When a sound wave passes through a fluid(liquid or gas), the period of vibration is short compared to the relaxation time necessary for macroscopically small element of volume of the fluid to exchange energy with the rest of the fluid through heat flow. hence compressions of such an element of volume can be considered adiabatic.
by analyzing one- dimensional compressions and rarefactions of the system of the fluid contained in a slab of thickness dx, show that the pressure p(x, t) in the fluid depends on the position x and the time t so as to satisfy the wave equation
\partial 2 p /\partialt2 = u2\partial2 p / \partial x2
where the velocity of sound propagation u is a constant given by u=(\rhoks)^-1/2.
here \rho is the equilibrium density of the fluid and ks is its adiabatic compressibility ks = -v^-1(\partialv/\partialp)s, i. e., its compressibilty measured under conditions where the fluid is thermally insulated.