- #1

snoopies622

- 825

- 26

Let's suppose the sound is moving through water in a long cylindrical horizontal pipe. The premises of the derivation are

1.) For a given cylindrical slice of water of thickness [itex] \Delta x [/itex], the net horizontal (direction of the wave motion) force acting on the water is proportional to the hoizontal pressure gradient times [itex] \Delta x [/itex], so [tex] \rho \frac {dv}{dt} = \frac {- \partial P}{\partial x} [/tex]

2.) the mass flux through any infinitely thin cylindrical slice is constant, or

[tex] \frac {d ( \rho v) }{dt} =0 [/tex]

And from these premises one can arrive at [tex] v^2 = \frac {dP }{d \rho } [/tex].

What I don't understand is why the second premise is true, since neither the water density nor water speed is constant. Or perhaps I don't understand the second premise: Is it supposed to be for an infinitely thin slice, or for a cylinder of thickness [itex] \Delta x [/itex], or something else?

Thanks.