# Amplitude of sound waves

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A quote from a book:
Fundamentals of Physics said:
A sound wave causes a longitudinal displacement s of a mass element in a medium as given by $s=s_mcos(kx-\omega t)$ where $s_m$ is the displacement amplitude (maximum displacement) from equilibrium, $k=\frac{2\pi}{\lambda}$, and $\omega=2\pi f$ , $\lambda$ and $f$ being the wavelength and frequency, respectively, of the sound wave.
At first, I was under the impression that $s_m$ was constant. But shouldn't $s_m$ be inversely proportional to the distance?

The book never says this anywhere directly, but it seems to be implied by two other equations (for the intensity) given in the chapter:
$$I=\frac{P_{source}}{4\pi R^2}=\frac{1}{2}\rho v\omega ^2s_m^2$$
Wouldn't this mean that either the frequency or the amplitude decreases with distance? But the frequency can't change without a corresponding change in wavelength, so doesn't this mean the displacement-amplitude (and therefore also the pressure-amplitude) decreases with the distance?

I just want to make sure, because all the writing in the chapter seemed to imply to me that the amplitude was constant, yet these two equations for the intensity seem to say otherwise.

## Answers and Replies

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Orodruin
Staff Emeritus
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The wave provided is a plane wave solution, which does not decrease with distance travelled unless damped. It only describes a wave generated at a point source far away from the source (much further than the wavelength and the distance over which you are considering the wave).

Edit: Of course, if you solve the wave equation for a spherical wave, you will end up with a decreasing amplitude.

Nathanael
Homework Helper
Thank you very much.

Homework Helper
One more question; When you say that equation describes a planar wave, is this just because the amplitude is constant?

In other words, would the equation that describes a spherical wave be the same except with a variable amplitude?
(Something like $\frac{c}{R}cos(kR-\omega t)$?)

Last edited:
Orodruin
Staff Emeritus