Acoustics – Why Two Sources of Same-Amplitude Sound Louder Than One?

acousticsinterferencesuperposition

Sorry if the question is stupid. The last time I touched physics was in school and I have never come back to it since. But this question popped up just recently in my life and I realized I don't know how to answer it.

As far as I remember, the amplitude of a sound wave defines the loudness of the sound – the bigger the amplitude is, the louder is the sound.

And I also remember that the frequency of a sound wave defines its pitch – the faster is the oscillation, the higher is the sound pitch.

Now imagine we take two oboes, each one playing the same note and with the same level of loudness. Both are producing a sound with the same amplitude. This means that the loudness of the resulting sound of two oboes will still be the same, but in the reality two oboes will sound louder than one. Why is it so?

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It is not always the case that the oboes will create louder sound. They may also create no sound! This is based on wave interference. When two waves, that have the same amplitude and frequency have their troughs and crests aligned, they can be added together (This is known as constructive interference) according to a principle known as superposition, and create a sound that is twice as loud. However, if the crests of one wave align with the trough of the other one, they will cancel (This is known as destructive interference) and there will be no sound. A visual representation is as follows

If you would like more details about the topic, you can head to the Wikipedia page.

Related Solutions

[Physics] Why are two voices singing the same note louder than one

This is a neat question. Did you know that adding two Sine waves of the same frequency but different phase together always produces another Sine wave? Of course you can imagine two perfectly out-of-phase Sine waves that "cancel" by adding to a line but in that case you can just imagine the result as a Sine wave with 0 amplitude.

Using gnuplot with the following commands I plotted the sum of 4 Sine waves:

a = rand(0)
b = rand(0)
c = rand(0)
d = rand(0)
plot sin(x + a * (2 * pi)) + sin(x + b * (2 * pi)) + sin(x + c * (2 * pi)) + sin(x + d * (2 * pi))

And on the first go I got this:

add 4 sine waves together

Notice the shape and frequency is the same but the amplitude has increased. If you add them randomly together again you'd get something like:

add 4 sine waves together again

Notice that the amplitude is still greater than 1 but the phase has shifted.

What's happening is that each Sine wave you add increases the maximum possible amplitude so the more you add the better the chance of getting a wave with amplitude greater than 1.

Chris White posted a fantastic answer about the brightness of the sun where he goes into a lot more details about the math of adding waves and the statistics of it.

Acoustics – Why Adding Intensities is Necessary for Coherent Sound Sources

I remember working this out in the opposite direction, hoping to get a paradox: that the amplitude of the sound produced by a choir of 100 singers is only 10 times the amplitude of the sounds produced by one of them.

Each individual sound can be represented as a vector in 2-dimensional space, with length 1 (since all are equally loud) and random phase. The resultant sound heard will be the sum of all those vectors.

Adding the vectors one at a time is thus effectively undertaking a random walk in 2 dimensions. The following statements are therefore equivalent:

The expected distance travelled in an $n$-step random walk is $\sqrt{n}$.
The expected amplitude of $n$ coherent equally loud sound sources is $\sqrt{n}$ times the amplitude of one of them.

It is entirely up to you to decide which one of them you prove from which, but in this case you can assume the truth of 1 (it is a standard result) and deduce the truth of 2, which, squared, is the result you asked for in your question.

(And my paradox wasn't a paradox because I had forgotten that loudness is proportional to the square of amplitude, so 100 choristers are 100 times as loud as one of them).

Incoherent sound sources yield essentially the same answer because you can think of them as coherent sources with varying phase - so again you end up with a total of random phases.

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Related Solutions

[Physics] Why are two voices singing the same note louder than one

Acoustics – Why Adding Intensities is Necessary for Coherent Sound Sources

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