Often with speakers, none of the approximation work and we simply have to work with the distances to find the path difference, because the angle to the observer is not small and also the spacing of the speakers is also not small. In this example, the spacing of the speakers is relatively small in comparison to the distance away L, so we can use m λ = d sin θ. However the location of point Q is unclear so we will not assume that the small angle approximation (x/L) would work.
(a) Simple, v = f λ … 343 = 2500 f … λ = 0.1372 m
(b) Determine θ … mλ = d sin θ … (0.5) (0.1372) = (0.75) sin θ … θ = 5.25° (small enough to have used x/L)
Now find Y … tan θ = o / a … tan (5.25) = Y / 5 … Y = 0.459 m
(c) Another minimum, ‘dark spot’ (not dark since its sound), could be found at the same distance Y above point P on the opposite side. Or, still looking below P, you could use m = 1.5 and find the new value of Y.
(d) i) Based on the formulas and analysis from point b, it can clearly be see that decreasing d, would make angle θ increase, which would increase Y
ii) Since the speed of sound stays constant, increasing f, decreases the λ. Again from the formulas and analysis in part b we see that less λ means less θ and decreases the location Y.
