Talk:Nonlinear acoustics

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1 In layman's terms
2 Mathematical model
3 Hamilton & Blackstock Nonlinear Acoustics Text
4 Nonlinearity parameter references

[edit] In layman's terms

I think Introduction is a more suitable name for this section. More encyclopediac and the content is closer to a technical lead in to the topic, than anything that would appeal to the layman.

Also, different frequencies may be attentuated by different amounts does not describe a non-linear effect, I believe. It is a characteristic of linear systems. LightYear 06:19, 15 January 2007 (UTC)

[edit] Mathematical model

Working on this at the moment, but not ready to publish yet. LightYear 06:19, 15 January 2007 (UTC)

Traditionally, the propagation of an acoustic wave is modeled using an [[LTI_system_theory|LTI (Linear Time-Invariant)] system. One important characteristic of such a system is that if the input to the system is a sinusoid, then the output of the system will also be a sinusoid, perhaps with a different amplitude and a different phase, but always with the same frequency.

If our acoustic wave consists of a signal frequency, than it may be represented as $x (t) = A sin(2π f t)$ where $A$ is the amplitude, $f$ is the frequency and $t$ is the time variable. Under an LTI system model, we can represent the wave's medium by $H(\bullet)$ . Then,

H (x (t)) = B sin(2π f t + φ)

where $B$ is a new amplitude and $φ$ is a phase shift or more simply, a time delay. The important thing to note is that the output of this LTI system consists of a single sinusoid at the same frequency as the input.

In general, most real-life mediums for the passage of acoustic waves are not linear and cannot be accurately modeled by an LTI system. To accurately model such a medium, a non-linear model must be used. Consider first, a simplified derivation of the one-dimensional linear model.

Imagine the medium in the model consists of particles separated by springs. If we consider the motion of one of those particles as the wave energy impinges on it, then from Newton's Laws of Motion, we know that

$F = m \ddot{x} ,$

where $F$ is the resultant force on the particle, $m$ is its mass and $a = \frac{d^2 x}{dt^2} = \ddot{x}$ is its acceleration. In a linear model, we know from Hooke's Law that the resultant force on the particle is

F = - k x,

where $k$ is a contant that relates to the density of the material. Combining the two equations, we find

$m \ddot{x} + kx = 0$

$\ddot{x} + \frac{k}{m}x = 0 .$

Solving the linear differential equation and simplifying, we arrive at a solution for $x (t)$ , the motion of the particle.

$x(t) = \alpha \cos ( \sqrt{k/m} t ) + \beta \sin ( \sqrt{k/m} t )$

x (t) = A sin(ω t + φ),

where $α$ , $β$ , $A$ , $ω$ and $φ$ are all constants. The result represents the simple harmonic motion that the particle undergoes in the linear system. This gives the system the properties found in the LTI_system_model.

Now consider the non-linear system and in particular, the increasing density of the medium at the peaks of the incident acoustic wave. We may wish to model the tendancy of a wave to travel faster during these periods of higher density by considering a non-linear spring. Therefore, instead of assuming the particle is governed by Hooke's Law, which only applies to linear systems, we may instead model the force on the particular as a non-linear function of its displacement from its resting position. That is, the resultant force on the particle is

F = f (x),

for some non-linear function $f$ . Continuing the derivation using the new resultant force, we find

$m \ddot{x} + f(x) = 0$

$\ddot{x} + \frac{f(x)}{m} = 0 .$

Solutions to the resulting non-linear differential equation are not trivial.

In practice, generating a non-linear model is difficult, and there are no generic properties one can assume the model will exhibit. Finding an accurate, non-linear model of various mediums under acoustic wave propagation is a popular research topic. Many solutions can only be evaluated numerically...

While the summary of an LTI is useful, perhaps it could be a bit more brief and refer the reader to the full article if they are interested in a more complete description.Dudecon 19:54, 15 January 2007 (UTC)

Agree we don't want to concern ourselves with the linear model in this non-linear article much, but that's a pretty short summary and included a treatment not found in LTI_system_theory, which is also linked. LightYear 06:51, 16 January 2007 (UTC)

[edit] Hamilton & Blackstock Nonlinear Acoustics Text

The current text used at Penn State and the University of Texas (and probably other universities) for their nonlinear acoustics classes is by Hamilton & Blackstock. This article needs some serious attention. People looking to rewrite this article will find a lot of useful information in this book.

[edit] Nonlinearity parameter references

F. Plantier, J. L. Daridon, and B. Lagourette, Measurement of the B/A nonlinearity parameter under high pressure: Application to water, J. Acoust. Soc. Am. 111 (2), February 2002. 707-715. This paper measures the nonlinearity parameter B/A for pure water for pressures from 1 to 500 bar, and temperature 30-100 C. The parameter B/A at 1 bar and 30C is 5.38 +/- 0.12. Robert Hiller 01:52, 21 February 2007 (UTC)

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Talk:Nonlinear acoustics

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[edit] In layman's terms

[edit] Mathematical model

[edit] Hamilton & Blackstock Nonlinear Acoustics Text

[edit] Nonlinearity parameter references

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