2.1 Steady flow

Before we can approach the unsteady aeroacoustics, we must first solve the steady background flow. This is step I in the road map (figure 1). There are several ways to calculate the potential flow through a cascade of aerofoils. By utilising a conformal map from a canonical circular domain to a cascade of flat plates, the flow for an arbitrary incident angle can be calculated analytically (Robinson & Laurmann Reference Robinson and Laurmann1956). Whilst this solution is exact, the expression for the conformal mapping is not invertible and the flow may only be written implicitly. Another method that is ubiquitous in thin-aerofoil theory is the method of singularities: the rigid aerofoil surface may be modelled as a distribution of mass sources and vortices on the chord (for thin aerofoils) or the surface (for thick aerofoils). This theory has previously been applied to cascades (Falcão Reference Falcão1975; Schulten Reference Schulten1982). The vortex distribution is typically written as a Glauert sine series or, equivalently, a series of weighted Chebyshev polynomials. These methods have been combined by Evers & Peake (Reference Evers and Peake2002) to provide an analytical expression for the steady flow near the leading edges.

Recently, exact solutions for the flow around thin, slightly staggered aerofoils have been found by modelling the flow as a Riemann–Hilbert problem (Baddoo & Ayton Reference Baddoo and Ayton2018b). The resulting expression is a clear extension of the classical thin-aerofoil theory for a single blade (Thwaites Reference Thwaites1960) and is invertible: given a particular tangential surface velocity, the corresponding aerofoil geometry can be found. However, we want to take into account large stagger angles, so a different approach is required here, although this analytical result will be useful to benchmark any other solutions.

To aid application of the boundary conditions in the unsteady problem, we will transform from non-dimensional Cartesian coordinates $(x,y)$ to potential-streamline coordinates $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$ as is typical for realistic-geometry aerofoil interactions studied analytically (Myers & Kerschen Reference Myers and Kerschen1995, Reference Myers and Kerschen1997; Peake & Kerschen Reference Peake and Kerschen1997, Reference Peake and Kerschen2004; Ayton & Peake Reference Ayton and Peake2013; Ayton Reference Ayton2017). The relationship between coordinates is given by

(2.3)$$\begin{eqnarray}z_{\unicode[STIX]{x1D6FD}}=z_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D716}F(z_{\unicode[STIX]{x1D719}}),\end{eqnarray}$$

where $z_{\unicode[STIX]{x1D6FD}}=x+\text{i}\unicode[STIX]{x1D6FD}_{\infty }y$ are the non-dimensional coordinates in the Prandtl–Glauert space (where the flow is incompressible) and $z_{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D719}+\text{i}\unicode[STIX]{x1D713}$. The coordinate transformation is illustrated in figure 3. $F(z_{\unicode[STIX]{x1D719}})$ is the complex potential of the steady flow written as $q(z_{\unicode[STIX]{x1D719}})-\text{i}\unicode[STIX]{x1D707}(z_{\unicode[STIX]{x1D719}})=F^{\prime }(z_{\unicode[STIX]{x1D719}})$ where the amplitude of the steady flow is given by $1+\unicode[STIX]{x1D716}q$, in the direction making an angle $\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}$ with the horizontal. Consequently, we may write the non-dimensional perturbation to horizontal uniform flow as (Baddoo & Ayton Reference Baddoo and Ayton2018b)

(2.4)$$\begin{eqnarray}q(z_{\unicode[STIX]{x1D6FD}})-\text{i}\unicode[STIX]{x1D707}(z_{\unicode[STIX]{x1D6FD}})=-\text{i}\unicode[STIX]{x1D6FC}+\frac{1}{2(d+\text{i}s)}\int _{0}^{2}\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})\left(\coth \left(\frac{\unicode[STIX]{x03C0}\text{i}(\unicode[STIX]{x1D70F}-z_{\unicode[STIX]{x1D6FD}})}{d+\text{i}s}\right)-1\right)\,\text{d}\unicode[STIX]{x1D70F},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}$ is a distribution of vortices, sources and sinks arranged on the aerofoil chord and $d$ and $s$ are the blade spacing in the Prandtl–Glauert space.

Figure 3.

Illustration of the change of coordinate system used to simplify the application of boundary conditions. The complex aerofoil surface is mapped to a flat plate.

We now apply the modified Plemelj formulae (Baddoo & Ayton Reference Baddoo and Ayton2018b, appendix A) to each side of the chord of each aerofoil. Taking the difference of the modified Plemelj formulae on each aerofoil yields $\text{Im}[\unicode[STIX]{x1D6FE}(x)]=-2y_{th}^{\prime }(x)$, while taking the sum yields the singular integral equation

(2.5)$$\begin{eqnarray}\text{Im}\left[\frac{1}{d+\text{i}s}\unicode[STIX]{x2A0D}_{0}^{2}\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})\left(\coth \left(\frac{\unicode[STIX]{x03C0}\text{i}(\unicode[STIX]{x1D70F}-x)}{d+\text{i}s}\right)-1\right)\,\text{d}\unicode[STIX]{x1D70F}\right]=-2(y_{c}^{\prime }(x)-\unicode[STIX]{x1D6FC}),\end{eqnarray}$$

for $0<x<2$. In the small stagger case, exact solutions for this singular integral equation are available via the Riemann–Hilbert method (Baddoo & Ayton Reference Baddoo and Ayton2018b). However, when the stagger is not asymptotically small, a crucial symmetry is broken and the same method cannot be applied. Consequently, we solve the singular integral equation (2.5) numerically by expanding the real part of $\unicode[STIX]{x1D6FE}$ as a series of weighted Chebyshev polynomials (Rienstra Reference Rienstra1992). The coefficients of this series are determined by a spectral Chebyshev collocation method due to Erdogan & Gupta (Reference Erdogan and Gupta1972), as outlined in § S:3 of the supplementary material available at https://doi.org/10.1017/jfm.2019.1016. This method is extremely rapid and converges exponentially fast as the number of basis polynomials is increased.

By integrating expression (2.4), we obtain analytic forms for the streamfunction and velocity potential

(2.6)$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D719}(z_{\unicode[STIX]{x1D6FD}})+\text{i}\unicode[STIX]{x1D713}(z_{\unicode[STIX]{x1D6FD}})=z_{\unicode[STIX]{x1D6FD}}(1-\text{i}\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FC})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{\unicode[STIX]{x1D716}}{2\unicode[STIX]{x03C0}\text{i}}\int _{0}^{2}\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})\left(\log \left(\frac{\displaystyle \sinh \left(\frac{\unicode[STIX]{x03C0}\text{i}(\unicode[STIX]{x1D70F}-z_{\unicode[STIX]{x1D6FD}})}{d+\text{i}s}\right)}{\displaystyle \sinh \left(\frac{\unicode[STIX]{x03C0}\text{i}\unicode[STIX]{x1D70F}}{d+\text{i}s}\right)}\right)+\frac{\unicode[STIX]{x03C0}\text{i}z_{\unicode[STIX]{x1D6FD}}}{d+\text{i}s}\right)\,\text{d}\unicode[STIX]{x1D70F},\end{eqnarray}$$

where the constant of integration has been chosen such that $F(0)=0$ and the logarithmic branch is orientated along the aerofoil wake. Consequently, (2.6) allows us to calculate the spacing between blades in the new coordinate space as

(2.7)$$\begin{eqnarray}d_{\unicode[STIX]{x1D719}}+\text{i}s_{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D719}(d+\text{i}s)+\text{i}\unicode[STIX]{x1D713}(d+\text{i}s)=(d+\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FC}s)+\text{i}(s-\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FC}d).\end{eqnarray}$$

We immediately note that the angle of attack affects the blade spacing in $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space and will accordingly influence the cut-on and cut-off frequencies of the acoustic modes. These new parameters are illustrated in figure 5.

Exponential decompositions of the complex velocity field $q-\text{i}\unicode[STIX]{x1D707}$ are available in the upstream and downstream regions, and are detailed in §§ S:3 and S:6 of the supplementary material. Such a decomposition is essential for the application of the Wiener–Hopf method in § 3.2.

We define the location of the upper/lower side of the image of the trailing edge in $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space as $(c_{\unicode[STIX]{x1D719}}^{\pm },0)$. It is important to note that the image of the physical trailing edge is not unique, but the upper and lower sides are in fact mapped to two different points. Formula (2.6) yields analytic expressions for the trailing-edge images as

(2.8)$$\begin{eqnarray}c_{\unicode[STIX]{x1D719}}^{\pm }=2(1-\text{i}\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FC})-\frac{\unicode[STIX]{x1D716}}{2\unicode[STIX]{x03C0}\text{i}}\int _{0}^{2}\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})\left(\log \left(\frac{\displaystyle \sinh \left(\frac{\unicode[STIX]{x03C0}\text{i}(\unicode[STIX]{x1D70F}-2)}{d+\text{i}s}\right)}{\displaystyle \sinh \left(\frac{\unicode[STIX]{x03C0}\text{i}\unicode[STIX]{x1D70F}}{d+\text{i}s}\right)}\right)\pm \unicode[STIX]{x03C0}\text{i}+\frac{2\unicode[STIX]{x03C0}\text{i}}{d+\text{i}s}\right)\,\text{d}\unicode[STIX]{x1D70F},\end{eqnarray}$$

which are both real numbers since the physical trailing edge $(2,0^{\pm })$ lies on the same streamline as the origin. The difference between the location of the upper and lower trailing edges in $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space is $-\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4}$, which is the circulation around a single aerofoil, and may be calculated as

(2.9)$$\begin{eqnarray}\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4}=-\unicode[STIX]{x1D716}\int _{0}^{2}\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}=-\unicode[STIX]{x1D716}\unicode[STIX]{x03C0}\left(\unicode[STIX]{x1D6FE}_{r,0}+\frac{\unicode[STIX]{x1D6FE}_{r,1}}{2}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}_{r,0}$ and $\unicode[STIX]{x1D6FE}_{r,1}$ are the first two coefficients of the weighted Chebyshev series defined in (S:3.6) in the supplementary material.

Figure 4.

Contours of the velocity potential (blue) and streamfunction (red) for two cascades of aerofoils with gap-to-chord ratio $\unicode[STIX]{x1D6E5}/2=2/3$, stagger angle $\unicode[STIX]{x1D712}=40^{\circ }$ and angle of attack $10^{\circ }$. (a) NACA 0008; (b) NACA 5500.

In figure 4 we plot the velocity potential and streamfunction contours for two aerofoil profiles at angle of attack $10^{\circ }$: one with thickness (figure 4a) and one with camber (figure 4b). At this point, we highlight several features of the background flow in order to expedite the discussion of the aeroacoustic analysis in the following sections. Firstly, we note that the flow field satisfies the no-flux condition on the aerofoil surfaces and the Kutta condition at the aerofoil trailing edges. Secondly, the velocity potential and streamfunction are orthogonal everywhere and consequently form an orthogonal coordinate system. Thirdly, it can be seen that the thick aerofoils (figure 4a) scarcely deflect the flow whereas the cambered aerofoils (figure 4b) induce a significant flow deflection. The deflection angle is closely related to the aerofoil circulation (Baddoo & Ayton Reference Baddoo and Ayton2018b) and, accordingly, we observe a significant discontinuity in velocity potential across the wake in figure 4(b). In the following aeroacoustic analysis, this discontinuity must be regularised in order to ensure that the pressure field does not possess a jump across the wake.

Figure 5.

Diagram showing parameters in the transformed $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space.

2.2 Unsteady formulation

We now consider the unsteady interaction in the new, transformed coordinate system. The present analysis is restricted to inviscid linear disturbances of a two-dimensional irrotational compressible mean flow, using the formulation derived by Goldstein (Reference Goldstein1978) and further developed by Myers & Kerschen (Reference Myers and Kerschen1995). We assume that there are small amplitude vortical and entropic perturbations in the uniform flow far upstream of the cascade, which we denote by $\boldsymbol{v}^{\prime }$ and $s^{\prime }$ respectively. These disturbances are convected downstream by the mean flow and interact with the cascade, thus producing sound. Similar approaches have been employed (Peake & Kerschen Reference Peake and Kerschen1997, Reference Peake and Kerschen2004; Evers & Peake Reference Evers and Peake2002) in modelling gust–cascade interaction. Unlike the flat-plate case (Glegg Reference Glegg1999), the mean flow modifies both the boundary conditions and governing equation. We will see that extra forcing terms are induced in the governing equation, while the boundary conditions include terms depending on the surface velocity.

The total unsteady velocity perturbation may be decomposed into the form (Goldstein Reference Goldstein1978)

(2.10a)$$\begin{eqnarray}\boldsymbol{u}^{\prime }=\unicode[STIX]{x1D735}G^{\prime }+\boldsymbol{v}^{\prime },\end{eqnarray}$$

where $\boldsymbol{v}^{\prime }$ satisfies a modified form of the linearised momentum equation. By employing the method of characteristics, $\boldsymbol{v}^{\prime }$ may be found exactly throughout the whole flow domain. The irrotational scattered field, $G^{\prime }$, then satisfies (Goldstein Reference Goldstein1978)

(2.10b)$$\begin{eqnarray}\frac{\text{D}_{0}}{\text{D}t}\left(\frac{1}{c_{0}^{2}}\frac{\text{D}_{0}G^{\prime }}{\text{D}t}\right)-\frac{1}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x1D735}G^{\prime })=\frac{1}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{0}\boldsymbol{v}^{\prime }),\end{eqnarray}$$

where the material derivative $\text{D}_{0}/\text{D}t$ is taken with respect to the local mean flow. The quantities $\unicode[STIX]{x1D70C}_{0}$ and $c_{0}$ are the local values of density and speed of sound for the mean flow. The no-flux boundary condition on $G^{\prime }$ at any rigid surface is

(2.10c)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}G^{\prime }}{\unicode[STIX]{x2202}n}=-\boldsymbol{v}^{\prime }\boldsymbol{\cdot }\boldsymbol{n},\end{eqnarray}$$

where $\boldsymbol{n}$ is the unit normal vector.

The non-dimensional upstream velocity and entropy disturbances far upstream may be written as

(2.11)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \boldsymbol{v}^{\prime }\rightarrow \unicode[STIX]{x1D708}\left[\begin{array}{@{}c@{}}A_{t}\\ A_{n}\\ A_{s}\end{array}\right]\text{e}^{\text{i}k(\unicode[STIX]{x1D719}+k_{n}\unicode[STIX]{x1D713}+k_{s}x_{3}-t)},\\ \displaystyle s^{\prime }\rightarrow 2\unicode[STIX]{x1D708}B\text{e}^{\text{i}k(\unicode[STIX]{x1D719}+k_{n}\unicode[STIX]{x1D713}+k_{s}x_{3}-t)},\end{array}\right\}\end{eqnarray}$$

subject to $A_{t}+A_{n}k_{n}\unicode[STIX]{x1D6FD}_{\infty }+A_{s}k_{s}=0$, where $k=\unicode[STIX]{x1D714}b^{\ast }/U_{\infty }^{\ast }$ is the reduced frequency and time is normalised by $b^{\ast }/U_{\infty }^{\ast }$. The amplitude of the disturbance, $\unicode[STIX]{x1D708}$, is assumed to be small ($\unicode[STIX]{x1D708}\ll \unicode[STIX]{x1D716}$).

The transformation

(2.12)$$\begin{eqnarray}G^{\prime }(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})=h(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\exp [\text{i}k(k_{3}x_{3}-t-\unicode[STIX]{x1D6FF}M_{\infty }^{2}\unicode[STIX]{x1D719})+\unicode[STIX]{x1D716}M_{\infty }^{2}q],\end{eqnarray}$$

allowed Kerschen & Myers (Reference Kerschen and Myers1987) to express Goldstein’s wave equation as

(2.13)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}^{2}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}^{2}}+\frac{\unicode[STIX]{x2202}^{2}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}^{2}}+k^{2}w^{2}h+\unicode[STIX]{x1D716}(\unicode[STIX]{x1D6FE}+1)M_{\infty }^{4}\unicode[STIX]{x1D6FF}\left\{q\left[\frac{\unicode[STIX]{x2202}^{2}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}^{2}}+2\text{i}k\unicode[STIX]{x1D6FF}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}+k^{2}(w^{2}+\unicode[STIX]{x1D6FF}^{2})h\right]\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left.\frac{\unicode[STIX]{x2202}q}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\left[\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}-\text{i}k\unicode[STIX]{x1D6FF}h\right]\right\}-\unicode[STIX]{x1D716}2k^{2}w^{2}\unicode[STIX]{x1D6FD}_{\infty }^{2}qh=\unicode[STIX]{x1D716}kS(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\text{e}^{\text{i}k\unicode[STIX]{x1D6FA}},\end{eqnarray}$$

where

(2.14)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \begin{array}{@{}c@{}}\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}+k_{n}\unicode[STIX]{x1D713}+\unicode[STIX]{x1D716}g(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}),\quad w^{2}=(\unicode[STIX]{x1D6FF}M_{\infty })^{2}-(k_{s}/\unicode[STIX]{x1D6FD}_{\infty })^{2},\\ \displaystyle \unicode[STIX]{x1D6FF}=1/\unicode[STIX]{x1D6FD}_{\infty }^{2},\quad \tilde{A}_{t}=A_{t}-B,\end{array}\\ \displaystyle \begin{array}{@{}rcl@{}}\displaystyle S(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\; & =\; & \displaystyle 2\unicode[STIX]{x1D6FF}\left\{\text{i}(\tilde{A}_{t}-k_{n}A_{n}\unicode[STIX]{x1D6FD}_{\infty }^{3})q+\text{i}(k_{n}\tilde{A}_{t}\unicode[STIX]{x1D6FD}_{\infty }+A_{n}\unicode[STIX]{x1D6FD}_{\infty })\unicode[STIX]{x1D707}\right.\\ \; & \; & \displaystyle +\left.\frac{1}{k}\left[\tilde{A}_{t}M_{\infty }^{2}\frac{\unicode[STIX]{x2202}q}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}+A_{n}M_{\infty }^{2}\unicode[STIX]{x1D6FD}_{\infty }\frac{\unicode[STIX]{x2202}q}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right]\right\},\end{array}\end{array}\right\}\end{eqnarray}$$

and $g$ is Lighthill’s drift function, which represents the cumulative distortion of vortex filaments by the non-uniform mean flow,

(2.15)$$\begin{eqnarray}g(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})=\int _{-\infty }^{\unicode[STIX]{x1D719}}\left[\frac{1}{U_{0}^{2}(\unicode[STIX]{x1D702},\unicode[STIX]{x1D713})}-1\right]\,\text{d}\unicode[STIX]{x1D702}=-2\int _{-\infty }^{\unicode[STIX]{x1D719}}q(\unicode[STIX]{x1D702},\unicode[STIX]{x1D713})\,\text{d}\unicode[STIX]{x1D702}.\end{eqnarray}$$

The no-flux boundary condition applied to rigid surfaces (2.10c) becomes

(2.16)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}+\unicode[STIX]{x1D716}M_{\infty }^{2}\frac{\unicode[STIX]{x2202}q}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}h=-\left[\frac{A_{n}}{\unicode[STIX]{x1D6FD}_{\infty }}(1-\unicode[STIX]{x1D716}M_{\infty }^{2}q)-\unicode[STIX]{x1D716}2\tilde{A}_{t}\unicode[STIX]{x1D707}\right]\text{e}^{\text{i}k\unicode[STIX]{x1D6FA}},\end{eqnarray}$$

applied on the body surface.

We will use the modified pressure $p$, which is defined in terms of actual pressure $p^{\prime }$ as

(2.17)$$\begin{eqnarray}p=-p^{\prime }\text{e}^{\text{i}k(-k_{s}x_{3}+t)}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}(h\text{e}^{-\text{i}k\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}})\text{e}^{\text{i}k\unicode[STIX]{x1D719}}.\end{eqnarray}$$

2.3 Boundary conditions

We must specify an adequate number of boundary conditions in order for the problem to be well posed and physically valid. The boundary conditions are applied to each streamline that correspond to a blade surface, i.e. $\unicode[STIX]{x1D713}=ns_{\unicode[STIX]{x1D719}}$ for integer $n$. Upstream of the blade, we require that there are no discontinuities across this streamline. On the blade itself, the no-flux condition for a rigid surface (2.16) must be satisfied either side of the blade. Downstream of the blade, we require two conditions – firstly the jump in normal velocity either side of the wake must vanish, and secondly, there can be no pressure jump across the wake. Due to the disparity between the images of the upper and lower trailing edge, which is equal to the mean flow circulation, some care is required with these latter two boundary conditions. The full set of boundary conditions is summarised and illustrated in figure 6.

As previously mentioned in (2.8), there are two trailing edges $c_{\unicode[STIX]{x1D719}}^{\pm }$ in the parametric $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space. In the application of the boundary conditions, we define a single trailing edge at $c_{\unicode[STIX]{x1D719}}=c_{\unicode[STIX]{x1D719}}^{+}$. This produces $O(\unicode[STIX]{x1D716})$ errors, which will be corrected in our asymptotic expansion later.

In the following analysis, we use $\unicode[STIX]{x1D6E5}_{n}$ and $\unicode[STIX]{x1D6F4}_{n}$ to denote the difference and sum of a given quantity either side of the $n$th aerofoil’s streamline.

Figure 6.

Schematic illustrating where each boundary condition is applied.

2.3.1 Upstream boundary condition

The condition of no upstream discontinuities is simply

(2.18)$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{n}[h](\unicode[STIX]{x1D719})=0,\quad \text{for }\unicode[STIX]{x1D719}<0.\end{eqnarray}$$

2.3.2 No-flux boundary conditions

The flow on each side of the blade must satisfy the rigid surface condition (2.16) which, after ignoring $O(\unicode[STIX]{x1D716}^{2})$ terms, becomes

(2.19)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}h^{\pm }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D719})+\unicode[STIX]{x1D716}M_{\infty }^{2}\frac{\unicode[STIX]{x2202}q^{\pm }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D719})h^{\pm }(\unicode[STIX]{x1D719})\end{eqnarray}$$

(2.20)$$\begin{eqnarray}\displaystyle & & \displaystyle \quad =-\left[\frac{A_{n}}{\unicode[STIX]{x1D6FD}_{\infty }}(1-\unicode[STIX]{x1D716}M_{\infty }^{2}q^{\pm }(\unicode[STIX]{x1D719})+\unicode[STIX]{x1D716}\text{i}kg^{\pm }(\unicode[STIX]{x1D719}))-\unicode[STIX]{x1D716}2\tilde{A}_{t}\unicode[STIX]{x1D707}^{\pm }(\unicode[STIX]{x1D719})\right]\text{e}^{\text{i}k\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}},\quad 0<\unicode[STIX]{x1D719}<c_{\unicode[STIX]{x1D719}},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where we have assumed that $\unicode[STIX]{x1D716}k\ll 1$. This linearisation corresponds to step IIa in the road map.

2.3.3 Wake boundary conditions

We now derive the boundary conditions to be applied on the aerofoil wake. This corresponds to step IIb in the road map. The lower trailing edge lags behind the upper trailing edge in $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space by a distance of $-\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4}$, as illustrated in figures 5 and 6. Consequently, the condition of continuity of normal velocity in terms of our acoustic potential function is

(2.21)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h^{+}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D719})\text{e}^{-\text{i}kM_{\infty }^{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}}-\frac{\unicode[STIX]{x2202}h^{-}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D719}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4})\text{e}^{-\text{i}kM_{\infty }^{2}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D719}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4})}=0,\quad \text{for }\unicode[STIX]{x1D719}>c_{\unicode[STIX]{x1D719}}.\end{eqnarray}$$

We may Taylor expand the second term about $\unicode[STIX]{x1D719}$ to give

(2.22)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h^{+}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D719})-\left(\frac{\unicode[STIX]{x2202}h^{-}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D719})+\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4}\frac{\unicode[STIX]{x2202}^{2}h^{-}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D719})\right)(1-\text{i}kM_{\infty }^{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4})=0,\end{eqnarray}$$

where we have assumed that $\unicode[STIX]{x1D716}kM_{\infty }^{2}\ll 1$ and ignored $O(\unicode[STIX]{x1D716}^{2})$ terms. We accordingly obtain the condition on the jump in normal velocity in $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space as

(2.23)$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{n}\left[\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})=\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4}\left(\frac{\unicode[STIX]{x2202}^{2}h^{-}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D719})-\text{i}kM_{\infty }^{2}\unicode[STIX]{x1D6FF}\frac{\unicode[STIX]{x2202}h^{-}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D719})\right),\quad \text{for }\unicode[STIX]{x1D719}>c_{\unicode[STIX]{x1D719}}.\end{eqnarray}$$

Using a similar method, we reformulate the pressure continuity condition in $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space. Due to the lag in the lower trailing-edge location, we may write

(2.24)$$\begin{eqnarray}p^{\prime +}(\unicode[STIX]{x1D719})-p^{\prime -}(\unicode[STIX]{x1D719}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4})=0,\quad \text{for }\unicode[STIX]{x1D719}>c_{\unicode[STIX]{x1D719}}.\end{eqnarray}$$

Taylor expanding the latter term about $\unicode[STIX]{x1D719}$ and ignoring terms of $O(\unicode[STIX]{x1D716}^{2})$ yields

(2.25)$$\begin{eqnarray}p^{\prime +}(\unicode[STIX]{x1D719})-p^{\prime -}(\unicode[STIX]{x1D719})-\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4}\frac{\unicode[STIX]{x2202}p^{\prime -}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D719})=0.\end{eqnarray}$$

By substituting (2.17) and cancelling terms, we obtain

(2.26)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D6E5}_{n}[h](\unicode[STIX]{x1D719})\text{e}^{-\text{i}k\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}})\text{e}^{\text{i}k\unicode[STIX]{x1D719}}-\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}(h^{-}(\unicode[STIX]{x1D719})\text{e}^{-\text{i}k\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}})\text{e}^{\text{i}k\unicode[STIX]{x1D719}}\right)=0.\end{eqnarray}$$

We may integrate the above expression to obtain the condition on the jump in acoustic potential in $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space as

(2.27)$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{n}[h](\unicode[STIX]{x1D719})=2\unicode[STIX]{x03C0}\text{i}P\text{e}^{\text{i}k\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E4}\left(\frac{\unicode[STIX]{x2202}h^{-}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D719})-\text{i}kM_{\infty }^{2}\unicode[STIX]{x1D6FF}h^{-}(\unicode[STIX]{x1D719})\right),\quad \text{for }\unicode[STIX]{x1D719}>c_{\unicode[STIX]{x1D719}},\end{eqnarray}$$

where $P$ is a constant of integration that will be determined by the Wiener–Hopf technique.

3.1 Solution to $O(1)$ problem

The $O(1)$ system of equations ((3.2a), (3.5a), (3.6a), (3.7a), (3.8a), (3.9a)) was considered by Glegg (Reference Glegg1999) and Posson et al. (Reference Posson, Roger and Moreau2010). This corresponds to step IVb in the road map. The solution is written in terms of the $\unicode[STIX]{x1D719}$-Fourier transform of the jump in potential either side of the blade and wake,

(3.10)$$\begin{eqnarray}D_{0}(\unicode[STIX]{x1D6FE})=\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{\infty }\unicode[STIX]{x1D6E5}_{0}[h_{0}](\unicode[STIX]{x1D719})\text{e}^{\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}}\,\text{d}\unicode[STIX]{x1D719},\end{eqnarray}$$

and the solution is

(3.11)$$\begin{eqnarray}\displaystyle D_{0}(\unicode[STIX]{x1D6FE}) & = & \displaystyle \frac{-\text{i}A_{n}}{\unicode[STIX]{x1D6FD}_{\infty }(2\unicode[STIX]{x03C0})^{2}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})J_{+}(\unicode[STIX]{x1D6FE})J_{-}(\unicode[STIX]{x1D705}_{0}^{-})}-\mathop{\sum }_{n=0}^{\infty }\frac{({\mathcal{A}}_{0,n}+{\mathcal{C}}_{0,n})\text{e}^{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{-})c_{\unicode[STIX]{x1D719}}}}{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{-})}\cdot \frac{J_{-}(\unicode[STIX]{x1D703}_{n}^{-})}{J_{-}(\unicode[STIX]{x1D6FE})}\nonumber\\ \displaystyle & & \displaystyle -\,\mathop{\sum }_{n=0}^{\infty }\frac{{\mathcal{B}}_{0,n}}{(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{+})}\cdot \frac{J_{+}(\unicode[STIX]{x1D703}_{n}^{+})}{J_{+}(\unicode[STIX]{x1D6FE})},\end{eqnarray}$$

where all undefined terms may be found in the supplementary material. The function $D_{0}(\unicode[STIX]{x1D6FE})$ was found by Glegg (Reference Glegg1999), and the Fourier transform was inverted to provide the acoustic field in the upstream and downstream regions. The acoustic field in the inter-blade region was found by Posson et al. (Reference Posson, Roger and Moreau2010) by choosing an appropriate contour of integration dependent on the region in the channel where the acoustic field was to be evaluated.

3.2 Solution to $O(\unicode[STIX]{x1D716})$ problem

Having solved the $O(1)$ equation, we now move on to the $O(\unicode[STIX]{x1D716})$ equation. In particular, we seek to generate an integral equation that will be amenable to the Wiener–Hopf method. Following Glegg (Reference Glegg1999), we seek a solution using integral transforms. Currently, $h_{1}$ has a discontinuity across each blade and wake and consequently its derivatives are not integrable. Therefore, we introduce generalised derivatives (Lighthill Reference Lighthill1958) such that the Laplacian operator becomes

(3.12)$$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}h_{1}=\tilde{\unicode[STIX]{x1D6FB}}^{2}h_{1}-\mathop{\sum }_{n=-\infty }^{\infty }\unicode[STIX]{x1D6E5}_{n}[h_{1}](\unicode[STIX]{x1D719})\unicode[STIX]{x1D6FF}^{\prime }(\unicode[STIX]{x1D713}-ns_{\unicode[STIX]{x1D719}})-\mathop{\sum }_{n=-\infty }^{\infty }\unicode[STIX]{x1D6E5}_{n}\left[\frac{\unicode[STIX]{x2202}h_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D713}-ns_{\unicode[STIX]{x1D719}}),\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D6FB}}^{2}$ represents the Laplacian operator with discontinuities removed, the second term on the right represents the discontinuity of acoustic potential either side of the blade and wake and the third term on the right represents the discontinuity of normal velocity either side of the blade and wake. This step corresponds to step IVc in the road map. Crucially, the above is expressed in the $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space, not in physical space.

In Glegg (Reference Glegg1999), the last term in (3.12) disappears on the aerofoil surfaces because the blades are infinitesimally thin, so there is no jump in the normal velocity either side of the blade. However, although the blades are ‘thin’ in our $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space, they have been converted from thick blades in physical space, and therefore will induce some mass flux in order to satisfy the no-flux condition in the physical domain. Also, in Glegg (Reference Glegg1999), the last term in (3.12) disappears along the wake, because it is not a source of mass injection. Whilst this condition is true in our case in physical space, the mean flow circulation has induced a shift in the upper and lower trailing edges in the $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space, so some mass injection is required in the transformed space in order to have continuous normal velocity in the physical space.

The jump in the acoustic potential across a blade streamline is unknown and must be found as part of the solution. Conversely, the jump in $\unicode[STIX]{x1D713}$-derivative of the acoustic potential is known in terms of mean flow quantities and the $O(1)$ solution.

As is typical in cascade acoustics problems, we note that the periodicity of the background flow and gust furnishes a quasi-periodic condition on the physical unsteady potential,

(3.13)$$\begin{eqnarray}G^{\prime }(\unicode[STIX]{x1D719}+nd_{\unicode[STIX]{x1D719}},\unicode[STIX]{x1D713}+ns_{\unicode[STIX]{x1D719}})=G^{\prime }(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\text{e}^{\text{i}n\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}$ is the inter-blade phase angle of the gust in $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space so that $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}=k(d_{\unicode[STIX]{x1D719}}+k_{n}s_{\unicode[STIX]{x1D719}})$. The inter-blade phase angle in the physical domain, $\unicode[STIX]{x1D70E}$, is obtained using (2.7). In terms of the modified potential, this quasi-periodic condition becomes

(3.14)$$\begin{eqnarray}h_{1}(\unicode[STIX]{x1D719}+nd_{\unicode[STIX]{x1D719}},\unicode[STIX]{x1D713}+ns_{\unicode[STIX]{x1D719}})=h_{1}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\text{e}^{\text{i}n\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }},\end{eqnarray}$$

where the modified inter-blade phase angle is $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }=k(\unicode[STIX]{x1D6FF}d_{\unicode[STIX]{x1D719}}+k_{n}s_{\unicode[STIX]{x1D719}})$. We will use this relation repeatedly to reduce the problem from an infinite cascade to a strip of finite height. For example, by inserting (3.14) into the expression for the generalised Laplacian (3.12), we may rewrite the $O(\unicode[STIX]{x1D716})$ governing equation (3.2b) as

(3.15)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}^{2}h_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}^{2}}+\frac{\unicode[STIX]{x2202}^{2}h_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}^{2}}+k^{2}w^{2}h_{1} & = & \displaystyle \mathbb{S}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})+\mathop{\sum }_{n=-\infty }^{\infty }\unicode[STIX]{x1D6E5}_{0}[h_{1}](\unicode[STIX]{x1D719}-nd_{\unicode[STIX]{x1D719}})\unicode[STIX]{x1D6FF}^{\prime }(\unicode[STIX]{x1D713}-ns_{\unicode[STIX]{x1D719}})\text{e}^{\text{i}n\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{n=-\infty }^{\infty }\unicode[STIX]{x1D6E5}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719}-nd_{\unicode[STIX]{x1D719}})\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D713}-ns_{\unicode[STIX]{x1D719}})\text{e}^{\text{i}n\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }}.\end{eqnarray}$$

We define the Fourier integral transform and its inverse as

(3.16)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle F(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D707})=\frac{1}{(2\unicode[STIX]{x03C0})^{2}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\text{e}^{\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}+\text{i}\unicode[STIX]{x1D707}\unicode[STIX]{x1D713}}\,\text{d}\unicode[STIX]{x1D719}\,\text{d}\unicode[STIX]{x1D713},\\ \displaystyle f(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }F(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D707})\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}-\text{i}\unicode[STIX]{x1D707}\unicode[STIX]{x1D713}}\,\text{d}\unicode[STIX]{x1D6FE}\,\text{d}\unicode[STIX]{x1D707}.\end{array}\right\}\end{eqnarray}$$

Applying the transform to the left-hand side of (3.15) yields

(3.17)$$\begin{eqnarray}(-\unicode[STIX]{x1D6FE}^{2}-\unicode[STIX]{x1D707}^{2}+k^{2}w^{2})H_{1}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D707})=s_{0}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D707})+\frac{1}{2\unicode[STIX]{x03C0}}\mathop{\sum }_{n=-\infty }^{\infty }\{-\text{i}\unicode[STIX]{x1D707}D_{1}(\unicode[STIX]{x1D6FE})+G(\unicode[STIX]{x1D6FE})\}\text{e}^{\text{i}n(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }+\unicode[STIX]{x1D6FE}d_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D707}s_{\unicode[STIX]{x1D719}})},\end{eqnarray}$$

where

(3.18)$$\begin{eqnarray}\displaystyle & \displaystyle s_{0}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D707})=\frac{1}{(2\unicode[STIX]{x03C0})^{2}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\mathbb{S}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\text{e}^{\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}+\text{i}\unicode[STIX]{x1D707}\unicode[STIX]{x1D713}}\,\text{d}\unicode[STIX]{x1D719}\,\text{d}\unicode[STIX]{x1D713}, & \displaystyle\end{eqnarray}$$

(3.19)$$\begin{eqnarray}\displaystyle & \displaystyle D_{1}(\unicode[STIX]{x1D6FE})=\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{\infty }\unicode[STIX]{x1D6E5}_{0}[h_{1}](\unicode[STIX]{x1D719})\text{e}^{\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}}\,\text{d}\unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

(3.20)$$\begin{eqnarray}\displaystyle & \displaystyle G(\unicode[STIX]{x1D6FE})=\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{\infty }\unicode[STIX]{x1D6E5}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})\text{e}^{\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}}\,\text{d}\unicode[STIX]{x1D719}. & \displaystyle\end{eqnarray}$$

The first of these expressions (3.18) is known in terms of the background flow quantities and $O(1)$ solution. The last of these expressions (3.20) is also known, and may be evaluated using the no-flux condition on the blade (3.7b) and continuity of normal velocity either side of the wake ((3.8b), (S:1.6)) to obtain

(3.21)$$\begin{eqnarray}\displaystyle G(\unicode[STIX]{x1D6FE}) & = & \displaystyle \mathop{\sum }_{l=-\infty }^{\infty }\frac{{\mathcal{c}}_{\unicode[STIX]{x1D6E5},l}((-1)^{l}\text{e}^{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})c_{\unicode[STIX]{x1D719}}}-1)}{2\unicode[STIX]{x03C0}\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{l}^{-})}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D6E4}}{2\unicode[STIX]{x03C0}}\left(\mathop{\sum }_{\pm }\frac{\pm \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{0}^{-}}{\mathcal{H}}_{0,\unicode[STIX]{x1D705}_{0}^{-}}^{\pm \prime }\text{e}^{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})c_{\unicode[STIX]{x1D719}}}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-}}+\mathop{\sum }_{m=-\infty }^{\infty }\frac{\unicode[STIX]{x1D701}_{m}^{-}{\mathcal{H}}_{0,m}^{\prime }\text{e}^{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{-})c_{\unicode[STIX]{x1D719}}}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{-}}\right).\end{eqnarray}$$

The problem is now to find $D_{1}(\unicode[STIX]{x1D6FE})$ – the Fourier transform of the jump in acoustic potential either side of the blade and wake. To do so, we must obtain an expression for the sum of the normal derivative in order to apply the relevant boundary conditions.

We now invert the Fourier transform in (3.17) to obtain an integral equation into which we may insert the known boundary data,

(3.22)$$\begin{eqnarray}\displaystyle & & \displaystyle h_{1}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\frac{\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}-\text{i}\unicode[STIX]{x1D707}\unicode[STIX]{x1D713}}}{k^{2}w^{2}-\unicode[STIX]{x1D707}^{2}-\unicode[STIX]{x1D6FE}^{2}}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\left\{s_{0}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D707})+\frac{1}{2\unicode[STIX]{x03C0}}\mathop{\sum }_{n=-\infty }^{\infty }[-\text{i}\unicode[STIX]{x1D707}D_{1}(\unicode[STIX]{x1D6FE})+G(\unicode[STIX]{x1D6FE})]\text{e}^{\text{i}n(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }+\unicode[STIX]{x1D6FE}d_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D707}s_{\unicode[STIX]{x1D719}})}\right\}\,\text{d}\unicode[STIX]{x1D6FE}\,\text{d}\unicode[STIX]{x1D707}.\quad\end{eqnarray}$$

The $\unicode[STIX]{x1D707}$ integral can be carried out using contour integration by writing $\unicode[STIX]{x1D701}=\sqrt{k^{2}w^{2}-\unicode[STIX]{x1D6FE}^{2}}$ where the branch cut is taken such that $\text{Im}[\unicode[STIX]{x1D701}]>0$ when $\unicode[STIX]{x1D6FE}$ is in a strip in the spectral plane for the Wiener–Hopf method. This gives

(3.23)$$\begin{eqnarray}\displaystyle h_{1}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}) & = & \displaystyle \int _{-\infty }^{\infty }\left\{s_{1}(\unicode[STIX]{x1D6FE})+\frac{1}{2}\mathop{\sum }_{n=-\infty }^{\infty }\left[-D_{1}(\unicode[STIX]{x1D6FE})\text{sgn}(ns_{\unicode[STIX]{x1D719}}-\unicode[STIX]{x1D713})\right.\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\left.\frac{G(\unicode[STIX]{x1D6FE})}{\text{i}\unicode[STIX]{x1D701}}\right]\text{e}^{\text{i}n(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }+\unicode[STIX]{x1D6FE}d_{\unicode[STIX]{x1D719}})+\text{i}\unicode[STIX]{x1D701}|ns_{\unicode[STIX]{x1D719}}-\unicode[STIX]{x1D713}|}\right\}\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}}\text{d}\unicode[STIX]{x1D6FE},\end{eqnarray}$$

where we have used the quasi-periodicity (3.14) of the source terms to write

(3.24)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle s_{1}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D713})=\frac{1}{4\text{i}\unicode[STIX]{x03C0}\unicode[STIX]{x1D701}}\int _{0}^{s_{\unicode[STIX]{x1D719}}}S_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D713}_{1})\mathop{\sum }_{n=-\infty }^{\infty }\text{e}^{\text{i}n(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }+\unicode[STIX]{x1D6FE}d_{\unicode[STIX]{x1D719}})+\text{i}\unicode[STIX]{x1D701}|\unicode[STIX]{x1D713}_{1}+ns_{\unicode[STIX]{x1D719}}-\unicode[STIX]{x1D713}|}\,\text{d}\unicode[STIX]{x1D713}_{1},\\ \displaystyle S_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D713}_{1})=\int _{-\infty }^{\infty }\mathbb{S}(\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D713}_{1})\text{e}^{\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}_{1}}\,\text{d}\unicode[STIX]{x1D719}_{1}.\end{array}\right\}\end{eqnarray}$$

We differentiate (3.23) with respect to $\unicode[STIX]{x1D713}$ and take the limit $\unicode[STIX]{x1D713}\rightarrow 0^{\pm }$ to obtain

(3.25)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h_{1}^{\pm }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D719})=2\unicode[STIX]{x03C0}\int _{-\infty }^{\infty }\{-D_{1}(\unicode[STIX]{x1D6FE})j(\unicode[STIX]{x1D6FE})+G(\unicode[STIX]{x1D6FE})[k(\unicode[STIX]{x1D6FE})\mp 1]+S(\unicode[STIX]{x1D6FE})\}\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}}\,\text{d}\unicode[STIX]{x1D6FE},\end{eqnarray}$$

where

(3.26)$$\begin{eqnarray}\displaystyle & \displaystyle j(\unicode[STIX]{x1D6FE})=\frac{\text{i}\unicode[STIX]{x1D701}}{4\unicode[STIX]{x03C0}}\mathop{\sum }_{n=-\infty }^{\infty }\text{e}^{\text{i}n(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }+\unicode[STIX]{x1D6FE}d_{\unicode[STIX]{x1D719}})+\text{i}\unicode[STIX]{x1D701}|ns_{\unicode[STIX]{x1D719}}|}=\frac{\unicode[STIX]{x1D701}}{4\unicode[STIX]{x03C0}}\cdot \frac{\sin (\unicode[STIX]{x1D701}s_{\unicode[STIX]{x1D719}})}{\cos (\unicode[STIX]{x1D701}s_{\unicode[STIX]{x1D719}})-\cos (\unicode[STIX]{x1D6FE}d_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime })}, & \displaystyle\end{eqnarray}$$

(3.27)$$\begin{eqnarray}\displaystyle & \displaystyle k(\unicode[STIX]{x1D6FE})=\frac{1}{4\unicode[STIX]{x03C0}}\mathop{\sum }_{\substack{ n=-\infty \\ n\neq 0}}^{\infty }\text{sgn}(ns_{\unicode[STIX]{x1D719}})\text{e}^{\text{i}n(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }+\unicode[STIX]{x1D6FE}d_{\unicode[STIX]{x1D719}})+\text{i}\unicode[STIX]{x1D701}|ns_{\unicode[STIX]{x1D719}}|}=\frac{1}{4\unicode[STIX]{x03C0}\text{i}}\cdot \frac{\sin (\unicode[STIX]{x1D6FE}d_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime })}{\cos (\unicode[STIX]{x1D701}s_{\unicode[STIX]{x1D719}})-\cos (\unicode[STIX]{x1D6FE}d_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime })},\qquad & \displaystyle\end{eqnarray}$$

(3.28)$$\begin{eqnarray}\displaystyle S(\unicode[STIX]{x1D6FE}) & = & \displaystyle -\frac{1}{8\unicode[STIX]{x03C0}^{2}}\int _{0}^{s_{\unicode[STIX]{x1D719}}}S_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D713}_{1})\mathop{\sum }_{n=-\infty }^{\infty }\text{sgn}(\unicode[STIX]{x1D713}_{1}+ns_{\unicode[STIX]{x1D719}})\text{e}^{\text{i}n(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }+d_{\unicode[STIX]{x1D719}}\unicode[STIX]{x1D6FE})}\text{e}^{\text{i}\unicode[STIX]{x1D701}|\unicode[STIX]{x1D713}_{1}+ns_{\unicode[STIX]{x1D719}}|}\,\text{d}\unicode[STIX]{x1D713}_{1}\nonumber\\ \displaystyle & = & \displaystyle -\frac{1}{8\unicode[STIX]{x03C0}^{2}}\int _{0}^{s_{\unicode[STIX]{x1D719}}}S_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D713}_{1})\mathop{\sum }_{\pm }\frac{\text{e}^{\pm \text{i}\unicode[STIX]{x1D701}\unicode[STIX]{x1D713}_{1}}}{1-\text{e}^{\text{i}(\pm \unicode[STIX]{x1D701}s_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D719}}^{\prime }+\unicode[STIX]{x1D6FE}d_{\unicode[STIX]{x1D719}})}}\,\text{d}\unicode[STIX]{x1D713}_{1},\end{eqnarray}$$

and we have used ((S:5.8), (S:5.9), (S:5.10)) in the supplementary material to simplify the infinite sums. The analysis of the transforms of these source terms is carried out in § S:6 of the supplementary material where the source terms are decomposed into Fourier-type series in the upstream, downstream and inter-blade regions. The modal expansion corresponds to step IVd in the road map.

We note that if we take the difference of formulae (3.25) and use contour integration to calculate the resulting integral, we recover the conditions for the jump in normal velocity (3.6b) and (3.9b), as expected. Conversely, summing formulae (3.25) gives the governing integral equation for the acoustic field

(3.29)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})=4\unicode[STIX]{x03C0}\int _{-\infty }^{\infty }\{-D_{1}(\unicode[STIX]{x1D6FE})j(\unicode[STIX]{x1D6FE})+G(\unicode[STIX]{x1D6FE})k(\unicode[STIX]{x1D6FE})+S(\unicode[STIX]{x1D6FE})\}\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}}\,\text{d}\unicode[STIX]{x1D6FE}.\end{eqnarray}$$

We see that $h_{1}$ and its normal derivative $\unicode[STIX]{x2202}h_{1}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}$ can be expressed solely in terms of the unknown $D_{1}(\unicode[STIX]{x1D6FE})$. Thus our problem is now to find $D_{1}(\unicode[STIX]{x1D6FE})$ which solves the integral equation (3.29) subject to (3.5b), (3.6b) and (3.9b).

The linearity of the governing equation and boundary conditions allow the solution to be split into four parts,

(3.30)$$\begin{eqnarray}h_{1}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})=h_{1,\unicode[STIX]{x1D6F4}}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})+h_{1,\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})+h_{1,\unicode[STIX]{x1D6E4}}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})+h_{1,S}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}).\end{eqnarray}$$

There are several reasons for this splitting, chiefly that each component can be broadly attributed to a feature of the background flow and distortion of the $O(1)$ acoustic field. Firstly, $h_{1,\unicode[STIX]{x1D6F4}}$ depends predominantly on the value of the sum of the normal velocity either side of the blade. In Baddoo & Ayton (Reference Baddoo and Ayton2018b), it was shown that this sum is a function of camber and angle of attack, and thickness terms generally do not appear. Therefore, $h_{1,\unicode[STIX]{x1D6F4}}$ may roughly be viewed as the influence of camber and angle of attack on the acoustic field. Secondly, $h_{1,\unicode[STIX]{x1D6E5}}$ depends predominantly on the jump in normal velocity. In the flat-plate case, this term disappears due to the symmetry of the blades, whereas in the present case this term can be viewed as a mass source required to satisfy the no-flux condition. Similarly, this jump was shown in Baddoo & Ayton (Reference Baddoo and Ayton2018b) to depend predominantly on aerofoil thickness, and therefore $h_{1,\unicode[STIX]{x1D6E5}}$ may be roughly viewed as the influence of thickness on the acoustic field. However, the separation of geometrical effects is not total, as both $h_{1,\unicode[STIX]{x1D6F4}}$ and $h_{1,\unicode[STIX]{x1D6E5}}$ include terms proportional to the drift function, $g$, which does not have a straightforward dependence on aerofoil geometry. Thirdly, $h_{1,\unicode[STIX]{x1D6E4}}$ arises due to the circulation induced on each blade by the background flow. The sum and jump in its normal velocity vanish on the blade, but it has a specified pressure jump (in $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space) across the wake. This jump is necessary to ensure no pressure jump in the physical $(x,y)$-space. This disparity arises due to the circulation around the blade in mean flow. Fourthly, and finally, $h_{1,S}$ depends on the source terms in the equation only. Similarly to $h_{1,\unicode[STIX]{x1D6E4}}$, the sum and jump in its normal velocity on the blades disappear, but unlike the previous solutions it has a forcing term in the governing Helmholtz equation. This forcing depends on the background flow, and the distortion of the $O(1)$ solution by the background flow.

In addition to the above physical reasons, this splitting substantially reduces the mathematical complexity of the problem. Each component induces its own system of Wiener–Hopf problems, and these may be solved separately. The Kutta condition is enforced individually for each component. When the Fourier transform is inverted, the components may be inverted individually, which yields slightly more tractable expressions, each of which may be verified independently.

In the following sections, the governing equations and boundary conditions are stated for each of the four components and we solve for the relevant $D_{1,x}(\unicode[STIX]{x1D6FE})$ (defined in an analogous way to (3.19)). The Fourier transform of the jump in potential either side of the blade is solved using a similar Wiener–Hopf method to Glegg (Reference Glegg1999) and the solution is stated in each case. The analysis is carried out in § S:2 of the supplementary material, wherein the definitions of all new parameters may be found.

3.2.1 The $h_{1,\unicode[STIX]{x1D6F4}}$ problem and solution (road map step Va)

The governing equation is

(3.31a)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1,\unicode[STIX]{x1D6F4}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})=-4\unicode[STIX]{x03C0}\int _{-\infty }^{\infty }D_{1,\unicode[STIX]{x1D6F4}}(\unicode[STIX]{x1D6FE})j(\unicode[STIX]{x1D6FE})\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}}\text{d}\unicode[STIX]{x1D6FE},\end{eqnarray}$$

subject to

(3.31b)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}_{0}[h_{1,\unicode[STIX]{x1D6F4}}](\unicode[STIX]{x1D719})=0,\quad \unicode[STIX]{x1D719}<0; & \displaystyle\end{eqnarray}$$

(3.31c)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F4}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1,\unicode[STIX]{x1D6F4}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})=\mathop{\sum }_{l=-\infty }^{\infty }{\mathcal{c}}_{\unicode[STIX]{x1D6F4},l}\text{e}^{-\text{i}\unicode[STIX]{x1D705}_{l}^{-}\unicode[STIX]{x1D719}},\quad 0<\unicode[STIX]{x1D719}<c_{\unicode[STIX]{x1D719}}; & \displaystyle\end{eqnarray}$$

(3.31d)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}_{0}[h_{1,\unicode[STIX]{x1D6F4}}](\unicode[STIX]{x1D719})=2\unicode[STIX]{x03C0}\text{i}P_{1,\unicode[STIX]{x1D6F4}}\text{e}^{-\text{i}\unicode[STIX]{x1D705}_{0}^{-}\unicode[STIX]{x1D719}},\quad \unicode[STIX]{x1D719}>c_{\unicode[STIX]{x1D719}}. & \displaystyle\end{eqnarray}$$

This problem corresponds to the homogeneous Helmholtz equation with symmetric boundary conditions on each blade (3.7b), along with some unknown vortex shedding into the wake. The decomposition of the boundary data into a Fourier series means that the method of solution is relevant to a range of applications where arbitrary boundary data are prescribed. The method of solution is described in § S:2.2 of the supplementary material. The solution is

(3.32)$$\begin{eqnarray}\displaystyle & & \displaystyle D_{1,\unicode[STIX]{x1D6F4}}(\unicode[STIX]{x1D6FE})=\mathop{\sum }_{l=-\infty }^{\infty }\frac{{\mathcal{T}}_{1,\unicode[STIX]{x1D6F4},l}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{l}^{-}}\cdot \frac{1}{J_{+}(\unicode[STIX]{x1D6FE})}-\mathop{\sum }_{l=-\infty }^{\infty }\frac{{\mathcal{S}}_{1,\unicode[STIX]{x1D6F4},l}\text{e}^{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{l}^{-})c_{\unicode[STIX]{x1D719}}}}{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{l}^{-})}\cdot \frac{J_{-}(\unicode[STIX]{x1D705}_{l}^{-})}{J_{-}(\unicode[STIX]{x1D6FE})}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\mathop{\sum }_{n=0}^{\infty }\frac{{\mathcal{B}}_{1,\unicode[STIX]{x1D6F4},n}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{+}}\cdot \frac{J_{+}(\unicode[STIX]{x1D703}_{n}^{+})}{J_{+}(\unicode[STIX]{x1D6FE})}-\mathop{\sum }_{n=0}^{\infty }\frac{({\mathcal{A}}_{1,\unicode[STIX]{x1D6F4},n}+{\mathcal{C}}_{1,\unicode[STIX]{x1D6F4},n})\text{e}^{\text{i}c_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{-})}}{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{-})}\cdot \frac{J_{-}(\unicode[STIX]{x1D703}_{n}^{-})}{J_{-}(\unicode[STIX]{x1D6FE})}.\end{eqnarray}$$

3.2.2 The $h_{1,\unicode[STIX]{x1D6E5}}$ problem and solution (road map step Vb)

The governing equation is

(3.33a)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1,\unicode[STIX]{x1D6E5}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})=4\unicode[STIX]{x03C0}\int _{-\infty }^{\infty }\{-D_{1,\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D6FE})j(\unicode[STIX]{x1D6FE})+G_{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D6FE})k(\unicode[STIX]{x1D6FE})\}\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}}\,\text{d}\unicode[STIX]{x1D6FE},\end{eqnarray}$$

where

(3.33b)$$\begin{eqnarray}G_{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D6FE})=\mathop{\sum }_{l=-\infty }^{\infty }\frac{{\mathcal{c}}_{\unicode[STIX]{x1D6E5},l}((-1)^{l}\text{e}^{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})c_{\unicode[STIX]{x1D719}}}-1)}{2\unicode[STIX]{x03C0}\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{l}^{-})}.\end{eqnarray}$$

This integral equation is solved subject to

(3.33c)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}_{0}[h_{1,\unicode[STIX]{x1D6E5}}](\unicode[STIX]{x1D719})=0,\quad \unicode[STIX]{x1D719}<0; & \displaystyle\end{eqnarray}$$

(3.33d)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F4}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1,\unicode[STIX]{x1D6E5}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})=0,\quad 0<\unicode[STIX]{x1D719}<c_{\unicode[STIX]{x1D719}}; & \displaystyle\end{eqnarray}$$

(3.33e)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}_{0}[h_{1,\unicode[STIX]{x1D6E5}}](\unicode[STIX]{x1D719})=2\unicode[STIX]{x03C0}\text{i}P_{1,\unicode[STIX]{x1D6E5}}\text{e}^{-\text{i}\unicode[STIX]{x1D705}_{0}^{-}\unicode[STIX]{x1D719}},\quad \unicode[STIX]{x1D719}>c_{\unicode[STIX]{x1D719}}. & \displaystyle\end{eqnarray}$$

This problem corresponds to the homogeneous Helmholtz equation with anti-symmetric boundary conditions along the blade. This term is not present in the flat-plate case, but the blade geometry induces a difference between upper and lower surface velocities, which generates anti-symmetric terms in the boundary data (3.6b). The method of solution is described in § S:2.3 of the supplementary material. The solution is

(3.34)$$\begin{eqnarray}\displaystyle & & \displaystyle D_{1,\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D6FE})=\mathop{\sum }_{l=-\infty }^{\infty }\frac{{\mathcal{T}}_{1,\unicode[STIX]{x1D6E5},l}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{l}^{-}}\cdot \frac{1}{J_{+}(\unicode[STIX]{x1D6FE})}-\mathop{\sum }_{l=-\infty }^{\infty }\frac{{\mathcal{S}}_{1,\unicode[STIX]{x1D6E5},l}\text{e}^{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{l}^{-})c_{\unicode[STIX]{x1D719}}}}{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{l}^{-})}\cdot \frac{J_{-}(\unicode[STIX]{x1D705}_{l}^{-})}{J_{-}(\unicode[STIX]{x1D6FE})}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\mathop{\sum }_{m=-\infty }^{\infty }\frac{{\mathcal{R}}_{1,\unicode[STIX]{x1D6E5},m}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{-}}\cdot \frac{1}{J_{+}(\unicode[STIX]{x1D6FE})}-\mathop{\sum }_{m=-\infty }^{\infty }\frac{{\mathcal{U}}_{1,\unicode[STIX]{x1D6E5},m}}{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{+})}\cdot \frac{1}{J_{-}(\unicode[STIX]{x1D6FE})}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\mathop{\sum }_{n=0}^{\infty }\frac{{\mathcal{B}}_{1,\unicode[STIX]{x1D6E5},n}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{+}}\cdot \frac{J_{+}(\unicode[STIX]{x1D703}_{n}^{+})}{J_{+}(\unicode[STIX]{x1D6FE})}-\mathop{\sum }_{n=0}^{\infty }\frac{\left({\mathcal{A}}_{1,\unicode[STIX]{x1D6E5},n}+{\mathcal{C}}_{1,\unicode[STIX]{x1D6E5},n}\right)\text{e}^{\text{i}c_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{-})}}{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{-})}\cdot \frac{J_{-}(\unicode[STIX]{x1D703}_{n}^{-})}{J_{-}(\unicode[STIX]{x1D6FE})}.\end{eqnarray}$$

3.2.3 The $h_{1,\unicode[STIX]{x1D6E4}}$ problem and solution (road map step Vc)

The governing equation is

(3.35a)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1,\unicode[STIX]{x1D6E4}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})=4\unicode[STIX]{x03C0}\int _{-\infty }^{\infty }\{-D_{1,\unicode[STIX]{x1D6E4}}(\unicode[STIX]{x1D6FE})j(\unicode[STIX]{x1D6FE})+G_{\unicode[STIX]{x1D6E4}}(\unicode[STIX]{x1D6FE})k(\unicode[STIX]{x1D6FE})\}\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}}\,\text{d}\unicode[STIX]{x1D6FE},\end{eqnarray}$$

where

(3.35b)$$\begin{eqnarray}G_{\unicode[STIX]{x1D6E4}}(\unicode[STIX]{x1D6FE})=\frac{\unicode[STIX]{x1D6E4}}{2\unicode[STIX]{x03C0}}\left(\mathop{\sum }_{m=-\infty }^{\infty }\frac{\unicode[STIX]{x1D701}_{m}^{-}{\mathcal{H}}_{0,m}^{\prime }\text{e}^{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{-})c_{\unicode[STIX]{x1D719}}}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{-}}+\mathop{\sum }_{\pm }\frac{\pm \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{0}^{-}}{\mathcal{H}}_{0,\unicode[STIX]{x1D705}_{0}^{-}}^{\pm \prime }\text{e}^{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})c_{\unicode[STIX]{x1D719}}}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-}}\right).\end{eqnarray}$$

This integral equation is solved subject to

(3.35c)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}_{0}[h_{1,\unicode[STIX]{x1D6E4}}](\unicode[STIX]{x1D719})=0,\quad \unicode[STIX]{x1D719}<0; & \displaystyle\end{eqnarray}$$

(3.35d)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F4}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1,\unicode[STIX]{x1D6E4}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})=0,\quad 0<\unicode[STIX]{x1D719}<c_{\unicode[STIX]{x1D719}}; & \displaystyle\end{eqnarray}$$

(3.35e)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{0}[h_{1,\unicode[STIX]{x1D6E4}}](\unicode[STIX]{x1D719}) & = & \displaystyle 2\unicode[STIX]{x03C0}\text{i}P_{1,\unicode[STIX]{x1D6E4}}\text{e}^{-\text{i}\unicode[STIX]{x1D705}_{0}^{-}\unicode[STIX]{x1D719}}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6E4}\left(\mathop{\sum }_{\pm }{\mathcal{H}}_{0,\unicode[STIX]{x1D705}_{0}^{-}}^{\pm \prime }\text{e}^{-\text{i}\unicode[STIX]{x1D705}_{0}^{-}\unicode[STIX]{x1D719}}+\mathop{\sum }_{m=-\infty }^{\infty }{\mathcal{H}}_{0,m}^{\prime }\text{e}^{-\text{i}\unicode[STIX]{x1D706}_{m}^{-}\unicode[STIX]{x1D719}}\right),\quad \unicode[STIX]{x1D719}>c_{\unicode[STIX]{x1D719}}.\end{eqnarray}$$

This problem corresponds to the homogeneous Helmholtz equation with homogeneous boundary conditions on the blade and prescribed data along the wake ((3.8b), (3.9b)) that necessitates no flux and no pressure jump across the wake in physical space. In this case it happens that these prescribed data can be expressed in a Fourier series of the acoustic modes of the cascade, but any type of Fourier series could have been considered. However, the acoustic potential function is not fully specified along the wake, and the unknown constant $P_{1,\unicode[STIX]{x1D6E4}}$ still needs to be determined via the Wiener–Hopf method to specify the strength of vortex shedding into the wake. The method of solution is described in § S:2.4 of the supplementary material. The solution is

(3.36)$$\begin{eqnarray}\displaystyle D_{1,\unicode[STIX]{x1D6E4}}(\unicode[STIX]{x1D6FE}) & = & \displaystyle -\frac{\unicode[STIX]{x1D6E4}}{2\unicode[STIX]{x03C0}\text{i}}\cdot \mathop{\sum }_{m=-\infty }^{\infty }\frac{(\unicode[STIX]{x1D706}_{m}^{-}-\unicode[STIX]{x1D705}_{0}^{-}){\mathcal{H}}_{0,m}^{\prime }\text{e}^{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{-})c_{\unicode[STIX]{x1D719}}}}{(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{-})}\cdot \frac{J_{-}(\unicode[STIX]{x1D706}_{m}^{-})}{J_{-}(\unicode[STIX]{x1D6FE})}\nonumber\\ \displaystyle & & \displaystyle -\,\mathop{\sum }_{m=-\infty }^{\infty }\frac{{\mathcal{U}}_{1,\unicode[STIX]{x1D6E4},m}\text{e}^{\text{i}\unicode[STIX]{x1D6FE}c_{\unicode[STIX]{x1D719}}}}{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{+})}\cdot \frac{1}{J_{-}(\unicode[STIX]{x1D6FE})}-\mathop{\sum }_{n=0}^{\infty }\frac{{\mathcal{B}}_{1,\unicode[STIX]{x1D6E4},n}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{+}}\cdot \frac{J_{+}(\unicode[STIX]{x1D703}_{n}^{+})}{J_{+}(\unicode[STIX]{x1D6FE})}\nonumber\\ \displaystyle & & \displaystyle -\,\mathop{\sum }_{n=0}^{\infty }\frac{{\mathcal{C}}_{1,\unicode[STIX]{x1D6E4},n}\text{e}^{\text{i}c_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{-})}}{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{-})}\cdot \frac{J_{-}(\unicode[STIX]{x1D703}_{n}^{-})}{J_{-}(\unicode[STIX]{x1D6FE})}.\end{eqnarray}$$

3.2.4 The $h_{1,S}$ problem and solution (road map step Vd)

The governing equation is

(3.37a)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1,S}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})=4\unicode[STIX]{x03C0}\int _{-\infty }^{\infty }\{-D_{1,S}(\unicode[STIX]{x1D6FE})j(\unicode[STIX]{x1D6FE})+S(\unicode[STIX]{x1D6FE})\}\text{e}^{-\text{i}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}}\,\text{d}\unicode[STIX]{x1D6FE},\end{eqnarray}$$

subject to homogeneous boundary conditions analogous to those in (3.33c), (3.33d) and (3.33e). This problem is the inhomogeneous Helmholtz equation subject to homogeneous boundary conditions in a cascade geometry. The only requirements on the source terms are that they can be expanded in the upstream and downstream regions as series of the form (S:6.1) and (S:6.3) respectively. Consequently, this method of solution can be applied to other problems that involve forcing terms in a cascade geometry, such as in Guzman Inigo et al. (Reference Guzman Inigo, Baddoo, Morgans and Ayton2019). By linearity, this solution can be combined with any of $h_{1,\unicode[STIX]{x1D6F4}}$, $h_{1,\unicode[STIX]{x1D6E5}}$ or $h_{1,\unicode[STIX]{x1D6E4}}$ to generate a solution to the inhomogeneous Helmholtz equation with inhomogeneous boundary conditions. The boundary conditions for $h_{1,S}$ are given by

(3.37b)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}_{0}[h_{1,S}](\unicode[STIX]{x1D719})=0,\quad \unicode[STIX]{x1D719}<0; & \displaystyle\end{eqnarray}$$

(3.37c)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F4}_{0}\left[\frac{\unicode[STIX]{x2202}h_{1,S}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\right](\unicode[STIX]{x1D719})=0,\quad 0<\unicode[STIX]{x1D719}<c_{\unicode[STIX]{x1D719}}; & \displaystyle\end{eqnarray}$$

(3.37d)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}_{0}[h_{1,S}](\unicode[STIX]{x1D719})=2\unicode[STIX]{x03C0}\text{i}P_{1,S}\text{e}^{-\text{i}\unicode[STIX]{x1D705}_{0}^{-}\unicode[STIX]{x1D719}},\quad \unicode[STIX]{x1D719}>c_{\unicode[STIX]{x1D719}}. & \displaystyle\end{eqnarray}$$

The method of solution is described in § S:2.5 of the supplementary material. The solution is

(3.38)$$\begin{eqnarray}\displaystyle & & \displaystyle D_{1,S}(\unicode[STIX]{x1D6FE})=\mathop{\sum }_{m=-\infty }^{\infty }\frac{{\mathcal{R}}_{1,S,m}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{-}}\cdot \frac{1}{J_{+}(\unicode[STIX]{x1D6FE})}-\mathop{\sum }_{m=-\infty }^{\infty }\frac{{\mathcal{U}}_{1,S,m}}{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D706}_{m}^{+})}\cdot \frac{\text{e}^{\text{i}\unicode[STIX]{x1D6FE}c_{\unicode[STIX]{x1D719}}}}{J_{-}(\unicode[STIX]{x1D6FE})}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\mathop{\sum }_{n=0}^{\infty }\frac{{\mathcal{B}}_{1,S,n}}{\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{+}}\cdot \frac{J_{+}(\unicode[STIX]{x1D703}_{n}^{+})}{J_{+}(\unicode[STIX]{x1D6FE})}-\mathop{\sum }_{n=0}^{\infty }\frac{({\mathcal{A}}_{1,S,n}+{\mathcal{C}}_{1,S,n})\text{e}^{\text{i}c_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{-})}}{\text{i}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D705}_{0}^{-})(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D703}_{n}^{-})}\cdot \frac{J_{-}(\unicode[STIX]{x1D703}_{n}^{-})}{J_{-}(\unicode[STIX]{x1D6FE})}.\quad\end{eqnarray}$$

4.1 Upstream region (I)

In the upstream region,

(4.3)$$\begin{eqnarray}\displaystyle h(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}) & = & \displaystyle \mathop{\sum }_{m=-\infty }^{\infty }{\mathcal{H}}_{m}^{+}\exp [-\text{i}(\unicode[STIX]{x1D706}_{m}^{+}\unicode[STIX]{x1D719}-\unicode[STIX]{x1D701}_{m}^{+}\unicode[STIX]{x1D713})]\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{M_{m}^{+}}\unicode[STIX]{x1D716}\mathbb{S}_{P,M_{m}^{+}}^{+}\mathbb{I}_{P,M_{m}^{+}}^{+}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}).\end{eqnarray}$$

4.2 Inter-blade upstream region (II)

In the inter-blade upstream region,

(4.4)$$\begin{eqnarray}\displaystyle h(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}) & = & \displaystyle \mathop{\sum }_{m=-\infty }^{\infty }\{({\mathcal{H}}_{a,m}^{+}\cos (\unicode[STIX]{x1D713}\unicode[STIX]{x1D701}_{m}^{+})+{\mathcal{H}}_{G,a,m}^{+}\sin (\unicode[STIX]{x1D701}_{m}^{+}\unicode[STIX]{x1D713}))\exp [-\text{i}\unicode[STIX]{x1D706}_{m}^{+}\unicode[STIX]{x1D719}]\nonumber\\ \displaystyle & & \displaystyle +\,({\mathcal{H}}_{{\mathcal{R}},a,m}^{-}\cos (\unicode[STIX]{x1D713}\unicode[STIX]{x1D701}_{m}^{-})+{\mathcal{H}}_{G,a,m}^{-}\exp [\text{i}\unicode[STIX]{x1D701}_{m}^{-}\unicode[STIX]{x1D713}])\exp [-\text{i}\unicode[STIX]{x1D706}_{m}^{-}\unicode[STIX]{x1D719}]\!\}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{n=0}^{\infty }({\mathcal{H}}_{\unicode[STIX]{x1D703}^{+},a,n}\exp [-\text{i}\unicode[STIX]{x1D703}_{n}^{+}\unicode[STIX]{x1D719}]+{\mathcal{H}}_{\unicode[STIX]{x1D703}^{-},a,n}\exp [-\text{i}\unicode[STIX]{x1D703}_{n}^{-}\unicode[STIX]{x1D719}])\cos (n\unicode[STIX]{x1D713})\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{l=-\infty }^{\infty }\{({\mathcal{H}}_{S,a,l}+{\mathcal{H}}_{\unicode[STIX]{x1D6F4},a,l})\cos (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}\left(\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}}\right))+{\mathcal{H}}_{S,b,l}\cos (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}\unicode[STIX]{x1D713})\nonumber\\ \displaystyle & & \displaystyle +\,({\mathcal{H}}_{S,c,l}+{\mathcal{H}}_{\unicode[STIX]{x1D6E5},a,l})\sin (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}(\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}}))+{\mathcal{H}}_{S,d,l}\sin (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}\unicode[STIX]{x1D713})\!\}\nonumber\\ \displaystyle & & \displaystyle \times \,\exp [-\text{i}\unicode[STIX]{x1D705}_{l}^{-}\unicode[STIX]{x1D719}]\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{l_{1},l_{2}}\unicode[STIX]{x1D716}\mathbb{S}_{l_{1},l_{2}}^{i}\mathbb{I}_{l_{1},l_{2}}^{i}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}).\end{eqnarray}$$

4.3 Inter-blade inner region (III)

In the inter-blade inner region,

(4.5)$$\begin{eqnarray}\displaystyle h(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}) & = & \displaystyle \mathop{\sum }_{n=0}^{\infty }\{({\mathcal{H}}_{\unicode[STIX]{x1D703}^{+},a,n}+{\mathcal{H}}_{\unicode[STIX]{x1D703}^{+},b,n})\exp [-\text{i}\unicode[STIX]{x1D703}_{n}^{+}\unicode[STIX]{x1D719}]\nonumber\\ \displaystyle & & \displaystyle +\,({\mathcal{H}}_{\unicode[STIX]{x1D703}^{-},a,n}+{\mathcal{H}}_{\unicode[STIX]{x1D703}^{-},b,n})\exp [-\text{i}\unicode[STIX]{x1D703}_{n}^{-}\unicode[STIX]{x1D719}]\!\}\cos (n\unicode[STIX]{x1D713})\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{l=-\infty }^{\infty }\{({\mathcal{H}}_{S,a,l}+{\mathcal{H}}_{\unicode[STIX]{x1D6F4},a,l})\cos (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}(\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}}))+({\mathcal{H}}_{S,b,l}+{\mathcal{H}}_{\unicode[STIX]{x1D6F4},b,l})\cos (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}\unicode[STIX]{x1D713})\nonumber\\ \displaystyle & & \displaystyle +\,({\mathcal{H}}_{S,c,l}+{\mathcal{H}}_{\unicode[STIX]{x1D6E5},a,l})\sin (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}(\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}}))+({\mathcal{H}}_{S,d,l}+{\mathcal{H}}_{\unicode[STIX]{x1D6E5},b,l})\sin (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}\unicode[STIX]{x1D713})\!\}\nonumber\\ \displaystyle & & \displaystyle \times \,\exp [-\text{i}\unicode[STIX]{x1D705}_{l}^{-}\unicode[STIX]{x1D719}]\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{l_{1},l_{2}}\unicode[STIX]{x1D716}\mathbb{S}_{l_{1},l_{2}}^{i}\mathbb{I}_{l_{1},l_{2}}^{i}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}).\end{eqnarray}$$

4.4 Inter-blade downstream region (IV)

In the inter-blade downstream region,

(4.6)$$\begin{eqnarray}\displaystyle h(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}) & = & \displaystyle \mathop{\sum }_{m=-\infty }^{\infty }\{\!(\!{\mathcal{H}}_{b,m}^{-}\cos ((\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}})\unicode[STIX]{x1D701}_{m}^{-})+{\mathcal{H}}_{G,b,m}^{-}\sin ((\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}})\unicode[STIX]{x1D701}_{m}^{-})\nonumber\\ \displaystyle & & \displaystyle +\,{\mathcal{H}}_{m,b}^{-}\exp [-\text{i}\unicode[STIX]{x1D701}_{m}^{-}\unicode[STIX]{x1D713}]\!)\exp [-\text{i}\unicode[STIX]{x1D706}_{m}^{-}\unicode[STIX]{x1D719}]\nonumber\\ \displaystyle & & \displaystyle +\,({\mathcal{H}}_{{\mathcal{U}},b,m}^{+}\cos ((\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}})\unicode[STIX]{x1D701}_{m}^{+})+{\mathcal{H}}_{G,b,m}^{+}\exp [-\text{i}\unicode[STIX]{x1D701}_{m}^{+}\unicode[STIX]{x1D713}])\exp [-\text{i}\unicode[STIX]{x1D706}_{m}^{+}\unicode[STIX]{x1D719}]\nonumber\\ \displaystyle & & \displaystyle +\,(\unicode[STIX]{x1D706}_{m}^{-}(\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}})-\unicode[STIX]{x1D701}_{m}^{-}\unicode[STIX]{x1D719}){\mathcal{H}}_{\unicode[STIX]{x1D6E4},m,b}^{-}\text{e}^{-\text{i}(\unicode[STIX]{x1D706}_{m}^{-}\unicode[STIX]{x1D719}+\unicode[STIX]{x1D701}_{m}^{-}\unicode[STIX]{x1D713})}\!\}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{n=0}^{\infty }(\!{\mathcal{H}}_{\unicode[STIX]{x1D703}^{+},b,n}\exp [-\text{i}\unicode[STIX]{x1D703}_{n}^{+}\unicode[STIX]{x1D719}]+{\mathcal{H}}_{\unicode[STIX]{x1D703}^{-},b,n}\exp [-\text{i}\unicode[STIX]{x1D703}_{n}^{-}\unicode[STIX]{x1D719}]\!)\cos (n\unicode[STIX]{x1D713})\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{l=-\infty }^{\infty }\{\!{\mathcal{H}}_{S,a,l}\cos (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}(\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}}))+({\mathcal{H}}_{S,b,l}+{\mathcal{H}}_{\unicode[STIX]{x1D6F4},b,l})\cos (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}\unicode[STIX]{x1D713})\nonumber\\ \displaystyle & & \displaystyle +\,{\mathcal{H}}_{S,c,l}\sin (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}(\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}}))+({\mathcal{H}}_{S,d,l}+{\mathcal{H}}_{\unicode[STIX]{x1D6E5},b,l})\sin (\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{l}^{-}}\unicode[STIX]{x1D713})\!\}\nonumber\\ \displaystyle & & \displaystyle \times \,\exp [-\text{i}\unicode[STIX]{x1D705}_{l}^{-}\unicode[STIX]{x1D719}]\nonumber\\ \displaystyle & & \displaystyle +\,\left(\mathop{\sum }_{\pm }{\mathcal{H}}_{\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D705}_{0}^{-},\pm ,b}^{-}\exp [\mp \text{i}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{0}^{-}}\unicode[STIX]{x1D713}]+{\mathcal{H}}_{\unicode[STIX]{x1D705}_{0}^{-},b}^{-}\cos ((\unicode[STIX]{x1D713}-s_{\unicode[STIX]{x1D719}})\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{0}^{-}})\right)\exp [-\text{i}\unicode[STIX]{x1D705}_{0}^{-}\unicode[STIX]{x1D719}]\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{l_{1},l_{2}}\unicode[STIX]{x1D716}\mathbb{S}_{l_{1},l_{2}}^{i}\mathbb{I}_{l_{1},l_{2}}^{i}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}).\end{eqnarray}$$

4.5 Downstream region (V)

In the downstream region,

(4.7)$$\begin{eqnarray}\displaystyle h(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}) & = & \displaystyle \mathop{\sum }_{m=-\infty }^{\infty }{\mathcal{H}}_{m}^{-}\exp [-\text{i}(\tilde{\unicode[STIX]{x1D706}}_{m}^{-}\unicode[STIX]{x1D719}+\tilde{\unicode[STIX]{x1D701}}_{m}^{-}\unicode[STIX]{x1D713})]\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{M_{m}^{-}}\unicode[STIX]{x1D716}\mathbb{S}_{P,M_{m}^{-}}^{-}\mathbb{I}_{P,M_{m}^{-}}^{-}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})+\mathop{\sum }_{N_{\unicode[STIX]{x1D719},n}^{-}}\unicode[STIX]{x1D716}\mathbb{S}_{N,N_{\unicode[STIX]{x1D719},n}^{-}}^{-}\mathbb{I}_{N,N_{\unicode[STIX]{x1D719},n}^{-}}^{-}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{\pm }\unicode[STIX]{x1D716}\mathbb{S}_{N,\unicode[STIX]{x1D705}_{0}^{-},\pm }^{-}\mathbb{I}_{N,\unicode[STIX]{x1D705}_{0}^{-},\pm }^{-}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})+\mathop{\sum }_{\pm }{\mathcal{H}}_{\unicode[STIX]{x1D705}_{0}^{-},\pm }^{-}\text{e}^{-\text{i}(\unicode[STIX]{x1D705}_{0}^{-}\unicode[STIX]{x1D719}\pm \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D705}_{0}^{-}}\unicode[STIX]{x1D713})},\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{m=-\infty }^{\infty }{\mathcal{H}}_{\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D706}_{m}^{-}}^{-}\unicode[STIX]{x1D713}\exp [-\text{i}(\tilde{\unicode[STIX]{x1D706}}_{m}^{-}\unicode[STIX]{x1D719}+\tilde{\unicode[STIX]{x1D701}}_{m}^{-}\unicode[STIX]{x1D713})].\end{eqnarray}$$

4.6 Fourier inversion discussion

It may be verified by hand that the acoustic potential satisfies the boundary conditions of no discontinuities upstream (3.5b), the specified normal velocity on either side of the blade ((3.6b), (3.7b)), the specified jump in normal velocity (in $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})$-space) across the wake (3.8b) and the specified jump in acoustic potential across the wake (3.9b). The pressure jump also vanishes (except in the $h_{1,\unicode[STIX]{x1D6E4}}$ case, where a finite jump is permitted), at the trailing edge, signifying the fulfilment of the unsteady Kutta condition. It can also be verified that the function satisfies the inhomogeneous convected Helmholtz equation, and has the correct forcing due to the source terms. In § 5 it will also be observed that $h$ is continuous across the different regions.