# Intensity of sound vs speed of sound

I am finding out the relationship between intensity of sound and speed of sound. To do that, I am using this equation: $$I =\frac{\Delta P^2_\text{max}}{2 \rho v}$$

where $\Delta P_\text{max}$ is pressure amplitude, $I$ is intensity of sound, $\rho$ is density of medium, and $v$ is wave speed.

I got this equation from this website, Physics.info, my only problem is that I cannot figure out how to measure the pressure amplitude. How do I do that?

From pressure variations to $P_{max}$ there is a simple way using RMS (root mean square, consult this), i.e. for a sine pressure wave of amplitude $A$: $P_{max}=\frac{\sqrt{2}}{2}A$.
Lets denote Intensity from (I).Then we know $$I=\frac{1}{2}\rho vA^2\omega^2$$ Hence the "A" here is amplitude and "v" is velocity of wave and $\omega$ is the angular velocity of wave. So we know that $\Delta P_{max}=\beta Ak$ From this we could say $$A=\Delta P_{max}\frac{\beta}{Ak}$$(bulk modulus of the material is $\beta$ So we get $$I=\frac{1}{2}\rho v \left(\frac{(\Delta P_{max})^2}{\beta^2 k^2}\right)\omega^2$$ For Pressure amplitude consider Imagine a half vertically filled cylinder having moved the Liquid a $\Delta x$ So $\Delta V=A_1\Delta x$(Initial Volume ) and futhermore $\Delta y$ distance by it now next change $\Delta V=A_1\Delta y$(volume change). The equation $$\beta=\frac{\Delta P}{\left(-\frac{\Delta V}{V}\right)}$$ $$\Rightarrow\beta=\frac{\Delta P}{\left(-\frac{\Delta y}{\Delta x}\right)}$$ $$\Rightarrow\Delta P=(-\beta)\left(\frac{\Delta y}{\Delta x}\right)\tag1$$ we know by wave equation $$y=A\sin(\omega t-kx)$$ from here $$\frac{\Delta y}{\Delta x}=-kA\cos(\omega t-kx)$$ Putting this in equation 1 $$\Rightarrow\Delta P=(\beta Ak)(\cos(\omega t-kx))$$ from here $$(\omega t-kx)=0$$ Then $$\Delta P_{max}=\beta Ak$$