Derivation of sound wave propagation equation

Question

I am trying to derive the sound propagation equation like this.

$$\frac{\partial^2 P'}{\partial t^2} = c^2\left(\frac{\partial^2 P'}{\partial x^2}+\frac{\partial^2 P'}{\partial y^2}+\frac{\partial^2 P'}{\partial z^2}\right)$$

In order to get above equation,

1. I integrated the continuity equation and the momentum equations and got the equation below.

   $$\frac{\partial^2 \rho'}{\partial t^2} = \frac{\partial^2 P'}{\partial x^2}+\frac{\partial^2 P'}{\partial y^2}+\frac{\partial^2 P'}{\partial z^2}$$

   where $\rho'$ is density fluctuation and $P'$ is pressure fluctuation.

2. My text said if you time sound speed to both sides you can get the governing equation.

   $$c^2 = \frac{\partial P}{\partial\rho}$$

   $$c^2\frac{\partial^2 \rho'}{\partial t^2} = c^2\left(\frac{\partial^2 P'}{\partial x^2}+\frac{\partial^2 P'}{\partial y^2}+\frac{\partial^2 P'}{\partial z^2}\right)$$

But I couldn't understand the left side because,

$$c^2\frac{\partial^2\rho'}{\partial t^2}=\frac{\partial(\overline P+P')}{\partial(\overline\rho+\rho')}\frac{\partial^2\rho'}{\partial t^2}=\frac{\partial P'}{\partial(\overline\rho+\rho')}\frac{\partial^2\rho'}{\partial t^2}$$

Would you tell me how you can get pressure fluctuation derivatives from sound speed?

Answer 1

From 1. use $P' = \rho' \left( \dfrac{\partial P}{\partial \rho} \right)$ and replace density time derivative with pressure time derivative to get the desired wave equation