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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) reply

I've added a simple description of the KZK model. That's probably all that is necessary in a Wikipedia article. Please review and consider removing the expert and references tags. If there are no concerns in a month or so, I'll remove the tags. LightYear ( talk) 03:52, 10 November 2008 (UTC) reply

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.

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) reply

what abour siesmic waves? CorvetteZ51 13:39, 4 June 2007 (UTC) reply

Have added levitation link. What do acoustics have to do with siesmic waves? LightYear ( talk) 00:44, 1 December 2008 (UTC) reply

The following content by Libertariodemi requires some cleanup. In particular:

• refs.
• wikilinks.
• integration into article flow.
• make clear Pressure is ${\displaystyle P}$ and density is ${\displaystyle \rho }$?
• clarification of ${\displaystyle \partial _{\rho }}$. Appears to be differentiation, but notation does not appear at Notation for differentiation for example.
• relationship of EOS to existing content on Taylor series.
• rephrase to sound less like conversation missing its context.
• replace references to non-existant figure with something useful?
• remove LaTeX (eg. '$') syntax.
• 'stata condition' == 'state condition'?
• 'along the state pressure' == 'from the ambient state pressure'?

It is possible to write the equation of state (EOS) as a Taylor series expansion of Pressure as a function of density around the ambient state ${\displaystyle (P_{0},\rho _{0})}$ and under isentropic condition. An isentropic process (iso = "equal" (Greek); entropy = "disorder") is one during which the entropy of the system remains constant; for acoustic the isentropic assumption is valid when we neglect the heat flow (adiabatic medium) and the dissipation in the medium. Hence is important to underline that we are going to express the EOS as a Taylor second order approximation which is appropriate until is possible to neglect the third order terms.

${\displaystyle \,P(\rho )=P_{0}+\partial _{\rho }P(\rho )(\rho -\rho _{0})+{\frac {1}{2!}}\partial _{\rho }^{2}P(\rho )(\rho -\rho _{0})^{2}}$

So we can write that equation as :

${\displaystyle P(\rho )-P_{0}=c_{0}^{2}(\rho -\rho _{0})+{\frac {1}{2}}{\frac {c_{0}^{2}}{\rho _{0}}}{\frac {B}{A}}(\rho -\rho _{0})^{2}}$

where :

${\displaystyle c_{0}^{2}=\partial _{\rho }P(\rho )}$ is the sound speed for small signals

${\displaystyle A=\rho _{0}\partial _{\rho }P(\rho )}$

${\displaystyle {B}=\rho _{0}^{2}\partial _{\rho }^{2}P(\rho )}$

${\displaystyle {\frac {B}{A}}={\frac {\rho _{0}}{c_{0}^{2}}}\partial _{\rho }^{2}P(\rho )}$ And remember that all the derivates must be calculated around the ambient stata condition ${\displaystyle (P_{0},\rho _{0})}$.

There is also another equation that can be used in order to model NonLinear Acoustic phenomena and it is the Westervelt Equation where ${\displaystyle P'=P-P_{0}}$ so it represents the pressure variation along the state pressure:

${\displaystyle \partial _{k}^{2}P'={\frac {1}{c_{0}^{2}}}\partial _{t}^{2}P'-\rho _{0}\kappa _{0}^{2}\beta \partial _{t}^{2}P'^{2}.}$

This is the lossless equation but we can simply add one term in order to consider also a medium with loss proprieties. Anyhow the nonlinear effect is caused by the term that we can call the Nonlinear Term :

${\displaystyle \rho _{0}\kappa _{0}^{2}\beta \partial _{t}^{2}P'^{2}}$

Now considering that term in which lays the nonlinear effect caused to the pressure field we can apply a Fourier transform to understand what happens in the frequency domain. Considering ${\displaystyle \beta }$ a time independent coefficient from the theory on Fourier transform we can write

${\displaystyle \rho _{0}\kappa _{0}^{2}\beta \partial _{t}^{2}P'^{2}\Leftrightarrow -\rho _{0}\kappa _{0}^{2}\beta \omega ^{2}{\hat {P'}}(\omega )*{\hat {P'}}(\omega ).}$

Hence if we consider a Gaussian Modulated Pulse (G.M.P.) as the function that represents ${\displaystyle P'}$ in a time domain we can see the signal in the frequency domain just applying a Fourier transform. Than we can do the same thing with P², As it is possible to see from the pictures that the nonlinear term produces a first harmonic component that comes from the convolution in the frequency domain. It is also important to underline that the spectrum of P'² has to be multiply by f² ( Dotted line in Figure ) that will 'filter' the low frequency component generated by the nonlinear term. Starting from this construction and assuming now the pressure as a signal formed by a sum of two gaussian centered one at f0 and at 2f0 we will end up with a frequency nonlinear term . In this way is easy to get an idea on the harmonics formation during the propagation of a sound wave trough a tissue.

LightYear ( talk) 00:44, 1 December 2008 (UTC) reply

This article has come a long ways in the past seven years. Looks great! Keep it up everybody. Dudecon ( talk) 02:13, 2 August 2014 (UTC) reply

As a physicist, I am struggling to understand some concepts that are being presented as fundamental to this "branch of physics".

I don't have access to [1] and am using Enflo and Hedberg^[2] as my reference.

It states: "The state of the fluid at a point  r = (x₁,x₂,x₃) at time t is determined by six variables: pressure (p), density (ρ), temperature (T) and three velocity components  v = (v₁,v₂, v₃)"

Now p, ρ and T are the scalar properties of a fluid regarded as a continuum, whilst velocity, being a vector property of position and time, is a property of Lagrangian mechanics.

Continuum mechanics treats fluid as a continuous medium, ignoring the discrete nature of its constituent molecules. This is valid when the characteristic length scale of the phenomenon of study is much larger than the mean free path of the molecules (typically on the order of nanometers). It works well for describing large-scale fluid flows, like those of water in pipes or rivers.

Lagrangian mechanics can be used to track individual fluid particles and their movement. This approach would be relevant to analyse the behaviour of specific particles, or to study phenomena on the scale of the mean free path, such as Brownian motion or molecular interactions.

Both perspectives can coexist and describe the same fluid system depending on the level of detail and scale of interest. For example, a continuum model can be used to predict the overall flow pattern of a river (large scale) and then a Lagrangian model could be used to track the movement of individual pollutants within the flow (smaller scale).

In summary:

Continuum mechanics - Valid for large scales (>> mean free path), describing overall flow behaviour.

Lagrangian mechanics - Relevant for small scales (comparable to mean free path), focusing on individual particles.

However, it appears that nonlinear acoustics combines these two valid but mutually incompatible realms of mechanics to argue its case.

Bearing in mind the divergent scales of the two realms, I don't think it is scientifically valid to apply both continuum and Lagrangian mechanics explain a single phenomenon.

[1] Hamilton, M.F.; Blackstock, D.T. (1998). Nonlinear Acoustics. Academic Press. ISBN 0-12-321860-8.

[2] Enflo, B. O.; Hedberg, C. M. (2002) Theory of Nonlinear Acoustics in Fluids  ISBN: 0-306-48419-6 p.12 Gpsanimator ( talk) 21:20, 16 December 2023 (UTC) reply

Nonlinear acoustics (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes.

This opening sentence makes no sense. The term sufficiently demands the question...sufficiently large for what? Gpsanimator ( talk) 06:15, 7 January 2024 (UTC) reply