acoustic theory is the field relating to mathematical description of sound waves. it is derived from fluid dynamics. see acoustics for the engineering approach. the propagation of sound waves in a fluid (such as water) can be modeled by an equation of motion (conservation of momentum) and an equation of continuity (conservation of mass). with some simplifications, in particular constant density, they can be given as follows: : \begin{align} \rho_0 \frac{\partial \mathbf{v}}{\partial t} + \nabla p & = 0 \qquad \text{(momentum balance)} \\ \frac{\partial p}{\partial t} + \kappa~\nabla \cdot \mathbf{v} & = 0 \qquad \text{(mass balance)} \end{align} where p(\mathbf{x}, t) is the acoustic pressure and \mathbf{v}(\mathbf{x}, t) is the acoustic fluid velocity vector, \mathbf{x} is the vector of spatial coordinates x, y, z, t is the time, \rho_0 is the static mass density of the medium and \kappa is the bulk modulus of the medium. the bulk modulus can be expressed in terms of the density and the speed of sound in the medium (c_0) as : \kappa = \rho_0 c_0^2 ~. if the acoustic fluid velocity field is irrotational, \nabla \times \mathbf{v}=\mathbf{0}, then the acoustic wave equation is a combination of these two sets of balance equations and can be expressed as douglas d. reynolds. (1981). engineering principles in acoustics, allyn and bacon inc., boston. : \cfrac{\partial^2 \mathbf{v}}{\partial t^2} - c_0^2~\nabla^2\mathbf{v} = 0 \qquad \text{or} \qquad \cfrac{\partial^2 p}{\partial t^2} - c_0^2~\nabla^2 p = 0, where we have used the vector laplacian, \nabla^2 \mathbf{v} = \nabla(\nabla \cdot \mathbf{v}) - \nabla \times (\nabla \times \mathbf{v}) . the acoustic wave equation (and the mass and momentum balance equations) are often expressed in terms of a scalar potential \varphi where \mathbf{v} = \nabla\varphi. in that case the acoustic wave equation is written as : \cfrac{\partial^2 \varphi}{\partial t^2} - c_0^2~\nabla^2 \varphi = 0 and the momentum balance and mass balance are expressed as : p + \rho_0~\cfrac{\partial\varphi}{\partial t} = 0 ~;~~ \rho + \cfrac{\rho_0}{c_0^2}~\cfrac{\partial\varphi}{\partial t} = 0 ~. ==derivation of the governing equations== the derivations of the above equations for waves in an acoustic medium are given below. ===conservation of momentum=== the equations for the conservation of linear momentum for a fluid medium are : \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \nabla \cdot\boldsymbol{s} + \rho\mathbf{b} where \mathbf{b} is the body force per unit mass, p is the pressure, and \boldsymbol{s} is the deviatoric stress. if \boldsymbol{\sigma} is the cauchy stress, then : p := -\tfrac{1}{3}~\text{tr}(\boldsymbol{\sigma}) ~;~~ \boldsymbol{s} := \boldsymbol{\sigma} + p~\boldsymbol{\mathit{1}} w
Acoustics Fluid_dynamics