Singe Slit Diffractions Question?

Question

Darren stands 3.0 m in front of a door (0.75 m wide) which leads to a room with a playing stereo. He finds that when he moves 2.0 m to the right, like in the picture below, the sound begins to drop off suddenly. Using single-slit diffraction theory, what's the wavelength of the sound waves?

Answer 1

Let's suppose that by the ridiculuous expression "begins to drop off suddenly" is actually meant "is at minimum loudness". Then we can say that there's a half-wavelength path length difference between either edge of the slit and the center when the observer's y value = 2. So we have the geometrical relationship theta = arctan(y/D) = arctan(2/3), and lambda = a*sin(theta), where a is the door (slit) width of 0.75 m. The terms used here are those of the ref., which explains the theory of diffraction displacements well.

The answer depends on whether you want the exact solution given above, or the small-angle approximation that says lambda = ay/D. If you choose the latter, the answer is exactly 0.5 m, which I suspect is the answer the problem is looking for. Since it's anything but a small angle, however, I'll give you the exact one as well, 0.461025 m. Take your pick.