# 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* [^1] and his subsequent
texts on the subject Nonlinear Acoustics.[^2] It is worthwhile to note
that in Hamilton's text on Nonlinear Acoustics,[^3] Beyer attributes the
concept to an earlier work by Fox and Wallace (1954).[^4]

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, *ρ~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.[^5][^6][^7] 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*)*~ρ~o~,s~o~~* =
*c~o~*^2^, has been applied to shown: *A* = *ρ*~o~ *c*~o~^2^. 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*~o~^2^ 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:[^8]

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

## 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,[^9] 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~o~^2^* =
*γ*(*P~o~*+*D*) / *ρ~o~*. Including this expression to simplify
*∂p*/*∂ρ*, and continuing on to calculate *∂^2^p* / *∂ρ^2^* gives:

\
$$\frac{\partial P}{\partial \rho} = c_o^2\left(\frac{\rho}{\rho_o} \right )^{\gamma-1}$$

$$\frac{\partial^2P}{\partial \rho^2} = c_o^2  \frac{\left(\gamma-1 \right )}{\rho_o}  \left(\frac{\rho}{\rho_o} \right )^{\gamma-2}$$

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.[^10]

\
:{\| class=\"wikitable\" style=\"text-align: center;\" \|+ Table 2:
Sample *B/A* values for fluids. \|- ! Material !! *B/A* !! Ref. Temp.
^o^C !! Reference \|- \| style=\"width: 160px;\" \| Diatomic Gases (Air)
\|\| style=\"width: 60px;\" \| 0.4 \|\| 20 \|\| [^11] \|- \| Distilled
Water \|\| 5.0 \|\| 20 \|\| [^12] \|- \| Distilled Water \|\| 5.4 \|\|
40 \|\| [^13] \|- \| Salt Water \|\| 5.3 \|\| 20 \|\| [^14] \|- \|
Ethanol \|\| 10.5\|\| 20 \|\| [^15] \|}\
:{\| class=\"wikitable\" style=\"text-align: center;\" \|+ Table 2:
Sample *B/A* values for organic materials. \|- ! Material !! *B/A* !!
Ref. Temp. ^o^C !! Reference \|- \| style=\"width: 160px;\" \| Glycerol
\|\| style=\"width: 60px;\" \| 9.1 \|\| 30 \|\| [^16] \|- \| Hemoglobin
(50%) \|\| 7.6 \|\| 30 \|\| [^17] \|- \| Liver \|\| 6.5 \|\| 30 \|\|
[^18] \|- \| Fat \|\| 9.9 \|\| 30 \|\| [^19] \|- \| Collagen \|\| 4.3
\|\| 25 \|\| [^20] \|}

## References


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