Campos *et al.* [1] derived an acoustic–vortical wave equation for the study of acoustics within a sheared flow, *U*_{0}(*y*)*e*_{x}, over an acoustic lining with a constant cross flow, *V*_{0}*e*_{y}, through the lining. Unfortunately, their derivation makes inconsistent assumptions, and the resulting wave equation is therefore incorrect. This comment points out the error, and derives a corresponding equation using the same approximations as Campos *et al.*

The situation considered is as shown in figure 1*a*. A mean flow *v*_{0}=*U*_{0}(*y*)*e*_{x}+*V* _{0}*e*_{y} flows across an acoustic liner located along *y*=0. This mean flow should satisfy the inviscid governing equations of conservation of mass, momentum and entropy,
*v*_{0} is solenoidal, **∇**⋅*v*_{0}=0, and hence conservation of mass (0.1a) gives *v*_{0}⋅**∇***ρ*_{0}=0. The momentum equation (0.1b) implies that
*ρ*_{0}*U*_{0}*V* _{0}(d*U*_{0}/d*y*)≠0 because *U*_{0}, *V* _{0} and d*U*_{0}/d*y* are all supposed non-zero. Therefore, the base flow assumed by Campos *et al.* [1] does not satisfy the governing equations. This leads to ambiguity in deriving a wave equation based on this mean flow, as will be seen below.

The expressions for the partial derivatives of *p*_{0} given in (0.2) are in general incompatible, meaning that (assuming (1/*ρ*_{0})(d*U*_{0}/d*y*) is not everywhere constant) there is no function *p*_{0} satisfying (0.2). In the linear shear case specifically considered by Campos *et al.* [1], shown in figure 1*b*, *y*<*L* but *y*>*L*, leading to a mean pressure jump across *y*=*L* for *x*≠0. In order to proceed, the approximation made by Campos *et al.* [1] is that
*p*_{0} with *x* may be considered to be small, and therefore *p*_{0} may be approximated as constant. While this is true for the *numerical value* of *p*_{0}, it is not true for the *derivative* ∂*p*_{0}/∂*x*, which remains non-zero and potentially of significant magnitude. This is in effect a Boussinesq approximation, where both *p*_{0} and ∂*p*_{0}/∂*x* are considered constant and non-zero. Unfortunately, Campos *et al.* [1] assume both that **∇***p*_{0}=0 (in deriving their eqn (2.18)) and that *v*_{0}⋅**∇***v*_{0}=*V* _{0}d*U*_{0}/d*y**e*_{x} (in deriving their eqn (2.21*a*)), which are inconsistent with conservation of *x*-momentum (0.1b above). Their derived wave equation (eqn (2.25) of [1]) is therefore incorrect, as seen in §2 below.

Even with the Boussinesq assumption (0.3), the mean flow entropy equation (0.1c) remains unsatisfied. The mean flow assumed by Campos *et al.* [1] therefore requires a rather unphysical steady external cooling in order to be realized, as will be seen next.

## 1. A consistent (albeit artificial) mean flow

In order to have a consistent mean flow, and therefore a unique linearization of the governing equations about that mean flow, we will assume here a steady heat source *Q*_{0}. While this is certainly not the only possible consistent extension of [1], it is the simplest extension that allows the same velocity and density as Campos *et al.* [1], eqns (2.5) and (2.11a). The full governing equations are then
*γ*=*c*_{p}/*c*_{v} is the ratio of specific heats.

Tildes here denote a total quantity, which is considered as a sum of a steady mean flow and a small unsteady perturbation, e.g. *p*_{0} gives the consistent solution *v*_{0}=*U*_{0}(*y*)*e*_{x}+*V* _{0}*e*_{y} with *ρ*_{0} constant, *p*_{0} approximated as constant and
*Q*_{0} is needed for this flow to exist in practice.

It should be noted that the sound speed *α*=*γ*), and thus we may not neglect gradients of *et al.* [1], eqn (2.13) did neglect gradients of

## 2. Derivation of the ‘wave equation’ following [1]

Linearizing the governing equations (1.1) about the steady mean flow (1.3) gives
*D*_{0}/*D*_{0}*t*=∂/∂*t*+*v*_{0}⋅**∇** is the material derivative with respect to the mean flow, the velocity perturbation is ** v**′=

**′**

*u*

*e*_{x}+

*v*′

*e*_{y}, and

*et al.*[1], and originates from ∂

*p*

_{0}/∂

*x*given in (0.2). Were it true that

**∇**

*p*

_{0}≡0, as assumed by Campos

*et al.*[1], then both the term marked * in (2.1b) and the term marked † in (2.1c) would be identically zero. Campos

*et al.*[1] included the term marked * and excluded the term marked †, showing that their wave equation is inconsistently derived.

We now follow the procedure of Campos *et al.* [1]. Noting that
*D*_{0}*ρ*′/*D*_{0}*t* from (2.1c) and then taking *x*-momentum perturbation equation (2.1b) to substitute for *D*_{0}*u*′/*D*_{0}*t* and then taking *D*_{0}(2.3)/*D*_{0}*t* yields
*V* _{0}=0 recovers^{1} the Pridmore-Brown equation [2], while setting *U*_{0}(*y*) to be constant recovers the convected wave equation. By assuming a linear shear *U*_{0}(*y*)=*κy* and a perfect gas, equation (2.5) ‘simplifies’ to
*et al.* [1], eqn (2.25). We must therefore conclude that the wave equation [1] is based on is erroneous.

It may, in certain cases, be possible to rearrange (2.5) or (2.8) into a wave equation in only one variable, say *p*′; however, doing so would yield an unhelpfully complicated equation of even higher order for this rather artificial mean flow, and is therefore not pursued further here.

## 3. A linear shear boundary layer

Campos *et al.* [1] considered a linear boundary layer of thickness *L* between the liner and a region of uniform flow, as shown in figure 1*b*. As already commented, the mean flow pressure *p*_{0} is discontinuous at *y*=*L*. Moreover, equation (2.5) includes second and third derivatives of *U*_{0}(*y*), so that delta functions *δ*(*y*−*L*) and *δ*′(*y*−*L*) would be introduced in this case, which would cause *p*′ to have a discontinuous derivative when crossing *y*=*L*. This flow profile is therefore not equivalent to solving a uniform flow *U*_{0}(*y*)=*κy* and matching the solutions together at *y*=*L*, as was done by Campos *et al.* [1], eqn (3.19).

## Competing interests

I have no competing interests.

## Funding

The author is funded by a Royal Society University Research Fellowship (UF100844), and gratefully acknowledges their support.

## Footnotes

- Received February 28, 2016.
- Accepted July 20, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.