Question

Two loudspeakers emit sound waves along the x-axis. A listener in front of both speakers hears a maximum sound intensity when speaker 2 is at the origin and speaker 1 is at x = 0.540m . If speaker 1 is slowly moved forward, the sound intensity decreases and then increases, reaching another maximum when speaker 1 is at x =0.930m .

What is the phase difference between the speakers?

Answer 1

In order to answer this question we need to know exactly what the position of the listener is in relation to the two speakers. To say that the listener is 'in front of both speakers' is not sufficiently detailed.

The question will be simplest if we assume that the listener is positioned on the x-axis, and at a point sufficiently distant from the origin that both speakers always have a smaller x-coordinate than that of the listener. That is what I shall assume. We also need to interpret 'speaker 1 is slowly moved forward'. I will assume that this means that S1 is moved along the x-axis in the positive x direction. Speak to your teacher about this, suggesting (politely) that he or she should be more precise in wording her questions.

Under the conditions I have described, after the first maximum, successive maxima will be heard when S1 moves 1 full wavelength along the x axis in the positive x direction, so the distance between the x-coordinates of S2 for the 1st and 2 maxima will be one wavelength (L), giving L = 0.39m

If the speakers were in-phase, the separation for the 1st maximum would be 0.39m also, but it is in fact 0.54m - a difference of 0.15m. This implies that the phase of S2 differs from that of S1 by an amount equal to 360*0.15/0.33 = 138.46deg.