# Relation between Displacement Amplitude and Intensity of Sound Waves

How do we derive a relation between the displacement amplitude and the intensity of a sound wave? I saw a formula on the internet regarding the same i.e

But I can't find a way that would help me in deriving this expression.

All related suggestions are appreciated.

• Would you define the symbols please. Jun 12, 2021 at 10:26

Sound wave velocity: $$v_s = \sqrt{\frac{B}{\rho}};\tag{1}$$ where $$\rho$$ the mass density of medium, $$B$$ is the boulk modulous defined as ratio of pressure change $$\Delta P$$ with percentage change of volume ($$\frac{\Delta V}{V}$$) $$B = -\frac{\Delta P }{\Delta V / V} = -V\frac{d P}{d V}.$$ Assume the area of the cross-section of propagation is $$A$$, and $$V= A dx$$, the volume unit without deformation. The deformation $$\Delta V = A \Delta s$$, $$s$$ is the disantace displacement. $$\Delta P (x) = B \frac{\Delta V}{V}\big\vert_x = B\frac{\partial \Delta s}{\partial x}.\tag{2}$$

Sinusoidal equation for displacement: $$\Delta s(x, t) = \Delta s_0 \sin (kx-\omega t). \tag{3}$$ where angular frequecy $$\omega = 2\pi f = k v_s$$.

Using Eq.(2) converting displacement $$\Delta s(x,t)$$ to pressure deviation $$\Delta P(x,t)$$

\begin{align} \Delta P(x,t) &=B\frac{\partial \Delta s}{\partial x} \\ &= B k\Delta s_0 \cos k(x-vt). \end{align}

The work done per wave length $$F = A\Delta P$$:

\begin{align} W &= \int_{0}^{\frac{2\pi}{k}} F d\Delta s \\ &= \int_{0}^{\frac{2\pi}{k}} A B k\Delta s_0 \cos (kx-\omega t) k\Delta s_0 \cos (kx-\omega t) dx\\ &=A B k^2 \Delta s_0^2 \int_{0}^{\frac{2\pi}{k}} \cos^2 (kx-\omega t) dx\\ &= \frac{\pi}{k} A B k^2 \Delta s_0^2 \end{align}

Intensity (period $$T =\frac{2\pi}{\omega}$$, $$B = \rho\,v_s^2$$, and $$v_s = \frac{\omega}{k}$$ ) \begin{align} I &= \frac{W}{A T} =\frac{\pi}{k}\frac{ B k^2 \Delta s_0^2}{2\pi/\omega}\\ &= \frac{1}{2} \rho v_s^2 k \omega \Delta s_0^2 \\ &=\frac{1}{2} \rho v_s \omega^2 \Delta s_0^2\\ &=2\pi^2 \,\rho\, v_s f^2 \Delta s_0^2 \end{align} In the last equation, $$\omega$$ is replaced by frequency $$\omega = 2\pi f$$.