m/s. Students sit facing the speakers in a row of sear 5.4 m away. Along the row of students, what is the spacing between the locations on either side of the center line between the speakers where no sound is heard because of destructive interference? The angle may be too large to use small angle approximation.
This is what I've come up with my known variables.
d=1.9m (Distance between the two "slits"
f=1220 Hz
c=343 m/s
L=5.4m (distance from "slits" to the "screen")
y=? unknown spacing
And I'm also assuming I'll be using the formula for dark fringe due to the fact they want
the spacing where no sound will be heard.
I've also calculated the wavelength, λ, to be 0.281147541 m from the following equation:
λ= c/f

The equation I'm using for dark fringe is as follows:
(m + 1/2)(λ)=(d)(sinθ)
Then to find the unknown y, I was using the following equation:
y=(L)(tanθ)

I'm not quite sure where to go from here and if I'm even doing this right, especially due to theta being too large to approximate.

 

[Delphi51]

A very interesting question, Jones. You certainly can't use L = 5.4. In the formula L is actually the distance from the slits to the point on the "screen" you are interested in. Usually it is approximately the same as the perpendicular distance from slits to screen, but certainly not this time. You might get away with using (n+½)λ = d*sin(θ) but I doubt it. I would start with the diagram

and work it out from basics. Say that the red distance is ½λ more than the blue distance; use the distance between points formula for those distances. More solutions for n + ½λ.

I get 2 solutions for n = 0, one at about x = 0.6 and the other at about x = -2.
I don't trust the negative value. Rather, the left side should be worked out on the basis of symmetry with the right side. Now I wonder about the word "spacing" in the question. Surely they will not be equally spaced? Perhaps they do want some kind of approximation based on the formula.