# The amplitude of sound waves from two nearby speakers

• anhhuyalex
In summary: This will result in destructive interference. In summary, when two waves are in phase with each other, it means that they have the same frequency and their crests and troughs align. However, in three dimensional space, there can still be areas of destructive interference due to differences in the distance traveled by the waves. This is what happens at point P, where the waves from the two speakers are 90 degrees out of phase and cancel each other out, leading to a lower amplitude compared to point Q where they are in phase and constructively interfere.
anhhuyalex

## Homework Statement

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

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

a. Ap<Aq
b. Ap=Aq
c. Ap>Aq
d. Ap<0, Aq>0
e. Ap and Aq 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 Ap>Aq 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?

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.

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## What is the amplitude of sound waves?

The amplitude of sound waves refers to the maximum displacement of the wave from its rest position. In other words, it measures the intensity or strength of the sound wave.

## How is the amplitude of sound waves from two nearby speakers affected?

The amplitude of sound waves from two nearby speakers is affected by the distance between the speakers, the power of the speakers, and the frequency of the sound waves being emitted.

## What is constructive interference and how does it affect the amplitude of sound waves from two nearby speakers?

Constructive interference occurs when two sound waves have the same frequency and are in phase (peaks and valleys align). This results in an increase in the amplitude of the sound waves, making the sound louder.

## What is destructive interference and how does it affect the amplitude of sound waves from two nearby speakers?

Destructive interference occurs when two sound waves have the same frequency but are out of phase (peaks and valleys do not align). This results in a decrease in the amplitude of the sound waves, making the sound quieter or cancelling it out completely.

## How can the amplitude of sound waves from two nearby speakers be adjusted or controlled?

The amplitude of sound waves from two nearby speakers can be adjusted or controlled by changing the distance between the speakers, adjusting the power or volume of the speakers, or changing the frequency of the sound waves being emitted.

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