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Acoustic theory

Classical acoustic theory derives from fluid mechanics, and centers on the mathematical description of sound waves. See acoustics for the engineering approach.

In approaching the description of a sound wave the mathematics never gives the whole story. The subtleties of thermodynamics are difficult enough to recommend a gradual familiarization with some related problems of vibration such as arise in mechanical sound production: motion of a spring[?] vibration of a string[?] equation of motion harmonic.

Besides the the math tools that we will use again, the preceding examples help inform the beginner's physical intuition with analogies to the periodic compression domains.

The propagation of sound waves in air can be modeled by an equation of motion and an equation of continuity. With some simplifications, they can be given as follows:

$\rho_0 \frac{\partial}{\partial t} \mathbf{v}(\mathbf{x}, t) + \nabla p(\mathbf{x}, t) = 0$
$\frac{\partial}{\partial t} p(\mathbf{x}, t) + \rho_0 c^2 \nabla \cdot \mathbf{v}(\mathbf{x}, t) = 0$
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 density of air and $c$ is the speed of sound in air.

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