snoopies622
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I've having trouble understanding a derivation of the speed of sound waves, which is actually similar to another derivation I found a couple days ago.
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 \Delta x, the net horizontal (direction of the wave motion) force acting on the water is proportional to the hoizontal pressure gradient times \Delta x, so \rho \frac {dv}{dt} = \frac {- \partial P}{\partial x}
2.) the mass flux through any infinitely thin cylindrical slice is constant, or
\frac {d ( \rho v) }{dt} =0
And from these premises one can arrive at v^2 = \frac {dP }{d \rho }.
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 \Delta x, or something else?
Thanks.
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 \Delta x, the net horizontal (direction of the wave motion) force acting on the water is proportional to the hoizontal pressure gradient times \Delta x, so \rho \frac {dv}{dt} = \frac {- \partial P}{\partial x}
2.) the mass flux through any infinitely thin cylindrical slice is constant, or
\frac {d ( \rho v) }{dt} =0
And from these premises one can arrive at v^2 = \frac {dP }{d \rho }.
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 \Delta x, or something else?
Thanks.