Pressure change in pipe due to sudden closure of valve

[phantomvommand]

Homework Statement

Please see the attached photo

Relevant Equations

Conservation of Mass
Impulse forumla

Water is flowing in the pipe with velocity v0. Upon sudden closure of the valve at T, a pressure wave travels in the -ve x direction with speed c. The task is to find $\alpha$, where $\Delta P = \rho_0 c (\Delta v) \alpha$.
The 1st step is to set up an equation using conservation of mass. (picture is below)
I do not understand why we need to use a moving frame c to find the speed of the water particles. Why is simply stating $\rho_0 v_0 = \rho_1 v_1$ incorrect?

 

[haruspex]

Why is simply stating ρ0v0=ρ1v1 incorrect?

One obvious reason is that v0 is positive and v1 is negative, so it cannot be true.

Consider positions x, y, x', y' in the pipe, where,
at time t, the wave front is at x
at time t+dt, the wave front is at y'
at time t+dt, the water that had been at x has moved to y
the water that is at y' at t+dt had been at y at time t
Draw a picture or two.
So the water that was between x and y at time t is between x' and y' at t+dt.
By mass conservation, $(x-y)\rho_0=(x'-y')\rho_1$.
Also $x'-x=v_1 \delta t$, $y'-y=v_0 \delta t$ and $y'-x=c \delta t$.

I leave the working to you.