The amplitude of sound waves from two nearby speakers

[anhhuyalex]

Homework Statement

In the figure below, two speakers S₁ and S₂ emit sound waves of wavelength 2m, in phase with each other.

Let A_p be the amplitude of the resulting wave at point P, and A_q be he amplitude of the resultant wave at point Q. How does A_p compare to A_q?

a. A_p<A_q
b. A_p=A_q
c. A_p>A_q
d. A_p<0, A_q>0
e. A_p and A_q vary with time, so no comparison can be made

Homework Equations

Pythagorean Theorem

The Attempt at a Solution

My guess is that since amplitude varies with time, there would be times when A_p>A_q and vice versa, so the answer would be E. However, my textbook says that the answer is A because at Q sound waves from both speakers traveled 4m, so there would be constructive interference, but at point P sound waves from S₁ traveled 4m whereas sound waves from S₂ traveled 5m by the Pythagorean theorem, so there is destructive interference. Can you explain to me why there would be destructive interference at all in two waves that were said to be "in phase with each other?" Specifically, can you explain what that phrase and the phrase "out of phase" mean when we're talking about waves?

 

[paisiello2]

A wave that is out of phase with another wave is essentially one that has been shifted over. Recall the equation for a wave in two dimensions as a function of time t:

A*sin(ωt + ø)

where ø is the phase. Now if two waves with the same frequency ω both also have ø=0 then the waves are said to be in phase with one another. The amplitude A can still be the same or different.

However, in 3 dimensions the wave looks like a series of concentric circles when viewed from the top. This means any two waves that are overlapping even if they are in phase will always have areas of interference. In some areas the crest meets the crest and in other areas the trough meets the trough. In both cases the interference is said to be constructive or additive.

Where the crest of one wave meets the trough of another wave, however, the waves will cancel each other out and said to be deconstructive or subtractive.

The wiki entry here has some good illustrations of what this looks like:
https://en.wikipedia.org/wiki/Interference_(wave_propagation )

Another way to think about the mathematics is that using the geometry of 3 dimensions means the waves are effectively shifted by an amount 2∏x/λ:

A*sin(ωt + ø + 2∏x/λ) = A*sin(ωt + 2∏x/λ) for ø=0

where λ is the wave length of the wave and x is the relative position along the wave in a given dimension. So in that sense they could be considered out of phase with one another at some values of x and in phase with one another for other values of x.

So in this case x =4m for the first wave and x =5m for the second wave which is a difference of λ/2 or half a wave length. So they are effectively 90 degrees out of phase at this point.