Answer to Question #104182 in Trigonometry for Tamara

Question

What is “sound”? How can sound waves be described mathematically? Be sure to link correct mathematical terms to correct physics terms – for example, how are the amplitude, period length, and other mathematical properties of sound waves related to pitch, tone, volume, and other physical properties of the sound wave? What is a “beat” pattern, and when does it happen? Why does a beat happen?

Answer 1

Sound is defined as an oscillation in pressure, stress, particle displacement, particle velocity, etc., propagated in a medium with internal forces ( elastic or viscous) or the superposition of such propagated oscillation. 

Sound waves are longitudinal waves in nature.

The frequency of the sound is not related to its loudness.

The pitch of sound depends on the frequency of vibration of the waves.

Amplitude$^2 \propto$ Loudness

The pitch or note of a sound that we experience is determined by its wavelength or its frequency. The shorter the wavelength, the higher the frequency becomes, and the higher the pitch that we hear.

Mathematical equation of a sound wave is:

$\frac { \partial^2P}{\partial x^2 }-\frac{1}{c^2}\frac{\partial^2P}{\partial t^2}=0$

Its solution is:

$P=P_0sin(kx\pm \omega t)$

When two sound waves of different frequency approach your ear, the alternating constructive and destructive interference causes the sound to be alternatively soft and loud - a phenomenon which is called "beating" or producing beats.

The frequency difference must be up to $10 Hz$ .

Number of beats produced$=f_1-f_2$

where $f_1$ and $f_2$ are frequency of interfering waves.