Show that intensity of sound is directly proportional to the square of the amplitude.

The intensity of a sound: The intensity of sound wave at a place is defined as the rate of flow of energy per unit area held normally (perpendicularly) to the direction of the wave propagation. it is denoted by I and given by.

\(I=\frac{E_0}{A\times t}\)

Where E₀ is the sound energy, A is the area on which it passes, and t is the time. 

The displacement y of a vibrating layer of the air due to the wave propagation is given by

\(y=a\sin\left(\omega t\right)\) --------- (i)

Where a is the amplitude of vibration. Then the velocity v at any instant is  

\(v=\frac{dy}{dt}=\frac{\operatorname d\left(a\sin\left(\omega t\right)\right)}{\operatorname dt}\)
∴ \(v=a\omega\cos\left(\omega t\right)\) ------------ (ii)

The kinetic energy E of the layer is given as,

\(E=\frac12mv^2=\frac12m\omega^2a^2\cos^2\left(\omega t\right)\)     [∵ v = ωa]  ------------- (iii)

where m is the mass of the layer. When the KE becomes maximum, the PE becomes zero and vice versa, for an oscillating system. Thus, the maximum value of KE is equal to the total energy E₀ of the system.

In equation (iii), E will be maximum, when \(\cos\left(\omega t\right)=1\)

∴ \(E_0=\frac12m\omega^2a^2\)

Then, intensity of sound is given by

\(I=\frac12\frac{m\omega^2a^2}{A\times t}\)
Or, \(I=\frac12\frac{\rho V\omega^2a^2}{A\times t}\)
Or, \(I=\frac12\frac{\rho AI\omega^2a^2}{A\times t}\)
Or, \(I=\frac12\rho v\omega^2a^2\)    \(\left(\because m=V\rho,V=A\times l\;and\;v=\frac lt\right)\)
Or, \(I=\frac12v\rho\left(2\pi f\right)^2a^2=2\pi^2f^2v\rho a^2\)       \(\left[\because\omega=2\pi f\right]\)

From definition, this is the intensity of the wave.

i.e., \(I=\frac12v\rho\left(2\pi f\right)^2a^2=2\pi^2f^2v\rho a^2\) ⇒ \(I\propto A^2\)

In other words, the intensity of a sound wave of a given frequency is directly proportional to the square of the amplitude of the viubration.