The
speed of
sound in a
fluid medium is given by
c = sqrt(dp/dρ)
where p denotes the pressure and ρ the density of the fluid. Since the bulk modulus of elasticity can be expressed as
E = ρ(dp/dρ)
we then have
c = sqrt(E/ρ)
For an isentropic process (which is assumed when we talk about acoustic waves), the bulk modulus is simply E = kp, where k is the ratio of the specific heat at constant temperature to the specific heat at constant volume. (Remember that in an isentropic process, p/ρk is constant; for air under standard atmospheric conditions, k = 1.40). Thus,
c = sqrt(kp/ρ)
For an ideal gas, p = ρRT, where R is the ideal gas constant and T the temperature (in K). We have finally
c = sqrt(kRT)
which means that the speed of sound in an ideal gas is proportional to the square root of the temperature. Furthermore, we see that if a fluid would really be incompressible, then E would be infinite and so would be the speed of sound.
Primary source: Fundamentals of fluid mechanics, Munson, Young, Okiishi, Wiley Editor.