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Acoustics: Sound Fields, Transducers and Vibration, 2nd Edition by Tim Mellow, Leo Beranek

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2.2. Derivation of the wave equation

2.2.1. The equation of motion

If we write Newton's second law for a small volume of gas located in a homogeneous medium, we obtain the equation of motion, or the force equation as it is sometimes called. Imagine the small volume of gas to be enclosed in a box with weightless flexible sides (Fig. 2.1).
One-dimensional derivation  [1] Three-dimensional derivation  [2]
Let us suppose that the box is situated in a medium where the sound pressure p increases from left to right at a space rate of ∂p/∂x (see Fig. 2.1). Let us suppose that the box is situated in a medium (see Fig. 2.1) where the sound pressure p changes in space at a space rate of
grad p = p = i p x + j

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