that And the use of this equation in the continuity equation tells us that This simplifies to Thus the velocity potential obeys the wave equation in the limit of small disturbances. The boundary conditions required to solve for the potential come from the fact that the velocity of the fluid must be 0 normal to the fixed surfaces of the system. Taking the time derivative of this wave equation and multiplying all sides by the unperturbed density, and then using the fact that tells us that Similarly, we saw that . Thus we can multiply the above equation appropriately and see that Thus, the velocity potential, pressure, and density all obey the wave equation. Moreover, we only need to solve one such equation to determine all other three. In particular, we have For a moving medium Again, we can derive the small-disturbance limit for sound waves in a moving medium. Again, starting with We can linearize these into For Irrotational Fluids in a Moving Medium Given that we saw that If we make the previous assumptions of the fluid being ideal and the velocity being irrotational, then we have Under these assumptions, our linearized sound equations become Importantly, since is a constant, we have , and then the second equation tells us that Or just that Now, when we use this relation with the fact that , alongside cancelling and rearranging terms, we arrive at We can write this in a familiar form as This differential equation must be solved with the appropriate boundary conditions. Note that setting returns us the wave equation. Regardless, upon solving this equation for a moving medium, we then have See also Acoustic attenuation Sound Fourier analysis References Fluid dynamics Acoustics Sound