Acoustics

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Basic wave theory

Although not fundamentally difficult to understand, there are a number of alternate techniques used to analyze wave motion which could seem overwhelming to a novice at first. Therefore, only 1-D wave motion will be analyzed to keep most of the mathematics as simple as possible. This analysis is valid, with not much error, for the majority of pipes and enclosures encountered in practice.

Plane-wave pressure distribution in pipes

The most important equation used is the wave equation in 1-D form (See [1],[2], 1-D Wave Equation, for information).

Therefore, it is reasonable to suggest, if plane waves are propagating, that the pressure distribution in a pipe is given by:

$\mathbf{p}=\mathbf{Pi}e^{j[\omega t-kx]}+\mathbf{Pr}e^{j[\omega t+kx]}$

where Pi and Pr are incident and reflected wave amplitudes respectively. Also note that bold notation is used to indicate the possibility of complex terms. The first term represents a wave travelling in the +x direction and the second term, -x direction.

Since acoustic filters or mufflers typically attenuate the radiated sound power as much as possible, it is logical to assume that if we can find a way to maximize the ratio between reflected and incident wave amplitude then we will effectively attenuated the radiated noise at certain frequencies. This ratio is called the reflection coefficient and is given by:

$\mathbf{R}=\left( \frac{\mathbf{Pr}}{\mathbf{Pi}} \right)$

It is important to point out that wave reflection only occurs when the impedance of a pipe changes. It is possible to match the end impedance of a pipe with the characteristic impedance of a pipe to get no wave reflection. For more information see [1] or [2].

Although the reflection coefficient isn't very useful in its current form since we want a relation describing sound power, a more useful form can be derived by recognizing that the power intensity coefficient is simply the magnitude of reflection coefficient square [1]:

$R_{\pi}=\left|\mathbf{R}\right|^2$

As one would expect, the power reflection coefficient must be less than or equal to one. Therefore, it is useful to define the transmission coefficient as:

$T_{\pi}=\left(1-R_{\pi}\right)$

which is the amount of power transmitted. This relation comes directly from conservation of energy. When talking about the performance of mufflers, typically the power transmission coefficient is specified.