Answer: 0.729 m
Explanation:
In order to be able to hear a constructive interference, the difference between the distances from both speakers to the person must be a multiple of the wavelength of the sound.
We can find out the wavelength as we know the value of the frequency and the sound speed, as follows:
λ = v / f = 343 m/s / 514 Hz = 0.67 m.
So, if we call x to the distance perpendicular to the wall, right front of the left speaker, and h to the distance to the right speaker, we can write the following expression:
h – x = n λ (1)
As x, h, and the distance d between speakers determine a square triangle, we can apply Pitheagoras ‘theorem, as follows:
h2 = d2 + x 2
h2 – x2 = d2 (2)
(h+x)(h-x) = d2 (3)
Dividing both sides in (3) and (1), we get:
h + x = d2 / n λ (4)
Substracting both sides in (4) and (1), we have:
2 x = (d2/ n λ) - n λ = (d2 – n2 λ2) / 2 n λ (5)
We must choose the maximum value for n that satisfies x>0, as follows:
nmax = d / λ = 5.7 → nmax = 5
Solving for x, and replacing for the values of d, λ and n in (5), we get:
x = 16 – 25. (0.67)2 / 2.5.(0.67) = 0.729 m
Explanation:
