# Difference in deriving sound waves in liquids and gases

In our fluids course we derive the small amplitude approximation of waves in both a perfect gas and liquids using the Navier Stokes Equations, Mass Conservation Equation and then:

1. for a perfect gas, assuming the fluid elements are adiabatically compressed/expanded such that the pressure and density are related by $$p \rho^{-\gamma} = \text{const.}$$
2. for a liquid, using that the bulk modulus $$K=\rho \frac{dp}{d\rho}$$ is constant.

My question is, why is it required that we use such different approaches for liquids and gases - can't we approximate the liquid fluid elements as being adiabatically compressed/expanded, or can't we assume a constant bulk modulus of the gas?

A side question is, these '5th equations' (in addition to 3 Navier Stokes Equations and 1 Mass Conservation Equation) needed to solve for the 5 variables ($$v_x$$, $$v_y$$, $$v_z$$, $$p$$, $$\rho$$) are often referred to as 'energy equations'. Could someone enlighten me to the exact connection of these to energy conservation?

Side question first:

A side question is, these '5th equations' (in addition to 3 Navier Stokes Equations and 1 Mass Conservation Equation) needed to solve for the 5 variables (vx, vy, vz, p, ρ) are often referred to as 'energy equations'. Could someone enlighten me to the exact connection of these to energy conservation?

These "5th equations" are actually called equations of state, or constitutive equations. "Energy equations", on the other hand, are used to incorporate thermal effects. For adiabatic systems, the appropriate energy equation is $$\dot s=0$$, where the dot is the time derivative and $$s$$ is the specific entropy. Combining this energy equation with the ideal gas law (equation of state) yields the equation you provide above: $$p\rho^{-\gamma}=\text{const}$$. Your equation for a liquid is purely an equation of state, as there is no explicit assumption about heat exchange. Practically, it is also adiabatic.

My question is, why is it required that we use such different approaches for liquids and gases - can't we approximate the liquid fluid elements as being adiabatically compressed/expanded, or can't we assume a constant bulk modulus of the gas?

In principle, yes, you can use either approach for either medium. The difference is the level of practicality. In liquids the equation of state can be relatively difficult to define explicitly, and so we assume that the equation of state, regardless of its actual form, is differentiable for the conditions we are interested in. We are not able to calculate the bulk modulus from first principles, so we just say you need to measure that. This is a macroscopic approach.

A perfect gas, on the other hand, does have a well known and easy-to-work-with equation of state: the ideal gas law. Therefore, we are able to calculate the wave speed (and other quantities) from first principles with relative ease. This is a microscopic approach, is more powerful, and it provides a deeper understanding of the underlying physics, and is therefore more desirable.

From a pedagogical standpoint, I would want to have both of these approaches explained. First, the perfect gas would be explained to give a solid understanding of what is going on in a sound wave. At this point I would expect a discussion about the importance of an equation of state and the difference between microscopic and macroscopic derivations. Then, the liquid case would be provided as an example of what you can do without a fundamental equation of state.

• Excellent answer thank you! Commented Mar 24, 2021 at 15:07