2. Formulation and governing equations

We consider a thin symmetric aerofoil aligned parallel to the $x$ axis, with surface $y=\pm {\it\epsilon}y(x)$ , $0\leqslant x\leqslant 2$ . Here, lengths have been non-dimensionalised using the aerofoil semi-chord $b^{\ast }$ ( $^{\ast }$ denotes dimensional quantities). The aerofoil sits in a mean shear flow of velocity $\boldsymbol{U}$ , which is aligned parallel to the aerofoil chord at infinity (velocities are non-dimensionalised using $U_{\infty }^{\ast }$ , the uniform mean flow speed at infinity in the transverse direction, and we explicitly exclude the case $U_{\infty }^{\ast }=0$ ). We work in the orthogonal $({\it\phi},{\it\psi})$ coordinate system, where ${\it\psi}$ is the non-dimensional mean-flow stream function and ${\it\phi}$ is the non-dimensional pseudo-velocity potential, chosen such that surfaces of constant ${\it\phi}$ and ${\it\psi}$ are orthogonal. The origin in $({\it\phi},{\it\psi})$ space is located at the leading edge of the aerofoil. This coordinate system has the advantage of mapping the aerofoil surface onto the flat plate ${\it\psi}=0$ , $0\leqslant {\it\phi}\leqslant {\it\phi}_{e}$ , where ${\it\phi}_{e}$ must be calculated from the mean-flow solution. Far upstream the steady shear flow velocity is $U_{0}({\it\psi})\boldsymbol{e}_{{\it\phi}}$ , where $\boldsymbol{e}_{{\it\phi}}$ is the unit vector in the ${\it\phi}$ direction, and the shear profile $U_{0}({\it\psi})$ is a given function (with the property that $U_{0}\rightarrow 1$ as ${\it\psi}\rightarrow \pm \infty$ ). The presence of the thin aerofoil distorts the incident mean flow, and we write the total mean velocity as $\left(U_{0}({\it\psi})+{\it\epsilon}q({\it\phi},{\it\psi})+O({\it\epsilon}^{2})\right)\boldsymbol{e}_{{\it\phi}}$ . The local Mach number is denoted $M({\it\psi})$ , which takes the value $M_{\infty }$ as ${\it\psi}\rightarrow \pm \infty$ ; in what follows we shall be considering low-Mach-number flow only, with $M=O({\it\epsilon}^{1/2})$ . This means that the steady flow around the aerofoil can be determined to $O({\it\epsilon})$ using incompressible thin-aerofoil theory, and it follows that (see Thwaites Reference Thwaites1960)

(2.1) $$\begin{eqnarray}q(z)=\frac{U_{0}(0)}{{\rm\pi}}\text{Re}\left[\int _{0}^{2}\frac{\text{d}y(x)/\text{d}x}{z-x}\,\text{d}x\right],\end{eqnarray}$$

where $z={\it\phi}+\text{i}{\it\psi}$ . Furthermore, the corrections to the otherwise uniform steady pressure, density and sound speed due to the presence of the aerofoil are of $O({\it\epsilon}M_{\infty }^{2})=O({\it\epsilon}^{2})$ , and to $O({\it\epsilon})$ are therefore ignored.

Let the incident gust have typical amplitude which is much less than ${\it\epsilon}$ , allowing linearisation about the mean flow, and dimensional frequency ${\it\omega}^{\ast }$ . In what follows we non-dimensionalise time using $b^{\ast }/U_{\infty }^{\ast }$ , to give non-dimensional hydrodynamic frequency ${\it\omega}={\it\omega}^{\ast }b^{\ast }/U_{\infty }^{\ast }$ , and we also introduce the non-dimensional acoustic frequency $k={\it\omega}M_{\infty }$ , where $M_{\infty }=U_{\infty }^{\ast }/c_{\infty }^{\ast }$ . We suppose $k$ is large, with preferred limit $k=O({\it\epsilon}^{-1})$ . The unsteady velocity, pressure and density are written in the form

(2.2) $$\begin{eqnarray}\{u,v,p,{\it\rho}\}({\it\phi},{\it\psi},t)=\{u,v,p,{\it\rho}\}({\it\phi},{\it\psi})\text{e}^{-\text{i}{\it\omega}t},\end{eqnarray}$$

and we make the one further assumption that the flow is isentropic, which means that the pressure and density fluctuations are connected by ${\it\rho}=M_{\infty }^{2}p$ . The idea now is to substitute this unsteady perturbation into the equations of mass and (inviscid) momentum conservation, linearised about the steady base flow. In order to transform these equations into $({\it\phi},{\it\psi})$ space, we use the well-known results for orthogonal curvilinear coordinates, see for example Batchelor (Reference Batchelor1967, Appendix 2). The metric elements, given by Finnigan (Reference Finnigan1983), for $({\it\phi},{\it\psi})$ -space are

(2.3a,b ) $$\begin{eqnarray}h_{{\it\phi}}={\it\zeta}|\boldsymbol{U}|,\quad h_{{\it\psi}}=|\boldsymbol{U}|,\end{eqnarray}$$

where ${\it\zeta}$ is defined by

(2.4) $$\begin{eqnarray}\frac{{\rm\Omega}}{|\boldsymbol{U}|^{2}}=\frac{\partial \log {\it\zeta}}{\partial {\it\psi}},\end{eqnarray}$$

and ${\rm\Omega}=-{\rm\nabla}_{\boldsymbol{x}}^{2}{\it\psi}$ is the mean vorticity ( $\boldsymbol{{\rm\nabla}}_{\boldsymbol{x}}$ denotes the differential operator with respect to non-dimensional physical coordinates). This leads to

(2.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}{\it\omega}u+{\it\zeta}|\boldsymbol{U}|^{2}\frac{\partial u}{\partial {\it\phi}}-{\rm\Omega}v+{\it\zeta}|\boldsymbol{U}|\frac{\partial p}{\partial {\it\phi}}=-u{\it\zeta}\frac{\partial }{\partial {\it\phi}}\left(\frac{|\boldsymbol{U}|^{2}}{2}\right), & \displaystyle\end{eqnarray}$$

(2.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle -\,\text{i}{\it\omega}v+{\it\zeta}|\boldsymbol{U}|^{2}\frac{\partial v}{\partial {\it\phi}}+|\boldsymbol{U}|\frac{\partial p}{\partial {\it\psi}}=v{\it\zeta}\frac{\partial }{\partial {\it\phi}}\left(\frac{|\boldsymbol{U}|^{2}}{2}\right)-2v\left({\rm\Omega}+\frac{\partial }{\partial {\it\psi}}\left(\frac{|\boldsymbol{U}|^{2}}{2}\right)\right), & \displaystyle\end{eqnarray}$$

(2.5c ) $$\begin{eqnarray}\displaystyle & \displaystyle -\,\text{i}{\it\omega}{\it\rho}+{\it\zeta}|\boldsymbol{U}|\frac{\partial u}{\partial {\it\phi}}+|\boldsymbol{U}|\frac{\partial v}{\partial {\it\psi}}+{\it\zeta}|\boldsymbol{U}|^{2}\frac{\partial {\it\rho}}{\partial {\it\phi}}=\frac{v}{|\boldsymbol{U}|}\left({\rm\Omega}+\frac{\partial }{\partial {\it\psi}}\left(\frac{|\boldsymbol{U}|^{2}}{2}\right)\right)+u{\it\zeta}\frac{\partial |\boldsymbol{U}|}{\partial {\it\phi}} & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Since we are considering the case of a parallel shear flow disturbed by a thin aerofoil, we make the expansions

(2.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}|\boldsymbol{U}|^{2}=U_{0}({\it\psi})+{\it\epsilon}N_{1}({\it\phi},{\it\psi}), & \displaystyle\end{eqnarray}$$

(2.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\rm\Omega}=-U_{0}\frac{\text{d}U_{0}}{\text{d}{\it\psi}}+{\it\epsilon}N_{2}({\it\phi},{\it\psi}), & \displaystyle\end{eqnarray}$$

(2.6c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}|\boldsymbol{U}|=1+{\it\epsilon}N_{3}({\it\phi},{\it\psi}), & \displaystyle\end{eqnarray}$$

$N_{1,2,3}=O(1)$

$N_{2}\equiv 0$

$N_{1,3}$

$O({\it\epsilon})$

${\it\epsilon}N_{4,5,6}$

(2.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle -\,\text{i}ku+M({\it\psi})\frac{\partial u}{\partial {\it\phi}}+\frac{M({\it\psi})}{M_{\infty }}\frac{\text{d}M}{\text{d}{\it\psi}}v+{\it\epsilon}{\it\sigma}_{1}({\it\phi},{\it\psi})=-M_{\infty }\frac{\partial p}{\partial {\it\phi}}, & \displaystyle\end{eqnarray}$$

(2.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle -\,\text{i}kv+M({\it\psi})\frac{\partial v}{\partial {\it\psi}}+{\it\epsilon}{\it\sigma}_{2}({\it\phi},{\it\psi})=-M({\it\psi})\frac{\partial p}{\partial {\it\psi}}, & \displaystyle\end{eqnarray}$$

(2.7c ) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}kM_{\infty }p+\frac{\partial u}{\partial {\it\phi}}+\frac{M({\it\psi})}{M_{\infty }}\frac{\partial v}{\partial {\it\psi}}+M({\it\psi})M_{\infty }\frac{\partial p}{\partial {\it\phi}}+{\it\epsilon}{\it\sigma}_{3}({\it\phi},{\it\psi})=0, & \displaystyle\end{eqnarray}$$

(2.8a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\sigma}_{1}({\it\phi},{\it\psi})=-M_{\infty }\left(N_{4}-N_{1}\frac{\partial u}{\partial {\it\phi}}-N_{3}\frac{\partial p}{\partial {\it\phi}}\right), & \displaystyle\end{eqnarray}$$

(2.8b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\sigma}_{2}({\it\phi},{\it\psi})=-M_{\infty }\left(N_{5}-N_{1}\frac{\partial v}{\partial {\it\phi}}-q\frac{\partial p}{\partial {\it\psi}}\right), & \displaystyle\end{eqnarray}$$

(2.8c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\sigma}_{3}({\it\phi},{\it\psi})=-\!\left(N_{6}-N_{3}\frac{\partial u}{\partial {\it\phi}}-q\frac{\partial v}{\partial {\it\psi}}-N_{1}M_{\infty }^{2}\frac{\partial p}{\partial {\it\phi}}\right). & \displaystyle\end{eqnarray}$$

${\it\phi}$

(2.9) $$\begin{eqnarray}F({\it\alpha},{\it\psi})=\frac{1}{2{\rm\pi}}\int _{-\infty }^{\infty }\text{e}^{-\text{i}{\it\alpha}{\it\phi}}f({\it\phi},{\it\psi})\,\text{d}{\it\phi},\end{eqnarray}$$

with capital letters denoting transformed functions. Now taking the Fourier transform of (2.7) and rearranging, we obtain a single equation for the transformed pressure in the form

(2.10) $$\begin{eqnarray}\frac{1}{M}\frac{\partial }{\partial {\it\psi}}\left(M\frac{\partial P}{\partial {\it\psi}}\right)+\frac{2{\it\alpha}}{k-{\it\alpha}M}\frac{\text{d}M}{\text{d}{\it\psi}}\frac{\partial P}{\partial {\it\psi}}+\frac{M_{\infty }^{2}}{M^{2}}\left((k-{\it\alpha}M)^{2}-{\it\alpha}^{2}\right)P={\it\epsilon}{\rm\Sigma}({\it\alpha},{\it\psi}),\end{eqnarray}$$

where

(2.11) $$\begin{eqnarray}{\rm\Sigma}({\it\alpha},{\it\psi})=-\frac{{\rm\Sigma}_{2}}{M}-\text{i}(k-{\it\alpha}M)\frac{M_{\infty }}{M^{2}}{\rm\Sigma}_{3}-\frac{2{\it\alpha}\text{d}M/\text{d}{\it\psi}}{M(k-{\it\alpha}M)}{\rm\Sigma}_{2}-\frac{\text{i}{\it\alpha}M_{\infty }}{M^{2}}{\rm\Sigma}_{1}.\end{eqnarray}$$

Recall that ${\rm\Sigma}_{i}$ denotes the Fourier transform of ${\it\sigma}_{i}$ . Equations (2.7) and (2.10) are key results, and the rest of this paper is concerned with determining their solution.

3. Form of the incident gust

The form of the incident gust in parallel shear flow has been given by Goldstein (Reference Goldstein1978b ). Our flow is weakly non-parallel, thanks to the presence of the aerofoil, and this effect appears in two ways in (2.7): first, in the use of $({\it\phi},{\it\psi})$ coordinates, which captures the curvature of the mean streamlines, and second in the presence of the terms ${\it\sigma}_{1,2,3}$ representing the interaction of the unsteady flow with the non-uniform mean flow. Even so, Goldstein’s method and solutions can be applied in our case, and we need only briefly outline his approach and state the key results here. Although the equations we have presented already are valid for arbitrary mean shear distributions, $M({\it\psi})$ , at this point we restrict attention to the case in which $M({\it\psi})$ is a symmetric function, which will simplify both the form of the gust and our subsequent acoustic calculations. We also suppose that the shear layer has a single maximum or minimum at ${\it\psi}=0$ , which limits the number of critical layers where $M({\it\psi})=k/{\it\alpha}$ , and makes the construction of the gust solution easier.

Let a triple of Fourier-transformed solutions to (2.7) be denoted $\boldsymbol{Z}=\{P,U,V\}$ . Equation (2.7) has two linearly independent solutions; one of them will be denoted $\boldsymbol{Z}_{1}$ say, and we construct a second linearly independent solution, $\boldsymbol{Z}_{out}$ say, which has the property that it consists of only outgoing waves as ${\it\psi}\rightarrow \pm \infty$ . These two solutions must typically be computed numerically for a given mean shear profile, however we shall be able to calculate an asymptotic approximation for $\boldsymbol{Z}_{out}$ which is sufficient to obtain the acoustic solution. Thanks to the symmetry of the shear layer, both solutions can be written in the form

(3.1) $$\begin{eqnarray}\boldsymbol{Z}_{\ast }({\it\alpha},{\it\psi})=\{P_{\ast }({\it\alpha},|{\it\psi}|),(\text{sgn}{\it\psi})U_{\ast }({\it\alpha},|{\it\psi}|),V_{\ast }({\it\alpha},|{\it\psi}|)\}.\end{eqnarray}$$

The gust solution is now written as

(3.2) $$\begin{eqnarray}{\bf\zeta}_{g}({\it\phi},{\it\psi})\text{e}^{-\text{i}wt}=\{p_{g}({\it\phi},{\it\psi}),u_{g}({\it\phi},{\it\psi}),v_{g}({\it\phi},{\it\psi})\}\text{e}^{-\text{i}wt},\end{eqnarray}$$

and we simply state the Goldstein (Reference Goldstein1978b ) result for the transverse gust velocity $v_{g}$ here,

(3.3) $$\begin{eqnarray}\displaystyle v_{g} & = & \displaystyle \int _{{\it\psi}}^{\infty }\text{e}^{\text{i}k{\it\phi}/M({\it\eta})}\tilde{{\rm\Omega}}({\it\eta})\left[\frac{V_{1}(k/M({\it\eta}),{\it\psi})-{\it\gamma}({\it\eta})V_{out}(k/M({\it\eta}),{\it\psi})}{U_{1}(k/M({\it\eta}),{\it\eta})}\right]\,\text{d}{\it\eta}\nonumber\\ \displaystyle & & \displaystyle -\,\int _{-\infty }^{{\it\psi}}\text{e}^{\text{i}k{\it\phi}/M({\it\eta})}\tilde{{\rm\Omega}}({\it\eta}){\it\gamma}({\it\eta})\frac{V_{out}(k/M({\it\eta}),{\it\psi})}{U_{1}(k/M({\it\eta}),{\it\eta})}\,\text{d}{\it\eta}\quad \text{for}~{\it\psi}>0.\end{eqnarray}$$

In (3.3) we have ${\it\gamma}({\it\eta})={\rm\Gamma}_{+}^{\pm }(k/M({\it\eta}))$ for ${\it\eta}\gtrless 0$ with

(3.4) $$\begin{eqnarray}{\rm\Gamma}_{+}^{\pm }({\it\alpha})=\frac{P_{1}({\it\alpha},0_{\pm })V_{out}({\it\alpha},0_{-})-V_{1}({\it\alpha},0_{\pm })P_{out}({\it\alpha},0_{-})}{P_{out}({\it\alpha},0_{+})V_{out}({\it\alpha},0_{-})-P_{out}({\it\alpha},0_{-})V_{out}({\it\alpha},0_{+})},\end{eqnarray}$$

and ${\it\eta}$ is a function defined as the inverse of $f({\it\psi})=k/M({\it\psi})$ (note that this is well-defined since we consider symmetric a shear flow with a single turning point at ${\it\psi}=0$ ), so

(3.5) $$\begin{eqnarray}{\it\psi}={\it\eta}^{\pm }(k/M({\it\psi}))\quad \text{for}~{\it\psi}\gtrless 0.\end{eqnarray}$$

In (3.3), the function $\tilde{{\rm\Omega}}({\it\psi})$ is an arbitrary vorticity distribution that is fixed by the form of incident gust at upstream infinity. Given the symmetry of our problem we shall choose to work solely in the upper half-plane, ${\it\psi}>0$ , from this point on.

We mention that whilst the general form of the solution (3.3) is taken directly from Goldstein’s work, the actual value of the solution is different, because terms dependent on $\boldsymbol{Z}_{1}$ and $\boldsymbol{Z}_{out}$ rely on the solutions to our perturbed governing equations, rather than Goldstein’s flat-plate equations. The assumptions in Goldstein (Reference Goldstein1978b ) are consistent with our perturbed equations, which ensures we are able use this form of the gust solution.

At this point we will expand the unsteady flow quantities, and their Fourier transforms, in the form

(3.6) $$\begin{eqnarray}f=f^{0}+{\it\epsilon}\sqrt{k}f^{1}+O({\it\epsilon}).\end{eqnarray}$$

This choice of expansion is inspired by the work of Myers & Kerschen (Reference Myers and Kerschen1997) for an aerofoil in uniform flow, who showed that the leading effect of the aerofoil shape on the amplitude of the unsteady flow is to introduce an $O({\it\epsilon}\sqrt{k})$ correction. This effect arises from the interaction between the incident gust and the large mean-flow gradients close to the leading edge (with the flow at the leading edge being represented by an inverse square-root singularity in thin aerofoil theory). This interaction produces the $O({\it\epsilon}\sqrt{k})$ term both close to the leading edge and throughout the flow. We now expand the gust solution (3.3) in the form $v_{g}({\it\phi},{\it\psi})=v_{g}^{0}({\it\phi},{\it\psi})+{\it\epsilon}\sqrt{k}v_{g}^{1}({\it\phi},{\it\psi})+O({\it\epsilon})$ , and expressions for $v_{g}^{0,1}({\it\phi},{\it\psi})$ are given in appendix A. In the next section we will describe how the gust interacts with the leading-edge region to generate sound.

4. Leading-edge inner solution

Here we investigate the sound generated by the interaction of the gust with the leading edge of the aerofoil (region (i) in figure 1). We move to a leading-edge inner coordinate system, $({\rm\Phi},{\rm\Psi})=(k{\it\phi},k{\it\psi})$ , recalling that $k\gg 1$ is the high-frequency parameter, and write the scattered pressure as $p_{a}({\rm\Phi},{\rm\Psi})=p_{a}^{0}({\rm\Phi},{\rm\Psi})+{\it\epsilon}\sqrt{k}p_{a}^{1}({\rm\Phi},{\rm\Psi})+O({\it\epsilon})$ , with the suffix $_{a}$ denoting that this part of the solution contains the acoustic field generated by the gust–aerofoil interaction. The leading-order solution, $p_{a}^{0}({\rm\Phi},{\rm\Psi})$ , represents the effect of the blocking of the transverse momentum of the incident gust by the solid aerofoil surface approximated as a flat plate, while the perturbation $p_{a}^{1}({\rm\Phi},{\rm\Psi})$ represents the effects of thickness.

In the inner region the magnitude of the perturbation to the mean velocity, $q$ , is determined by substituting inner (polar) variables into (2.1) and expanding, to give

(4.1) $$\begin{eqnarray}q(R,{\it\theta})=-\frac{\text{i}\sqrt{k}}{2\sqrt{R}}\cos \frac{{\it\theta}}{2}.\end{eqnarray}$$

Note how the perturbation to the mean flow, which is of size $O({\it\epsilon})$ in the outer region, has been promoted to size $O({\it\epsilon}\sqrt{k})$ in the inner region, thanks to the presence of the inverse square-root singularity at the leading edge. This is what gives rise to the expansion (3.6).

4.1. General solution for inner leading-edge acoustic pressure

In this inner region it appears that the aerofoil is a semi-infinite flat plate ${\rm\Phi}>0,{\rm\Psi}=0$ , and hence we use the Wiener–Hopf method (Noble Reference Noble1998) to solve for the leading-edge inner acoustic solution. We write the solution as

(4.2a ) $$\begin{eqnarray}\displaystyle p_{a}({\rm\Phi},{\rm\Psi}) & = & \displaystyle \text{sgn}({\rm\Psi})\int _{-\infty }^{\infty }\text{e}^{\text{i}{\it\alpha}{\rm\Phi}/k}A({\it\alpha})P_{out}({\it\alpha},|{\rm\Psi}|)\text{d}{\it\alpha},\end{eqnarray}$$

(4.2b ) $$\begin{eqnarray}\displaystyle v_{a}({\rm\Phi},{\rm\Psi}) & = & \displaystyle \int _{-\infty }^{\infty }\text{e}^{\text{i}{\it\alpha}{\rm\Phi}/k}A({\it\alpha})V_{out}({\it\alpha},|{\rm\Psi}|)\text{d}{\it\alpha},\end{eqnarray}$$

demanding outgoing-wave behaviour at infinity. We also enforce the boundary conditions that $v_{a}=-v_{g}$ on ${\rm\Phi}>0,~{\rm\Psi}=0$ (in order to cancel the incident gust transverse velocity on the aerofoil surface), and that the pressure is continuous across ${\rm\Psi}=0$ for ${\rm\Phi}<0$ . These two conditions lead to the integral equations

(4.3a ) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }\text{e}^{\text{i}{\it\alpha}{\rm\Phi}/k}A({\it\alpha})V_{out}({\it\alpha},0)\text{d}{\it\alpha}=-v_{g}({\rm\Phi},0)\quad \text{for}~{\rm\Phi}>0 & & \displaystyle\end{eqnarray}$$

(4.3b ) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }\text{e}^{\text{i}{\it\alpha}{\rm\Phi}/k}A({\it\alpha})P_{out}({\it\alpha},0)\text{d}{\it\alpha}=0\quad \text{for}~{\rm\Phi}<0. & & \displaystyle\end{eqnarray}$$

The solution to this Wiener–Hopf problem is described in detail in appendix B, and we write

(4.4) $$\begin{eqnarray}p_{a}^{0,1}({\rm\Phi},{\rm\Psi})=\int _{0}^{\infty }p_{a}^{0,1}({\rm\Phi},{\rm\Psi}\mid {\it\eta})\text{d}{\it\eta},\end{eqnarray}$$

where the expression for $p_{a}^{0,1}({\rm\Phi},{\rm\Psi}\mid {\it\eta})$ can be found in appendix A. We have therefore found the first two terms in the inner region.

When evaluating the far-field pressure, rather than integrating over ${\it\eta}$ as required by (4.4), we follow Goldstein (Reference Goldstein1978b ) and simply evaluate our expressions for $p({\it\phi},{\it\psi}\mid {\it\eta})$ at ${\it\eta}=0$ . This is motivated by the assumption that the gust vorticity distribution, $\tilde{{\rm\Omega}}$ , is sharply peaked at ${\it\eta}=0$ , so that the integration is dominated by the contribution from ${\it\eta}=0$ . A sharp peak of vorticity at ${\it\eta}=0$ is characteristic of turbulent shear layers; see Goldstein (Reference Goldstein1978b ) for further details.

4.2. Outer limit of the inner solution

4.2.1. Solution for $P^{0}$

Taking (2.10) to $O(1)$ gives an equation for the Fourier transform of the leading-order pressure, $P^{0}$ ;

(4.5) $$\begin{eqnarray}\frac{\partial ^{2}P^{0}}{\partial {\it\psi}^{2}}-\frac{2M^{\prime }{\it\alpha}}{{\it\alpha}M-k}\frac{\partial P^{0}}{\partial {\it\psi}}+\frac{M^{\prime }}{M}\frac{\partial P^{0}}{\partial {\it\psi}}+\frac{M_{\infty }^{2}}{M^{2}}\left[\left({\it\alpha}M-k\right)^{2}-{\it\alpha}^{2}\right]P^{0}=0.\end{eqnarray}$$

All of the terms in (4.5) balance provided ${\it\alpha}=O(k)$ , and we therefore define ${\it\beta}\equiv {\it\alpha}/k$ with ${\it\beta}=O(1)$ . This is a valid scaling of ${\it\alpha}$ in the inner region, since the Fourier phase, $-\text{i}{\it\phi}{\it\alpha}$ , can then be written as $-\text{i}{\rm\Phi}{\it\beta}$ in inner variables, allowing for $O(1)$ variations in ${\rm\Phi}$ to be analysed. In inner coordinates, (4.5) becomes to leading order

(4.6) $$\begin{eqnarray}\frac{\partial ^{2}P^{0}}{\partial {\rm\Psi}^{2}}+\frac{M_{\infty }^{2}}{M_{0}^{2}}\left[({\it\beta}M_{0}-1)^{2}-{\it\beta}^{2}\right]P^{0}=0,\end{eqnarray}$$

where $M_{0}=M(0)$ is the Mach number in the inner region. Equation (4.6) has an outgoing-wave solution

(4.7) $$\begin{eqnarray}P_{out}^{0}({\it\alpha},{\rm\Psi})=C^{0}({\it\alpha})\exp \!\left[\text{i}\sqrt{(1-{\it\alpha}M_{0}/k)^{2}-({\it\alpha}/k)^{2}}\frac{M_{\infty }}{M_{0}}|{\rm\Psi}|\right],\end{eqnarray}$$

where

(4.8) $$\begin{eqnarray}C^{0}({\it\alpha})=c\left[\left(1-\frac{{\it\alpha}M_{0}}{k}\right)^{2}-\left(\frac{{\it\alpha}}{k}\right)^{2}\right]^{-1/4}\end{eqnarray}$$

and $c$ is an arbitrary constant. The reasons we include the factor defined in (4.8) are two-fold. First, the factor is included in order to match with the form of solution used by Goldstein (Reference Goldstein1978b ) in parallel shear flow: Goldstein developed a WKB solution and the factor appears there as the usual WKB amplitude. Second, the factor is included in order to recover the leading-order directivity known to be present in leading-edge scattering of both vorticity and sound; we will return to this point later in this subsection.

Taking the form of solution (4.8), substituting into (A 2a ) and using the method of stationary phase (Bender & Orszag Reference Bender and Orszag1978), we find that the outer limit of the inner acoustic solution is

(4.9) $$\begin{eqnarray}p_{a}^{0}(r,{\it\theta}\mid {\it\eta})\sim -\!\left(\frac{\text{i}}{2{\rm\pi}kr}\right)^{1/2}\frac{\sin {\it\theta}\text{e}^{\text{i}kr{\it\lambda}_{0}({\it\theta})}}{(1-M_{0}^{2}\sin ^{2}{\it\theta})^{3/4}}\frac{\tilde{{\rm\Omega}}({\it\eta})\tilde{Q}^{0}({\it\eta})M({\it\eta})}{1-{\it\beta}_{0}M({\it\eta})}\frac{{\it\kappa}^{0}(k/M({\it\eta}))_{+}C^{0}(k{\it\beta}_{0})}{{\it\kappa}^{0}(k{\it\beta}_{0})_{+}V_{out}^{0}(k{\it\beta}_{0},0)},\end{eqnarray}$$

where $(r,{\it\theta})$ are polar coordinates in $({\it\phi},{\it\psi})$ -space centred on the leading edge. We note that the arbitrary constant, $c$ , is cancelled out in the term $C^{0}/V_{out}^{0}$ in light of (B 5). In (4.9) we have introduced the phase function, ${\it\lambda}$ , defined as

(4.10) $$\begin{eqnarray}{\it\lambda}({\it\beta},M)={\it\beta}\cos {\it\theta}+\frac{M_{\infty }}{M}\sqrt{(1-M{\it\beta})^{2}-{\it\beta}^{2}},\end{eqnarray}$$

which has the point of stationary phase

(4.11) $$\begin{eqnarray}{\it\beta}^{s}(M)=-\frac{1}{1-M^{2}}\left(M-\frac{\cos {\it\theta}}{\sqrt{\cos ^{2}{\it\theta}+{\displaystyle \frac{M_{\infty }^{2}}{M^{2}}}\sin ^{2}{\it\theta}-M_{\infty }^{2}\sin ^{2}{\it\theta}}}\right).\end{eqnarray}$$

The functions ${\it\lambda}_{0}({\it\theta})$ and ${\it\beta}_{0}({\it\theta})$ are defined as

(4.12) $$\begin{eqnarray}{\it\lambda}_{0}({\it\theta})={\it\lambda}({\it\beta}_{0},M_{0}),\quad {\it\beta}_{0}={\it\beta}^{s}(M_{0}).\end{eqnarray}$$

The steady Mach number takes the value $M_{0}$ throughout the inner region, and the phase in (4.9) is therefore given by (4.10) and (4.11) with $M=M_{0}$ . Furthermore in (4.9), the function ${\it\kappa}_{+}^{0}({\it\eta})$ arises from the Wiener–Hopf solution of the inner problem, see (B 5), while the function $\tilde{Q}^{0}({\it\eta})$ appears in the form of the incident gust and is defined following (B 14).

We now return to the question of the choice of the factor in (4.8). We know from Tsai (Reference Tsai1992) and Ayton & Peake (Reference Ayton and Peake2013) that the leading-order outer solution for gust–aerofoil and sound–aerofoil interaction in uniform steady flow has directivity $\cos {\it\theta}/2$ . In steady shear flow, sound is produced at the leading edge both by the scattering of the vortical gust (as in Tsai Reference Tsai1992), and by the scattering of the gust self-noise (similar to Ayton & Peake Reference Ayton and Peake2013). Throughout the inner region the mean shear does not appear, and the mean flow is simply uniform with Mach number $M_{0}$ , and the $\cos {\it\theta}/2$ directivity must therefore be recovered in the outer limit of the inner solution in the present problem too. In fact, our choice of $C^{0}({\it\alpha})$ gives

(4.13) $$\begin{eqnarray}P^{0}\propto \cos ({\it\theta}/2)\left(1-M_{0}/2+(3+\cos 2{\it\theta})M_{0}^{2}/4+O(M_{0}^{3})\right),\end{eqnarray}$$

confirming the required directivity.

4.2.2. Solution for $P^{1}$

Taking $O({\it\epsilon}\sqrt{k})$ terms in (2.10) and converting to inner coordinates gives an equation for $P^{1}$ ;

(4.14) $$\begin{eqnarray}\frac{\partial ^{2}P^{1}}{\partial {\rm\Psi}^{2}}+\frac{M_{\infty }^{2}}{M_{0}^{2}}\left(1-2M_{0}{\it\beta}-{\it\beta}^{2}(1-M_{0}^{2})\right)P^{1}={\rm\Sigma}({\it\alpha},{\rm\Psi}).\end{eqnarray}$$

Here we have used the fact that to leading order ${\rm\Sigma}({\it\alpha},{\rm\Psi})\equiv \sqrt{k}{\rm\Sigma}({\it\alpha},k{\it\psi})$ , which follows from the inverse square-root singularity of the steady flow at the leading edge.

We solve (4.14) using the Green’s function

(4.15) $$\begin{eqnarray}\displaystyle G({\rm\Psi},{\rm\Psi}^{\prime }) & = & \displaystyle \frac{M_{0}}{2\text{i}M_{\infty }\sqrt{1-2M_{0}{\it\beta}-{\it\beta}^{2}(1-M_{0}^{2})}}\nonumber\\ \displaystyle & & \displaystyle \times \,\exp \left(\text{i}\sqrt{(1-M_{0}{\it\beta})^{2}-{\it\beta}^{2}}\displaystyle \frac{M_{\infty }}{M_{0}}|{\rm\Psi}-{\rm\Psi}^{\prime }|\right),\end{eqnarray}$$

which represents the desired outgoing wave field, to yield

(4.16) $$\begin{eqnarray}\displaystyle P_{out}^{1}(k{\it\beta},{\rm\Psi}) & = & \displaystyle \int _{0}^{\infty }\frac{M_{0}\exp \left(\text{i}\sqrt{(1-M_{0}{\it\beta})^{2}-{\it\beta}^{2}}\displaystyle \frac{M_{\infty }}{M_{0}}|{\rm\Psi}-{\rm\Psi}^{\prime }|\right)}{2\text{i}M_{\infty }\sqrt{(1-M_{0}{\it\beta})^{2}-{\it\beta}^{2}}}{\rm\Sigma}(k{\it\beta},{\rm\Psi}^{\prime })\,\text{d}{\rm\Psi}^{\prime }\nonumber\\ \displaystyle & & \displaystyle +\;C^{1}({\it\beta})\exp \left(\text{i}\sqrt{(1-M_{0}{\it\beta})^{2}-{\it\beta}^{2}}\displaystyle \frac{M_{\infty }}{M_{0}}|{\rm\Psi}|\right).\end{eqnarray}$$

From (2.11) we know that each term in ${\rm\Sigma}({\it\alpha},{\rm\Psi})$ will have a phase function $\sqrt{(1-M_{0}{\it\beta})^{2}-{\it\beta}^{2}}(M_{\infty }/M_{0})|{\rm\Psi}|$ (since each ${\it\sigma}_{i}$ is proportional to a linear combination of $u^{0}$ , $v^{0}$ and $p^{0}$ ), and further, since $Q$ is symmetric and ${\rm\Omega}$ is antisymmetric with respect to ${\it\psi}$ , we know that ${\rm\Sigma}$ is symmetric with respect to ${\it\psi}$ . Setting

(4.17) $$\begin{eqnarray}\hat{{\rm\Sigma}}({\it\alpha},{\rm\Psi})={\rm\Sigma}({\it\alpha},{\rm\Psi})\exp \left(\text{i}\sqrt{(1-M_{0}{\it\beta})^{2}-{\it\beta}^{2}}\displaystyle \frac{M_{\infty }}{M_{0}}|{\rm\Psi}|\right),\end{eqnarray}$$

so that $\hat{{\rm\Sigma}}$ is phase-less in the variable ${\rm\Psi}$ , and completing the ${\rm\Psi}^{\prime }$ integral in (4.16), we find that the outer limit of the inner solution is

(4.18) $$\begin{eqnarray}P_{out}^{1}(k{\it\beta},k{\it\psi})\sim \left(d(k{\it\beta},k{\it\psi})+C^{1}(k{\it\beta})\right)\exp \left(\text{i}\sqrt{(1-M_{0}{\it\beta})^{2}-{\it\beta}^{2}}\displaystyle \frac{M_{\infty }}{M_{0}}|{\rm\Psi}|\right),\end{eqnarray}$$

where $d(k{\it\beta},{\rm\Psi})$ is given by

(4.19) $$\begin{eqnarray}d(k{\it\beta},k{\it\psi})=\frac{\text{i}\hat{{\rm\Sigma}}(k{\it\beta},k{\it\psi})M_{0}}{k\left((1-M_{0}{\it\beta})^{2}-{\it\beta}^{2}\right)M_{\infty }}.\end{eqnarray}$$

Following the same arguments as in the previous subsection, we choose $C^{1}(k{\it\beta})=C^{0}(k{\it\beta})$ , and then repeating the method of stationary phase to invert the Fourier transform yields an outer expansion for $p_{a}^{1}$ in the form

(4.20) $$\begin{eqnarray}\displaystyle p_{a}^{1}(r,{\it\theta}\mid {\it\eta})\sim & - & \displaystyle \!\!\left({\displaystyle \frac{\text{i}}{2{\rm\pi}kr}}\right)^{1/2}{\displaystyle \frac{\sin {\it\theta}\text{e}^{\text{i}kr{\it\lambda}_{0}({\it\theta})}}{(1-M_{0}^{2}\sin ^{2}{\it\theta})^{3/4}}}{\displaystyle \frac{\tilde{{\rm\Omega}}({\it\eta})\tilde{Q}^{0}({\it\eta})M({\it\eta})}{1-{\it\beta}_{0}M({\it\eta})}}{\displaystyle \frac{{\it\kappa}^{0}(k/M({\it\eta}))_{+}C^{0}(k{\it\beta}_{0})}{{\it\kappa}^{0}(k{\it\beta}_{0})_{+}V_{out}^{0}(k{\it\beta}_{0},0)}}\nonumber\\ \displaystyle & & \displaystyle \times ~\left[{\displaystyle \frac{\tilde{Q}^{1}({\it\eta})}{\tilde{Q}^{0}({\it\eta})}}+M^{\prime }({\it\eta})\left({\displaystyle \frac{P_{out}^{1}(k/M({\it\eta}),0)}{P_{out}^{0}(k/M({\it\eta}),0)}}-{\displaystyle \frac{V_{out}^{1}(k/M({\it\eta}),0)}{V_{out}^{0}(k/M({\it\eta}),0)}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,{\displaystyle \frac{1}{C^{0}(k{\it\beta}_{0})}}\left(d(k{\it\beta}_{0},k{\it\psi})-d(k{\it\beta}_{0},0)\right)\right],\end{eqnarray}$$

where ${\it\beta}_{0}$ and ${\it\lambda}_{0}$ are given in (4.12). Comparing (4.20) and (4.9), we see that $p_{a}^{1}$ is obtained by multiplying $p_{a}^{0}$ by the correction factor given in square brackets in (4.20). This correction factor has arisen from two separate effects; the first two sets of terms inside the square brackets in (4.20) arise from the distortion of the incident gust by the non-uniform mean flow round the aerofoil; while the third set of terms, involving the function $d(k{\it\beta},k{\it\psi})$ , arises from the source terms in (2.10), i.e. from the interaction between the leading-order scattered field and the non-uniform mean flow near the leading edge. The correction term in (4.20) will have the important effect of introducing constructive and destructive interference between the two leading-edge fields $p_{a}^{0}$ and $p_{a}^{1}$ , and we return to this point in § 8.

Note that whilst ${\rm\Sigma}$ , as defined in (2.11), has appeared in our solution through (4.19), we only need to calculate the inner limit of ${\rm\Sigma}$ in order to establish (4.20). This is in exact parallel to the work of Myers & Kerschen (Reference Myers and Kerschen1997) and Tsai (Reference Tsai1992), who found that in a uniform stream the leading contribution of the volume terms only appears close to the leading edge where the mean flow gradients are large. Therefore, to calculate the outer limit of the inner leading-edge solution we only need to find the correction terms $N_{i},~i=1,\dots ,6$ appearing in (2.7) close to the aerofoil. We first note that, since $U=U_{0}({\it\psi})+{\it\epsilon}q({\it\phi},{\it\psi})$ , we have

(4.21) $$\begin{eqnarray}N_{1}=q+U_{0}N_{3},\end{eqnarray}$$

while by using (2.4) we obtain the relation

(4.22) $$\begin{eqnarray}\frac{q}{U_{0}^{2}}\frac{\text{d}U_{0}}{\text{d}{\it\psi}}+\frac{\partial q}{\partial {\it\psi}}\frac{1}{U_{0}}=\frac{\partial N_{3}}{\partial {\it\psi}}.\end{eqnarray}$$

In the leading-edge inner region, (4.22) can be integrated to yield

(4.23) $$\begin{eqnarray}N_{3}=\frac{q}{U_{0}(0)},\end{eqnarray}$$

where an arbitrary function of ${\it\phi}$ has been set to zero to ensure consistency with $N_{2}=0$ . It therefore follows that $N_{1}=2q$ . The quantities $N_{4,5,6}$ can be found immediately from expressions we have obtained for ${\it\zeta},q$ and ${\rm\Omega}$ , with

(4.24a−c ) $$\begin{eqnarray}N_{4}=u_{0}\frac{\partial q}{\partial {\it\phi}},\quad N_{5}=v_{0}\frac{\partial q}{\partial {\it\phi}}\left(1-2U_{0}(0)\right),\quad N_{6}=\frac{\partial q}{\partial {\it\phi}}\left(v_{0}+\frac{u_{0}}{U_{0}(0)}\right),\end{eqnarray}$$

again all evaluated in the inner region.

In summary, we have determined the first two terms in the outer limit of the inner pressure field, given by (4.9) and (4.20) which are integrated in (4.4). As mentioned previously, we assume the vorticity distribution is sharply peaked at ${\it\eta}=0$ , allowing us to use Laplace’s method to evaluate (4.4). We will write the sum of these two terms in the form

(4.25) $$\begin{eqnarray}\frac{\mathscr{P}_{l}({\it\theta})}{\sqrt{kr}}\exp (\text{i}kr{\it\lambda}({\it\beta}_{0})),\end{eqnarray}$$

and we will match this expression onto the outer solution in the next section.

5. Leading-edge outer solution

In this section we determine the leading-edge contribution to the acoustic pressure in the far field. The sound generated by the gust–aerofoil interaction in the leading-edge inner region, as determined in the previous section, propagates through the outer region, denoted by region (ii) in figure 1, and is distorted by the mean shear. The acoustic field of a point source in a mean shear has been determined by Durbin (Reference Durbin1983), and we use those results here.

The outer solution which matches with the outer limit of the inner solution takes the form

(5.1) $$\begin{eqnarray}p_{l}=\frac{s(r,{\it\theta})\mathscr{P}_{l}({\it\theta})}{\sqrt{kr}}\exp (\text{i}k{\it\varrho}^{0}+\text{i}k{\it\epsilon}{\it\varrho}^{1}).\end{eqnarray}$$

Here $\mathscr{P}_{l}({\it\theta})$ is the directivity of the inner solution as it emerges into the outer region, as defined in (4.25). The factor $s(r,{\it\theta})$ is the scaling factor derived by Durbin (Reference Durbin1983) to account for the distortion of the pressure amplitude due to variation in the ray tube area through the shear, and is given by

(5.2) $$\begin{eqnarray}s(r,{\it\theta})=\left({\displaystyle \frac{1-M_{0}^{2}}{1-M_{0}^{2}\sin ^{2}{\it\theta}}}\right)^{1/4}\left({\displaystyle \frac{M-M{\displaystyle \frac{\partial {\it\sigma}_{1}}{\partial {\it\phi}}}}{M_{0}-M_{0}\left.{\displaystyle \frac{\partial {\it\sigma}_{1}}{\partial {\it\phi}}}\right|_{r\rightarrow 0}}}\right)\left({\it\lambda}\sqrt{1-M_{0}^{2}}\cos {\it\mu}^{\prime }\frac{\partial {\it\mu}^{\prime }}{\partial {\it\mu}}\right)^{-1/2}.\end{eqnarray}$$

In (5.2), ${\it\mu}$ is the local ray angle (and ${\it\mu}^{\prime }$ is its value at the leading edge), see Durbin (Reference Durbin1983) equation (26b), while ${\it\lambda}$ is the local ray speed, see Durbin (Reference Durbin1983), following his equation (16). A factor in $s(r,{\it\theta})$ involving the local sound speed, present in Durbin (Reference Durbin1983), has been set to unity for our low-Mach-number flow. Note that $s(r,{\it\theta})\rightarrow 1$ as $r\rightarrow 0$ , while in the limit $r\rightarrow \infty$ , $s(r,{\it\theta})\rightarrow s({\it\theta})$ , where the latter can easily be calculated from (5.2).

We determine the first two terms phase terms, ${\it\varrho}^{0,1}$ , in (5.1) by substituting the ansatz (5.1) into an equation formed by rearranging (2.7) into a single equation for $p$ . We then take the real parts of the resulting equation at the first two asymptotic orders to form two eikonal equations for ${\it\varrho}^{0,1}$ . In what follows we will only require the acoustic pressure in the far field (i.e. $r\rightarrow \infty$ ), and we therefore write down expressions for the phase terms which are valid there. The first eikonal equation can easily be solved to give the first phase term in the form

(5.3) $$\begin{eqnarray}{\it\varrho}^{0}=kr{\it\lambda}({\it\beta}_{\infty },M_{\infty })\equiv kr{\it\lambda}_{\infty }({\it\theta}),\end{eqnarray}$$

where ${\it\beta}_{\infty }={\it\beta}^{s}(M_{\infty })$ .

The second eikonal equation is more complicated, since it includes contributions from the terms ${\it\sigma}_{1,2,3}$ in (2.7), which arise from the interaction between the leading-order unsteady flow and the steady-flow non-uniformity caused by the presence of the aerofoil. After some algebra we find that the second eikonal equation is

(5.4) $$\begin{eqnarray}\frac{\partial {\it\varrho}^{1}}{\partial {\it\phi}}+\frac{\partial {\it\varrho}^{1}}{\partial {\it\psi}}=\frac{1}{2}L({\it\phi},{\it\psi}),\end{eqnarray}$$

where the term $L({\it\phi},{\it\psi})$ involves the terms ${\it\sigma}_{1,2,3}$ . Specifically, we introduce the quantity

(5.5) $$\begin{eqnarray}M_{\infty }{\it\sigma}({\it\phi},{\it\psi})=-\text{i}k{\it\sigma}_{3}+M\frac{\partial {\it\sigma}_{3}}{\partial {\it\phi}}-\frac{\partial {\it\sigma}_{1}}{\partial {\it\phi}}-\frac{M}{M_{\infty }}\frac{\partial {\it\sigma}_{2}}{\partial {\it\psi}},\end{eqnarray}$$

which, in the light of (5.1), to leading-order in the outer region takes the form

(5.6) $$\begin{eqnarray}{\it\sigma}({\it\phi},{\it\psi})=\frac{k^{3/2}L({\it\phi},{\it\psi})\mathscr{P}({\it\theta})s({\it\theta})}{\sqrt{r}}\text{e}^{\text{i}k{\it\varrho}^{0}(r,{\it\theta})+\text{i}k{\it\epsilon}{\it\varrho}^{1}(r,{\it\theta})},\end{eqnarray}$$

where

(5.7) $$\begin{eqnarray}L({\it\phi},{\it\psi})=\left(\frac{\partial {\it\varrho}^{0}}{\partial {\it\phi}}\right)^{2}\left[\frac{q}{U_{0}}+\int _{{\it\psi}}^{\infty }\frac{2qU_{0}^{\prime }({\it\psi}^{\prime })}{U_{0}({\it\psi}^{\prime })^{2}}\,\text{d}{\it\psi}^{\prime }\right]+q\left(\frac{\partial {\it\varrho}^{0}}{\partial {\it\psi}}\right)^{2}.\end{eqnarray}$$

The solution of (5.4) can now be determined using the method of characteristics in the form

(5.8) $$\begin{eqnarray}{\it\varrho}^{1}(r,{\it\theta})=\frac{1}{2}\int _{0}^{{\it\phi}+{\it\psi}}L({\it\chi},{\it\psi})\,\text{d}{\it\chi},\end{eqnarray}$$

where ${\it\chi}={\it\phi}+{\it\psi}$ is the characteristic variable.

We have therefore completed the construction of the far-field solution for the noise emanating from the leading edge of the aerofoil, and we write finally the acoustic pressure as $r\rightarrow \infty$ in the form

(5.9) $$\begin{eqnarray}\frac{D_{l}({\it\theta})}{\sqrt{kr}}\exp \left(\text{i}kr{\it\lambda}_{\infty }({\it\theta})+\frac{1}{2}\text{i}k{\it\epsilon}\int _{0}^{{\it\phi}+{\it\psi}}L({\it\chi},{\it\psi})\,\text{d}{\it\chi}\right),\end{eqnarray}$$

where the leading-edge directivity is given by $D_{l}({\it\theta})=\mathscr{P}_{l}({\it\theta})s({\it\theta})$ . We emphasise that this solution is not valid in the mid field, where the mean flow is sheared; it is only valid in the far field, where $M\approx M_{\infty }$ .

6. Leading-edge transition solution

The transition solution (region (iii) in figure 1) accounts for the curvature of the surface of the aerofoil, in a very similar manner to the case of uniform flow considered by Tsai (Reference Tsai1992), and corrects for the boundary condition of zero normal velocity on the aerofoil surface that is violated by the leading-edge outer solution. We therefore suppose the transition solution takes the form

(6.1) $$\begin{eqnarray}p_{ltr}={\it\epsilon}G({\it\phi},{\it\xi})\exp \left(\displaystyle \frac{\text{i}k{\it\phi}}{1+M}+\frac{1}{2}\text{i}k{\it\epsilon}\int _{0}^{{\it\phi}}L({\it\phi}^{\prime },0)\,\text{d}{\it\phi}^{\prime }\right),\end{eqnarray}$$

where ${\it\xi}=\sqrt{k}{\it\psi}$ is the transition-region coordinate above the aerofoil in the direction normal to the surface. The choice of phase in (6.1) arises from taking ${\it\theta}=0$ (equivalently ${\it\psi}=0$ ) in (5.3) and (5.8).

In the transition region the leading-order expansion of (2.7) tells us that $G({\it\phi},{\it\xi})$ must satisfy

(6.2) $$\begin{eqnarray}\frac{M^{2}}{M_{\infty }^{2}}\frac{\partial ^{2}G}{\partial {\it\xi}^{2}}+2\text{i}\frac{\partial G}{\partial {\it\phi}}=0,\end{eqnarray}$$

subject to boundary condition

(6.3) $$\begin{eqnarray}-\left.{\it\epsilon}\sqrt{k}\frac{\partial G}{\partial {\it\xi}}\exp \left({\displaystyle \frac{\text{i}k{\it\phi}}{1+M}}+\frac{1}{2}\text{i}k{\it\epsilon}\int _{0}^{{\it\phi}}L({\it\phi},0)\text{d}{\it\phi}\right|_{{\it\xi}=0}\right)=-\left.\frac{\partial p_{l}}{\partial {\it\psi}}\right|_{{\it\psi}=0},\end{eqnarray}$$

which enforces zero total normal velocity on the aerofoil surface. We now take the Laplace transform of (6.2) with respect to ${\it\phi}$ , denoted by

(6.4) $$\begin{eqnarray}\tilde{G}(S,{\it\xi})=\int _{0}^{\infty }G({\it\phi},{\it\xi})\text{e}^{-S{\it\phi}}\,\text{d}{\it\phi},\end{eqnarray}$$

to find that

(6.5) $$\begin{eqnarray}\tilde{G}(S,{\it\xi})=B(S)\exp \left(-\text{e}^{-{\rm\pi}\text{i}/4}\sqrt{2S}\frac{M_{\infty }}{M}{\it\xi}\right),\end{eqnarray}$$

where

(6.6) $$\begin{eqnarray}\displaystyle B(S)={\displaystyle \frac{\text{e}^{3\text{i}{\rm\pi}/4}}{2\sqrt{2S}}}\int _{0}^{\infty } & {\displaystyle \frac{\text{e}^{-S{\it\phi}}}{\sqrt{{\it\phi}}}}s({\it\phi},0)\mathscr{P}({\it\phi},0)\left(L({\it\phi},0)+\displaystyle \int _{0}^{{\it\phi}}\left.{\displaystyle \frac{\partial L({\it\phi}^{\prime }+{\it\psi},{\it\psi})}{\partial {\it\psi}}}\right|_{{\it\psi}=0}\,\text{d}{\it\phi}^{\prime }\right)\text{d}{\it\phi}. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

This Laplace transform can be inverted numerically to determine the transition solution.

We mention briefly here that the terms ${\it\sigma}_{1,2,3}$ in (2.7) occur at higher order and do not appear in this transition solution explicitly, although they do appear implicitly through the forcing provided by the outer solution in (6.3). Physically this is because the dominant effects of curvature arise in the leading-edge inner region, where the aerofoil is most curved, and not along the upper and lower arcs of the aerofoil.

The total far-field acoustic pressure emanating from the leading edge is given as a sum of the outer field determined in the previous section and the transition solution determined in this section. The transition solution does not appear directly in the acoustics (note from (6.5) that the transition solution decays exponentially in the transverse direction away from the aerofoil surface). It does, however, introduce a pressure discontinuity across the aerofoil, which must be corrected downstream of the trailing edge across the wake. This is done by the introduction of trailing-edge inner and transition solutions, and the inner solution matches onto an outgoing trailing-edge acoustic field. This is described in the next section.

7. Trailing-edge inner and outer solutions

Here we determine the solution in the trailing-edge inner region and the trailing-edge contribution to the outer region, denoted by (iv) and (ii) in figure 1, respectively. The transition solution in the wake (region (v) in figure 1) is not required for the acoustic far field, and is very similar to solutions found in uniform flow by Myers & Kerschen (Reference Myers and Kerschen1997) and Tsai (Reference Tsai1992), and will therefore not be presented here.

We shift coordinates to be aligned with the trailing edge, defining $({\it\phi}_{t},{\it\psi}_{t})$ such that $({\it\phi},{\it\psi})=(2+{\it\phi}_{t}+{\it\epsilon}{\it\alpha}_{t},{\it\psi}_{t})$ . Here ${\it\alpha}_{t}=O(1)$ arises from the effect of thickness during the mapping of coordinates from physical space to $({\it\phi},{\it\psi})$ -space. By observing (2.1), ${\it\alpha}_{t}$ can be calculated in much the same way as was done by Tsai (Reference Tsai1992) for uniform flow. The transverse velocity of the incident gust solution at the trailing edge is still given by (A1).

7.1. Trailing-edge inner solution

We move to inner trailing-edge coordinates, $({\rm\Phi}_{t},{\rm\Psi}_{t})=k({\it\phi}_{t},{\it\psi}_{t})$ . The trailing-edge inner acoustic solution, equivalent to (4.2), satisfies a dual integral equation equivalent to (4.3), which is

(7.1a ) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }\text{e}^{\text{i}{\it\alpha}{\rm\Phi}_{t}/k}A({\it\alpha})P_{out}({\it\alpha},0)\,\text{d}{\it\alpha}=-{\rm\Delta}p({\rm\Phi}_{t})/2,\quad {\rm\Phi}_{t}>0, & & \displaystyle\end{eqnarray}$$

(7.1b ) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }\text{e}^{\text{i}{\it\alpha}{\rm\Phi}_{t}/k}A({\it\alpha})V_{out}({\it\alpha},0)\,\text{d}{\it\alpha}=-v_{g}({\rm\Phi}_{t},0),\quad {\rm\Phi}_{t}<0. & & \displaystyle\end{eqnarray}$$

Here all functions are written in terms of trailing-edge coordinates, ${\it\alpha}$ is redefined accordingly as the Fourier transform variable with respect to ${\it\phi}_{t}$ and ${\rm\Delta}p({\rm\Phi}_{t})$ is the inner approximation for the pressure jump across the trailing edge generated by the leading-edge solution. We separate the required inner solution, $p_{a}({\rm\Phi}_{t},{\rm\Psi}_{t})$ , into a term that corrects the pressure jump across the trailing edge, $p_{a,p}$ , and a term that corrects for the zero normal velocity condition on the surface of the aerofoil, $p_{a,H}$ . Using the notation from (4.2), we require

(7.2a ) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }\text{e}^{\text{i}{\it\alpha}{\rm\Phi}_{t}/k}A_{p}({\it\alpha})P_{out}({\it\alpha},0)\,\text{d}{\it\alpha}=-{\rm\Delta}p({\rm\Phi}_{t})/2,\quad {\rm\Phi}_{t}>0, & & \displaystyle\end{eqnarray}$$

(7.2b ) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }\text{e}^{\text{i}{\it\alpha}{\rm\Phi}_{t}/k}A_{p}({\it\alpha})V_{out}({\it\alpha},0)\,\text{d}{\it\alpha}=0,\quad {\rm\Phi}_{t}<0, & & \displaystyle\end{eqnarray}$$

and

(7.3a ) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }\text{e}^{\text{i}{\it\alpha}{\rm\Phi}_{t}/k}A_{H}({\it\alpha})P_{out}({\it\alpha},0)\text{d}{\it\alpha}=0,\quad {\rm\Phi}_{t}>0, & & \displaystyle\end{eqnarray}$$

(7.3b ) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }\text{e}^{\text{i}{\it\alpha}{\rm\Phi}_{t}/k}A_{H}({\it\alpha})V_{out}({\it\alpha},0)\text{d}{\it\alpha}=-v_{g}({\rm\Phi}_{t},0),\quad {\rm\Phi}_{t}<0. & & \displaystyle\end{eqnarray}$$

The solution of (7.2) and (7.3) is obtained using identical methods to those used at the leading edge in § 4, and is presented in appendix C. We use the solutions for $P_{out}^{0,1}$ as previously obtained in § 4.2, but translated to the trailing-edge inner coordinate system. Taking the outer limit of the inner solutions, (C 1), and using the method of steepest descents yields

(7.4) $$\begin{eqnarray}\displaystyle p_{a}^{t}(r_{t},{\it\theta}_{t}\mid {\it\eta}_{t}) & {\sim} & \displaystyle \left(\frac{\text{i}}{2{\rm\pi}kr_{t}}\right)^{1/2}\!\frac{{\it\kappa}_{t}^{0}(k{\it\beta}_{t0})_{-}\sin {\it\theta}_{t}}{(1-M_{0}^{2}\sin ^{2}{\it\theta}_{t})^{3/4}}\frac{\text{e}^{\text{i}kr_{t}{\it\lambda}_{t\,0}({\it\theta}_{t})}}{1-{\it\beta}_{t0}M({\it\eta}_{t})}\frac{C^{0}(k{\it\beta}_{t0})\tilde{{\rm\Omega}}({\it\eta}_{t})\tilde{Q}^{0}({\it\eta}_{t})M({\it\eta}_{t})}{{\it\kappa}_{t}^{0}(k/M({\it\eta}_{t}))_{-}V_{out}^{0}(k{\it\beta}_{t0},0)}\nonumber\\ \displaystyle & & \displaystyle \times \!\left[1+{\it\epsilon}\sqrt{k}\left\{\frac{\tilde{Q}^{1}({\it\eta}_{t})}{\tilde{Q}^{0}({\it\eta}_{t})}+M^{\prime }({\it\eta}_{t})\!\left(\frac{P_{out}^{1}(k/M({\it\eta}_{t}),0)}{P_{out}^{0}(k/M({\it\eta}_{t}),0)}-\frac{V_{out}^{1}(k/M({\it\eta}_{t}),0)}{V_{out}^{0}(k/M({\it\eta}_{t}),0)}\right)\right.\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\left.\frac{d_{t}(k{\it\beta}_{t0},{\rm\Psi}_{t})-d_{t}(k{\it\beta}_{t0},0)}{C^{0}(k{\it\beta}_{t0})}\right\}\right]\nonumber\\ \displaystyle & & \displaystyle +\left(\frac{\text{i}}{2{\rm\pi}kr_{t}}\right)^{1/2}\frac{{\it\kappa}_{t}^{0}(k{\it\beta}_{t0})_{-}\sin {\it\theta}_{t}\text{e}^{\text{i}kr_{t}{\it\lambda}_{t\,0}({\it\theta}_{t})}}{(1-M_{0}^{2}\sin ^{2}{\it\theta}_{t})^{3/4}}\frac{G_{t,p}(k{\it\beta}_{t0})P_{out}^{0}(k{\it\beta}_{t0})}{V_{out}^{0}(k{\it\beta}_{t0})}\end{eqnarray}$$

as $kr_{t}\rightarrow \infty$ , where ${\it\lambda}_{t0}$ is the trailing-edge equivalent of ${\it\lambda}_{0}$ , and ${\it\beta}_{t0}$ is the trailing-edge equivalent of ${\it\beta}_{0}$ as defined in (4.12). The final term in (7.4) is in fact $O(k^{-1})$ due to the scaling of the pressure jump term $G_{t,p}$ . It will be shown later that the term involving $d_{t}(k{\it\beta}_{t0},{\rm\Psi}_{t})$ in (7.4) is negligible to the orders retained here since in the trailing-edge region the terms ${\it\sigma}_{i}$ in (2.7) are negligible (because there is less curvature of the streamlines at the trailing edge than at the leading edge for the aerofoils we wish to consider, such as the Joukowski aerofoil or the NACA 4-digit series of aerofoils). The choice of $C^{0}$ is again given by (4.8), which now ensures that the trailing-edge inner solution has a $\sin {\it\theta}/2$ directivity pattern. This is the same directivity pattern found for sound– and gust–aerofoil interaction in steady uniform flow in Ayton & Peake (Reference Ayton and Peake2013) and Myers & Kerschen (Reference Myers and Kerschen1997), respectively. Once again we know that the shear flow directivity pattern should match the uniform flow directivity pattern to leading order, since in the trailing-edge inner region the aerofoil only experiences the local Mach number $M_{0}$ . Note that at the trailing edge we only need a match to $O(M_{0})=O(\sqrt{{\it\epsilon}})$ (as opposed to $O(M_{0}^{2})=O({\it\epsilon})$ required for the inner leading-edge solution (4.13)) since the trailing-edge scattered field of gust–aerofoil interaction is $O(k^{-1/2})$ smaller than the leading-edge field, and in the uniform limit the first two terms in (7.4) tend to zero since $({\it\kappa}_{t}^{0}(k/M({\it\eta}_{t}))_{-})^{-1}\rightarrow 0$ as $M({\it\psi})\rightarrow M_{0}$ .

The terms in square brackets in (7.4) represent the scattering of the pressure associated purely with the gust in the shear flow by the aerofoil (in uniform flow a gust is pressure-free, and these terms vanish). Whilst the contribution of these terms appears to be the same order as the contribution of the leading-edge solution (4.9), we in fact find that it is at least $O(M)$ smaller due to ${\it\kappa}_{t}^{0}(k/M({\it\eta}_{t}))_{-}$ having a singularity at ${\it\eta}_{t}=0$ . We mentioned at the end of § 4.1 that to evaluate the pressure $p_{a}(r,{\it\theta})$ given as an integral over ${\it\eta}$ of $p_{a}(r,{\it\theta}\mid {\it\eta})$ in (4.4), we consider only sharply peaked vorticity distributions where contributions from ${\it\eta}=0$ dominate. At ${\it\eta}=0$ , ${\it\kappa}_{t}^{0}(k/M({\it\eta}))_{-}=0$ , therefore before applying Laplace’s method we must take an expansion of ${\it\kappa}_{t}^{0}(k/M({\it\eta}_{t}))_{-}$ as ${\it\eta}\rightarrow 0$ . This expansion reduces the apparent order of the first term in (7.4) by at least $O(M)$ (the true scaling will depend on how the vorticity distribution depends on $k$ and ${\it\eta}$ ), thus the contribution from the scattering of the gust pressure by the trailing edge is at least $O(M)$ smaller than the leading-edge contribution to the far-field acoustics. The final term in (7.4) accounts for the rescattering of the leading-edge acoustic field by the trailing edge and, as expected by comparison with the uniform flow case, is $O(k^{-1/2})$ smaller than the leading-edge solution.

We write the outer limit of the trailing-edge inner solution (once integrated by ${\it\eta}_{t}$ as required in (4.4)) as

(7.5) $$\begin{eqnarray}\frac{1}{\sqrt{kr_{t}}}\left(M\mathscr{P}_{t_{1}}({\it\theta}_{t})+\frac{1}{\sqrt{k}}\mathscr{P}_{t_{2}}({\it\theta}_{t})\right)\exp (\text{i}kr_{t}{\it\lambda}_{t\,0}({\it\theta}_{t})),\end{eqnarray}$$

where $\mathscr{P}_{t_{2}}$ is formally the same order as $\mathscr{P}_{l}$ , but $\mathscr{P}_{t_{1}}$ could be smaller than $\mathscr{P}_{l}$ (depending on the choice of vorticity distribution). We set $\mathscr{P}_{t}=M\mathscr{P}_{t_{1}}+k^{-1/2}\mathscr{P}_{t_{2}}$ .

7.2. Trailing-edge outer solution

The trailing-edge outer solution is found in an identical way to the leading-edge outer solution, assuming a form

(7.6) $$\begin{eqnarray}p_{t}=A_{t}(r_{t},{\it\theta}_{t})\exp (\text{i}k{\it\varrho}_{t}^{0}(r_{t},{\it\theta}_{t})+\text{i}k{\it\epsilon}{\it\varrho}_{1}^{t}(r_{t},{\it\theta}_{t})+O({\it\epsilon})).\end{eqnarray}$$

We find that

(7.7a,b ) $$\begin{eqnarray}A_{t}(r_{t},{\it\theta}_{t})=D_{t}({\it\theta})(kr_{t})^{-1/2},\quad {\it\varrho}_{t}^{0}(r_{t},{\it\theta}_{t})=r{\it\lambda}_{t\infty }({\it\theta}_{t}),\end{eqnarray}$$

where ${\it\lambda}_{t\,\infty }({\it\theta}_{t})$ is the corresponding trailing-edge function to ${\it\lambda}_{\infty }({\it\theta})$ , and ${\it\varrho}_{t}^{1}$ is given by the corresponding trailing-edge formulation of (5.8). We match this to the trailing-edge inner solution by setting $D_{t}({\it\theta}_{t})$ equal to $\mathscr{P}_{t}({\it\theta}_{t})s_{t}(r_{t},{\it\theta}_{t})$ , where the first factor arises from the directivity emerging from the inner region in (7.4) and the second factor accounts for the variation in ray tube area as the sound propagates though the shear – see (5.2). The total far-field acoustic pressure emanating from the trailing edge then takes the form

(7.8) $$\begin{eqnarray}\frac{D_{t}({\it\theta}_{t})}{\sqrt{kr_{t}}}\exp \left(\text{i}kr_{t}{\it\lambda}_{t\infty }({\it\theta}_{t})+\frac{1}{2}\text{i}k{\it\epsilon}\int _{0}^{{\it\phi}_{t}+{\it\psi}_{t}}L_{t}({\it\chi},{\it\psi}_{t})\text{d}{\it\chi}\right).\end{eqnarray}$$

Again, this is only valid in the far field, where the Mach number approaches $M_{\infty }$ .

7.3. Total far-field solution

The total far-field solution is obtained by summing the outer leading-edge solution, from (4.9) and (4.20) substituted into (4.4), and the outer trailing-edge solution, from (7.4) substituted into (4.4). In the far field, the coordinate transformation between leading-edge and trailing-edge polar coordinates is given by

(7.9a,b ) $$\begin{eqnarray}r_{t}\approx r-(2+{\it\alpha}_{t}{\it\epsilon})\cos {\it\theta},\quad {\it\theta}_{t}\approx {\it\theta}-{\rm\pi}+{\rm\pi}\,\text{sgn}({\it\psi}),\end{eqnarray}$$

which allows the final solution to be expressed in terms of leading-edge variables $(r,{\it\theta})$ . The far-field acoustic pressure can then be written as

(7.10) $$\begin{eqnarray}\frac{1}{\sqrt{kr}}\left(D_{l}({\it\theta})+D_{t}({\it\theta})\text{e}^{\text{i}k{\it\varrho}_{s}(r,{\it\theta})}\right)\exp \left(\text{i}kr{\it\lambda}_{\infty }({\it\theta})+{\displaystyle \frac{1}{2}}\text{i}k{\it\epsilon}\displaystyle \int _{0}^{{\it\phi}+{\it\psi}}L({\it\chi},{\it\psi})\,\text{d}{\it\chi}\right),\end{eqnarray}$$

where $D_{l,t}$ are defined in (5.9) and (7.8). In the far field, the leading- and trailing-edge ray fields interact with a phase shift

(7.11) $$\begin{eqnarray}k{\it\varrho}_{s}(r,{\it\theta})=k({\it\varrho}_{t}^{0}(r_{t},{\it\theta}_{t})+{\it\epsilon}{\it\varrho}_{t}^{1}(r_{t},{\it\theta}_{t})-{\it\varrho}^{0}(r,{\it\theta})-{\it\epsilon}{\it\varrho}^{1}(r,{\it\theta})).\end{eqnarray}$$

The contribution to the phase shift given by the difference between the leading-order leading- and trailing-edge far-field phase terms is

(7.12) $$\begin{eqnarray}k({\it\varrho}_{t}^{0}(r_{t},{\it\theta}_{t})-{\it\varrho}^{0}(r,{\it\theta}))=-(2+{\it\alpha}_{t})k{\it\lambda}_{\infty }({\it\theta})\cos {\it\theta}\end{eqnarray}$$

in the far field. The $O({\it\epsilon}k)$ phase shift term, given by ${\it\epsilon}k({\it\varrho}_{t}^{1}-{\it\varrho}^{1})$ , is approximated by

(7.13) $$\begin{eqnarray}\frac{{\it\epsilon}k}{2}\int _{0}^{2}L({\it\chi},{\it\psi})\,\text{d}{\it\chi}\end{eqnarray}$$

in the far field. Numerically integrating (7.13) for the cases we choose in the following section, we find that this contribution is only non-negligible close to ${\it\theta}=0$ or ${\rm\pi}$ . Since the directivity function close to ${\it\theta}=0,{\rm\pi}$ is small we shall not include (7.13) in our final computed results.