Derive continuity equation for unsteady pipe flow

The continuity equation expresses conservation of mass in a pipe in which the water is slightly compressible and the pipe wall is elastic.

Set-up

Take a control volume of length $\delta s$ in a pipe of cross-section $A$, with density $\rho$ and velocity $v$ along the pipe axis $s$.

        mass in             mass out
  rho A v --> |<-- ds -->| --> rho A v + d(rho A v)/ds ds

Mass balance

The net mass flow into the element equals the rate of increase of mass inside it:

$$-\frac{\partial(\rho A v)}{\partial s}\delta s=\frac{\partial(\rho A\,\delta s)}{\partial t}$$ $$\frac{\partial(\rho A v)}{\partial s}+\frac{\partial(\rho A)}{\partial t}=0$$

Expanding and dividing by $\rho A$:

$$\frac{1}{\rho}\left(\frac{\partial\rho}{\partial t}+v\frac{\partial\rho}{\partial s}\right)+\frac{1}{A}\left(\frac{\partial A}{\partial t}+v\frac{\partial A}{\partial s}\right)+\frac{\partial v}{\partial s}=0$$

i.e.

$$\frac{1}{\rho}\frac{d\rho}{dt}+\frac{1}{A}\frac{dA}{dt}+\frac{\partial v}{\partial s}=0\qquad(1)$$

where $d/dt$ is the derivative following the fluid.

Compressibility of water

From the bulk modulus, $K=\dfrac{dp}{d\rho/\rho}$:

$$\frac{1}{\rho}\frac{d\rho}{dt}=\frac{1}{K}\frac{dp}{dt}\qquad(2)$$

Elasticity of pipe wall

For a thin-walled pipe of diameter $D$, thickness $e$ and modulus $E$, the hoop stress is $\sigma=pD/2e$ and the strain is $\varepsilon=\Delta D/D=\sigma/E$. As $A\propto D^2$:

$$\frac{1}{A}\frac{dA}{dt}=\frac{2}{D}\frac{dD}{dt}=\frac{2}{E}\frac{D}{2e}\frac{dp}{dt}=\frac{D}{eE}\frac{dp}{dt}\qquad(3)$$

Result

Substitute (2) and (3) in (1):

$$\frac{1}{K}\left(1+\frac{KD}{eE}\right)\frac{dp}{dt}+\frac{\partial v}{\partial s}=0$$

Multiply by $K/\left(1+\dfrac{KD}{eE}\right)$ and divide by $\rho$:

$$\frac{1}{\rho}\frac{dp}{dt}+\frac{K/\rho}{1+\dfrac{KD}{eE}}\frac{\partial v}{\partial s}=0$$

With celerity $c=\sqrt{\dfrac{K/\rho}{1+KD/(eE)}}$:

$$\boxed{\frac{1}{\rho}\frac{dp}{dt}+c^2\frac{\partial v}{\partial s}=0}$$

For a rigid pipe ($E\to\infty$), $c=\sqrt{K/\rho}$, which is the speed of sound in water (about 1480 m/s at 20 °C, with $K\approx2.2$ GPa); this is the form given in the question.