Wave Propagation Theory

Abstract

The wave equation in an ideal fluid can be derived from hydrodynamics and the adiabatic relation between pressure and density. The equation for conservation of mass, Euler’s equation (Newton’s second law), and the adiabatic equation of state are respectively

$$\begin{array}{rcl} & \frac{\partial \rho } {\partial t} = -\mathbf{\nabla }\cdot \rho \mathbf{v},&\end{array}$$

(2.1)

$$\begin{array}{rcl} & \frac{\partial \mathbf{v}} {\partial t} + \left (\mathbf{v} \cdot \nabla \right )\mathbf{v} = -\frac{1} {\rho }\,\nabla p(\rho ),&\end{array}$$

(2.2)

$$\begin{array}{rcl} & p = {p}_{0} + \rho ^{\prime}{\left [\frac{\partial p} {\partial \rho }\right ]}_{S} + \frac{1} {2}\,{(\rho ^{\prime})}^{2}{\left [\frac{{\partial }^{2}p} {\partial {\rho }^{2}}\right ]}_{S} + \cdots \,&\end{array}$$

(2.3)

and for convenience we define the quantity

$${c}^{2} \equiv {\left [\frac{\partial p} {\partial \rho }\right ]}_{S},$$

(2.4)

where c will turn out to be the speed of sound in an ideal fluid. In the above equations, ρ is the density, v the particle velocity, p the pressure, and the subscript S denotes that the thermodynamic partial derivatives are taken at constant entropy. The ambient quantities of the quiescent (time independent) medium are identified by the subscript 0. We use small perturbations for the pressure and density, p = p₀ + p′, ρ = ρ₀ + ρ′, and note that v is also a small quantity; that is, the particle velocity which results from density and pressure perturbations is much smaller than the speed of sound.