Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach. For sound waves of any magnitude of a disturbance in velocity, pressure, and density we have In the case that the fluctuations in velocity, density, and pressure are small, we can approximate these as Where is the perturbed velocity of the fluid, is the pressure of the fluid at rest, is the perturbed pressure of the system as a function of space and time, is the density of the fluid at rest, and is the variance in the density of the fluid over space and time. In the case that the velocity is irrotational (), we then have the acoustic wave equation that describes the system: Where we have Derivation for a medium at rest Starting with the Continuity Equation and the Euler Equation: If we take small perturbations of a constant pressure and density: Then the equations of the system are Noting that the equilibrium pressures and densities are constant, this simplifies to A Moving Medium Starting with We can have these equations work for a moving medium by setting , where is the constant velocity that the whole fluid is moving at before being disturbed (equivalent to a moving observer) and is the fluid velocity. In this case the equations look very similar: Note that setting returns the equations at rest. Linearized Waves Starting with the above given equations of motion for a medium at rest: Let us now take to all be small quantities. In the case that we keep terms to first order, for the continuity equation, we have the term going to 0. This similarly applies for the density perturbation times the time derivative of the velocity. Moreover, the spatial components of the material derivative go to 0. We thus have, upon rearranging the equilibrium density: Next, given that our sound wave occurs in an ideal fluid, the motion is adiabatic, and then we can relate the small change in the pressure to the small change in the density by Under this condition, we see that we now have Defining the speed of sound of the system: Everything becomes For Irrotational Fluids In the case that the fluid is irrotational, that is , we can then write and thus write our equations of motion as The second equation tells us