Engineering Acoustics/The Acoustic Parameter of Nonlinearity

The Acoustic Parameter of Nonlinearity

For many applications of nonlinear acoustic phenomena the magnitude of an acoustic medium's nonlinearity can be quantified using a single value known as the parameter of nonlinearity. This convention is often attributed to Robert T. Beyer, stemming from his influential 1960 paper titled Parameter of Nonlinearity in Fluids and his subsequent texts on the subject Nonlinear Acoustics. It is worthwhile to note that in Hamilton’s text on Nonlinear Acoustics, Beyer attributes the concept to an earlier work by Fox and Wallace (1954).

The mathematical grounds for the parameter of nonlinearity stem from the Taylor series relating the perturbed pressure, p ', to the perturbed density, ρ '. Physically, these small perturbation values are referenced to an ambient state defined by a density value, &rho _o, and a constant entropy, s = s_o.

$p'=A\left({\frac {\rho '}{\rho _{o}}}\right)+{\frac {B}{2!}}\left({\frac {\rho '}{\rho _{o}}}\right)^{2}+{\frac {C}{3!}}\left({\frac {\rho '}{\rho _{o}}}\right)^{3}+...$

where the coefficients A, B, C, give the magnitude for each term in the Taylor expansion. As the coefficients B, C, apply to squared and cubic terms they represent a nonlinearity in the relation between pressure, p ', and density ρ '. Values for A, B, and C can be determined experimentally using a several techniques. It is also possible to use the Taylor series definition of A, B, and C to calculate values when a constitutive relation between pressure and density is known. Using the ideal gas or Tait equation of state for this purpose is discussed in a subsequent section. The Taylor series coefficient definitions are:

$A=\rho _{o}\left({\frac {\partial P}{\partial \rho }}\right)_{\rho _{o},s_{o}}=\rho _{o}c_{o}^{2}$

$B=\rho _{o}^{2}\left({\frac {\partial ^{2}P}{\partial \rho ^{2}}}\right)_{\rho _{o},s_{o}}$

$C=\rho _{o}^{3}\left({\frac {\partial ^{3}P}{\partial \rho ^{3}}}\right)_{\rho _{o},s_{o}}$

where a definition for ambient sound speed, (∂p/∂ρ)_ρo,so = c_o², has been applied to shown: A = ρ_o c_o². For a majority of problems in nonlinear acoustics, utilizing the first two terms of this expansion is sufficient to represent the range of density perturbations encountered. In this case the series reduces to:

$p'=A\left({\frac {\rho '}{\rho _{o}}}\right)+{\frac {B}{2}}\left({\frac {\rho '}{\rho _{o}}}\right)^{2}$

This truncation leads to what is commonly referred to as the parameter of nonlinearity in a fluid, or B/A. By factoring A = ρ_o c_o² from both terms in the truncated series, the physical relevance of B/A becomes more apparent:

$p'=\rho 'c_{o}^{2}\left[1+{\frac {1}{2}}{\frac {B}{A}}\left({\frac {\rho '}{\rho _{o}}}\right)\right]$

This expression shows the ratio B/A quantifies the influence of nonlinearity on the local pressure perturbation for a given state, ρ' / ρ_o. Similarly, it can also be shown the parameter B/A quantifies the variation of local sound speed as a function of perturbed density according to:

$c=c_{o}\left[1+{\frac {1}{2}}{\frac {B}{A}}\left({\frac {\rho '}{\rho _{o}}}\right)\right]$

B/A In Relation to Power Law Equations of State

For power law equations of state (EOS), such as the Tait-Kirkwood EOS for liquids, or the isentropic compression of an ideal gas, the B/A parameter can be related to know power law coefficients. To demonstrate this relation, the partial derivatives of pressure with respect to density, ∂p/∂ρ, are calculated for the Tait EOS and are then applied in the Taylor series for perturbed pressure, p' . The end result when using the ideal gas EOS instead of Tait is identical to that shown.

${\frac {P+D}{P_{o}+D}}=\left({\frac {\rho }{\rho _{o}}}\right)^{\gamma }$

${\frac {\partial P}{\partial \rho }}=\left(P_{o}+D\right){\frac {\gamma }{\rho _{o}}}\left({\frac {\rho }{\rho _{o}}}\right)^{\gamma -1}$

Evaluating the first derivative at the ambient state (ρ = ρ_o) gives a useful equation for the linear sound speed: c_o² = γ(P_o+D) / ρ_o. Including this expression to simplify ∂p/∂ρ, and continuing on to calculate ∂²p / ∂ρ² gives:

${\frac {\partial P}{\partial \rho }}=c_{o}^{2}\left({\frac {\rho }{\rho _{o}}}\right)^{\gamma -1}$

${\frac {\partial ^{2}P}{\partial \rho ^{2}}}=c_{o}^{2}{\frac {\left(\gamma -1\right)}{\rho _{o}}}\left({\frac {\rho }{\rho _{o}}}\right)^{\gamma -2}$

Incorporating these first and second derivatives into a Taylor series of p' on ρ' gives equations derived from the power law EOS that can be compared to the previous series containing the B/A parameter.

$p'=c_{o}^{2}\rho '+{\frac {c_{o}^{2}\left(\gamma -1\right)}{2\rho _{o}}}\rho '^{2}+...$

$p'=c_{o}^{2}\rho '\left[1+{\frac {\left(\gamma -1\right)}{2}}\left({\frac {\rho '}{\rho _{o}}}\right)\right]+...$

This final equation showns the two Taylor series for p' are identical, with (γ-1) in place of B/A. Thus, for liquids obeying the Tait-Kirkwood EOS and ideal gasses under isentropic conditions with known adiabatic index:

${\frac {B}{A}}=\gamma -1$

Sample Values for B/A

Table 1 provides a sample of values for the B/A parameter in various gases, liquids, and biological materials. Reference temperatures are included along with each sample as the value of B/A for a particular material will vary with temperature. Several organic materials are included in Table 2 as nonlinear acoustic effects are particularly prominent in biomedical ultrasound applications.

Table 2: Sample B/A values for fluids.

Material	B/A	Ref. Temp. oC	Reference
Diatomic Gases (Air)	0.4	20
Distilled Water	5.0	20
Distilled Water	5.4	40
Salt Water	5.3	20
Ethanol	10.5	20

Table 2: Sample B/A values for organic materials.

Material	B/A	Ref. Temp. oC	Reference
Glycerol	9.1	30
Hemoglobin (50%)	7.6	30
Liver	6.5	30
Fat	9.9	30
Collagen	4.3	25