2.1. Elliptic-cylindrical coordinate system

In three-dimensional Euclidean space, the so-called elliptic-cylindrical coordinate system is an orthogonal coordinate system generated by projecting the two-dimensional elliptic coordinate system along the perpendicular $z$ -axis. The two-dimensional elliptic coordinate system, as shown in figure 4, consist of ellipses and hyperbolas with coincident foci $F_1$ and $F_2$ at $(-h_0,0)$ and $(h_0,0)$ , respectively. The definition of the elliptic-cylindrical coordinates $(\xi ,\eta ,\tilde {z})$ is then given by

(2.2) \begin{equation} \left \{\!\! \begin{array}{l} \tilde {x} = h_0 \cosh \xi \cos \eta , \\ \tilde {y} = h_0 \sinh \xi \sin \eta , \\ \tilde {z} = \tilde {z}, \end{array} \right . \end{equation}

where $\xi$ is non-negative real, and $-\unicode{x03C0} \lt \eta \leqslant \unicode{x03C0}$ . With the definition (2.2), all confocal ellipses could be expressed as $\xi = \textrm {constant}$ . Assuming that $\xi = \xi _0$ refers to the duct wall, we obtain

(2.3) \begin{equation} \xi _0 = \text{arctanh}\left (\frac {b_0}{a_0}\right )\!. \end{equation}

Figure 4.

The two-dimensional elliptic coordinate system, where the thin solid lines and dotted lines are elliptic and hyperbolic coordinate lines, respectively. The thick solid line represents the cross-section of the duct wall, which is an ellipse with semi-major axis $a_0=0.5$ m and semi-minor axis $b_0=0.3$ m.

2.2. Governing equations

By making the spatial dimensions $\{\tilde {x}, \tilde {y}, \tilde {z}\}$ , mean velocity $v_0$ , acoustic density $\tilde {\rho }$ , time $\tilde {t}$ , velocity potential $\tilde {\phi }$ , and acoustic pressure $\tilde {p}$ dimensionless on $h_0$ , $c_0$ , $\rho _0$ , $h_0/c_0$ , $h_0c_0$ and $c_0^2\rho _0$ , respectively, the convected wave equation for velocity potential $\phi$ is

(2.4) \begin{equation} \left (\frac {\partial }{\partial t}+M\frac {\partial }{\partial z}\right )^2\phi - \Delta \phi = 0, \end{equation}

where $M = v_0/c_0$ , and $\Delta$ is the three-dimensional Laplace operator with respect to the dimensionless spatial variables $x$ , $y$ and $z$ . By taking the elliptic-cylindrical coordinates (2.2), (2.4) is extended as

(2.5) \begin{equation} \left (\frac {\partial }{\partial t}+M\frac {\partial }{\partial z}\right )^2\phi - \left [\frac {2}{\cosh 2\xi - \cos 2\eta }\left (\frac {\partial ^2 \phi }{\partial \xi ^2}+\frac {\partial ^2 \phi }{\partial \eta ^2}\right )+\frac {\partial ^2 \phi }{\partial z^2}\right ] = 0. \end{equation}

From the velocity potential, we can obtain the acoustic pressure $p$ , the particle velocity $\boldsymbol{v}=(v_\xi ,v_\eta ,v_z)$ and the acoustic density $\rho$ as

(2.6) \begin{equation} p = -\left (\frac {\partial }{\partial t}+M\frac {\partial }{\partial z}\right )\phi ,\quad \boldsymbol{v} = \boldsymbol{\nabla }\phi ,\quad \rho = p, \end{equation}

where $\boldsymbol{\nabla}$ denotes the gradient operator with respect to $\xi$ , $\eta$ and $z$ . The continuity of the acoustic pressure and particle velocity across the line segment connecting the two foci $F_1$ and $F_2$ yields

(2.7) \begin{equation} \left (\frac {\partial }{\partial t}+M\frac {\partial }{\partial z}\right )\phi (0,\eta ,z,t) = \left (\frac {\partial }{\partial t}+M\frac {\partial }{\partial z}\right )\phi (0,-\eta ,z,t), \ \forall z, \end{equation}

and

(2.8) \begin{equation} \frac {\partial }{\partial \xi }\phi (0, \eta , z, t) = -\frac {\partial }{\partial \xi }\phi (0, -\eta , z, t), \ \forall z, \end{equation}

respectively. The natural boundary conditions (2.7) and (2.8) differ from previous conditions in a traditional circular duct because the radial variable $\xi =0$ represents a line segment rather than a point, as it does in a circular duct.

The total acoustic field can be expressed as the summation of incident wave $\psi _i$ and scattered wave $\psi _s$ , which indicates

(2.9) \begin{equation} \phi (\xi ,\eta ,z,t) = [\psi _i(\xi ,\eta ,z)+\psi _s(\xi ,\eta ,z)]\exp (-\textrm {i}\omega t). \end{equation}

2.3. Sound propagating inside an elliptic duct

To solve for the incident wave $\psi _i$ , we first need to understand the characteristics of sound propagation within an elliptic duct. Following Lowson & Baskaran (Reference Lowson and Baskaran1975), for the acoustic velocity potential field inside the duct, we assume

(2.10) \begin{equation} \phi (\xi ,\eta ,z,t) = \varPhi _\xi (\xi )\,\varPhi _\eta (\eta )\,\varPhi _z(z)\exp (-\textrm {i}\omega t). \end{equation}

Substituting (2.10) into (2.5), we obtain

(2.11) \begin{align} \frac {2}{\cosh 2\xi - \cos 2\eta } \left (\frac {1}{\varPhi _\xi }\frac {\textrm {d}^2 \varPhi _\xi }{\textrm {d} \xi ^2} + \frac {1}{\varPhi _\eta }\frac {\textrm {d}^2 \varPhi _\eta }{\textrm {d} \eta ^2}\right ) + 4q &= 0, \end{align}

(2.12) \begin{align} \frac {1}{\varPhi _z}\frac {\textrm {d}^2 \varPhi _z}{\textrm {d} z^2} - \frac {1}{\varPhi _z}\left (-\textrm {i}\omega +M\frac {\textrm {d}}{\textrm {d} z}\right )^2\varPhi _z - 4q &= 0, \\[9pt] \nonumber \end{align}

where $q$ is a constant arising naturally from the process of separation of variables. Equation (2.11) will be used later to calculate the modal patterns across the elliptic duct cross-section, and (2.12) is the characteristic equation that governs the axial propagation. At this stage, we are dealing with a non-spinning elliptic mode propagating inside the duct. This means that we should obtain a standing wave solution rather than a travelling wave solution in both the azimuthal and radial directions. Therefore, the solution to (2.11), which corresponds to the solution of the angular and radial Mathieu equations, should be real. Given the relationship between the Mathieu function and its parameter $q$ , $q$ should be non-negative real.

For (2.11), the separation of $\varPhi _\xi$ and $\varPhi _\eta$ yields the following two equations: the classical forms of the angular Mathieu differential equation

(2.13) \begin{equation} \frac {\textrm {d}^2\varPhi _\eta }{\textrm {d}\eta ^2} + (a - 2q\cos 2\eta ) \varPhi _\eta = 0, \end{equation}

and the radial Mathieu differential equation

(2.14) \begin{equation} \frac {\textrm {d}^2\varPhi _\xi }{\textrm {d}\xi ^2} - (a - 2q\cosh 2\xi ) \varPhi _\xi = 0, \end{equation}

where $a$ is the eigenvalue derived from the recurrence relations (Appendix A; see Berezman et al. (Reference Berezman, Kerimov, Skorokhodov and Shadrin1986) for details).

The general solutions of (2.13) and (2.14) with four arbitrary constants $C_1$ , $C_2$ , $C_3$ and $C_4$ are

(2.15) \begin{align} \varPhi _\eta (\eta ) &= C_1{\textit{ce}}_m(\eta ,q) + C_2{\textit{se}}_m(\eta ,q), \end{align}

(2.16) \begin{align} \varPhi _\xi (\xi ) &= C_3{\textit{Mc}}^{(1)}_m(\xi ,q) + C_4{\textit{Ms}}^{(1)}_m(\xi ,q), \end{align}

where $m$ is the acoustic mode number along the angular direction $\eta$ , which will take a non-negative integer (for $C_2,C_4 = 0$ ) or positive integer (for $C_2,C_4 \neq 0$ ) to satisfy the circumferential periodicity condition, ${\textit{ce}}_m$ and ${\textit{se}}_m$ are angular Mathieu functions of even and odd form, respectively, and ${\textit{Mc}}^{(1)}_m$ and ${\textit{Ms}}^{(1)}_m$ are first-kind radial Mathieu functions of even and odd form, respectively. The normalisation of Mathieu functions follows the convention described in McLachlan (Reference McLachlan1947). Because of the physical constraints in this problem, the radial variable $\xi$ will be non-negative real.

The angular Mathieu functions ${\textit{ce}}_m(\eta ,q)$ and ${\textit{se}}_m(\eta ,q)$ can be classified as those in table 1. Note that ${\textit{se}}_0(\eta ,q)$ is undefined, and the angular distribution of a plane wave modal pattern can be expressed simply by using ${\textit{ce}}_0(\eta ,q)$ .

Table 1.

Classification of angular Mathieu functions by the parity and periodicity with respect to argument $\eta$ , where $l = 0,1,2,\ldots$ and $q$ is the parameter.

Assume $\varPsi (\xi ,\eta ) = \varPhi _\eta (\eta )\,\varPhi _\xi (\xi )$ . The continuity conditions (2.7) and (2.8) yield

(2.17) \begin{align} \varPsi (0, \eta ) - \varPsi (0, -\eta ) &= 0, \quad \forall \eta , \end{align}

(2.18) \begin{align} \frac {\partial }{\partial \xi } \varPsi (0, \eta ) + \frac {\partial }{\partial \xi } \varPsi (0, -\eta ) &= 0, \quad \forall \eta . \end{align}

Substitute (2.17) and (2.18) into (2.15) and (2.16). Considering the oddness of ${\textit{Ms}}^{(1)}_m$ and ${\textit{se}}_m$ and the evenness of ${\textit{Mc}}^{(1)}_m$ and ${\textit{ce}}_m$ , we obtain

(2.19) \begin{align} C_2C_3\, {\textit{Mc}}^{(1)}_m(0,q) {\textit{se}}_m(\eta ,q) - C_2C_3\, {\textit{Mc}}^{(1)}_m(0,q) {\textit{se}}_m(-\eta ,q) &= 0,\quad \forall \eta , \end{align}

(2.20) \begin{align} C_1C_4\, {{\textit{Ms}}^{(1)}_m}'(0,q) {\textit{ce}}_m(\eta ,q) + C_1C_4\, {{\textit{Ms}}^{(1)}_m}'(0,q){\textit{ce}}_m(-\eta ,q) &= 0,\quad \forall \eta , \end{align}

where $({\cdot })'$ denotes the derivative of a radial Mathieu function with respect to its argument  $\xi$ . Hence $C_2C_3 = C_1C_4 = 0$ , and the general solution, with arbitrary complex constants $A_m = C_1C_3$ , $B_m = C_2C_4$ , is

(2.21) \begin{equation} \varPsi (\xi ,\eta ) = A_m\,{\textit{Mc}}_m^{(1)}(\xi ,q){\textit{ce}}_m(\eta ,q) + B_m{\textit{Ms}}_m^{(1)}(\xi ,q){\textit{se}}_m(\eta ,q). \end{equation}

Figure 5.

Representative elliptic duct mode patterns. The Cartesian coordinates $(x,y)$ has been made dimensionless on half-focal length $h_0=0.4$ m. The solid line represents the cross-section of the duct wall, which is an ellipse with semi-major axis $a_0=0.5$ m and semi-minor axis $b_0=0.3$ m.

Now consider the boundary conditions on the duct wall:

(2.22) \begin{equation} \frac {\partial }{\partial \xi }\varPsi (\xi _0, \eta ) = 0, \quad \forall \eta . \end{equation}

Substitution of (2.21) into (2.22) yields

(2.23) \begin{equation} A_m\,{{\textit{Mc}}^{(1)}_m}'(\xi _0,q){\textit{ce}}_m(\eta ,q) + B_m{{\textit{Ms}}^{(1)}_m}'(\xi _0,q){\textit{se}}_m(\eta ,q) = 0, \quad \forall \eta . \end{equation}

A set of non-trivial linearly independent solutions can be found by taking either

(2.24) \begin{equation} {{\textit{Mc}}^{(1)}_m}'(\xi _0,q) = 0, \quad \text{with }B_m = 0, \end{equation}

or

(2.25) \begin{equation} {{\textit{Ms}}^{(1)}_m}'(\xi _0,q) = 0, \quad \text{with }A_m = 0. \end{equation}

Equations (2.24) and (2.25) should be solved individually, and the solution $q_{\textit{mn}}$ refers to the $(n-1)$ th non-negative real value that satisfies its corresponding wall boundary conditions. It is clear that $q_{\textit{mn}}$ , the solution of boundary conditions (2.24) or (2.25), can only be non-negative real, considering the Mathieu function calculation process described in Appendix A. We should note that the solutions of (2.24) and (2.25) are usually not identical. The value of $q_{\textit{mn}}$ determines which one of the linearly independent terms in (2.21) should be kept. Without loss of generality, we discuss only the elliptic modes of even form. By taking $A_m = 1$ and $B_m = 0$ , we can express the velocity potential distribution of an elliptic mode of even form as

(2.26) \begin{equation} \varPsi _{\textit{mn}}(\xi ,\eta ) = {\textit{Mc}}^{(1)}_m(\xi ,q_{\textit{mn}})\,{\textit{ce}}_m(\eta ,q_{\textit{mn}}). \end{equation}

Modal patterns $\varPsi _{\textit{mn}}$ with low angular and radial modal order are shown in figure 5.

Substituting $q = q_{\textit{mn}}$ into (2.12), the dimensionless axial wave number for the elliptic duct mode $(m,n)$ is related to $q_{\textit{mn}}$ by the dispersion relation

(2.27) \begin{equation} \mu _{\textit{mn}}^\pm = \frac {\pm \sqrt {1-4q_{\textit{mn}}(1-M^2)/\omega ^2} - M}{1-M^2}, \quad 0\lt \xi \lt \xi _0. \end{equation}

Note that since (2.24) and (2.25) generally have distinct solutions, the even and odd modes of the same order $(m,n)$ propagate inside the elliptic duct with different axial wavenumbers.

Hence the wave propagating inside the duct can be expressed as

(2.28) \begin{align} \phi (\xi ,\eta ,z,t) & = \sum _{m=0}^{+\infty } \sum _{n=0}^{+\infty } \left[A_{\textit{mn}}^- \exp \left(\textrm {i}\omega \mu _{\textit{mn}}^- z\right)+A_{\textit{mn}}^+ \exp \left(\textrm {i}\omega \mu _{\textit{mn}}^+ z\right)\right]\, \varPsi _{\textit{mn}}(\xi ,\eta )\exp (\rm -i\omega t),\nonumber\\& \quad 0\lt \xi \lt \xi _0, \end{align}

where $A_{\textit{mn}}^\pm$ are complex amplitudes. Since we are dealing with forward emissions from a semi-infinite duct, we set $A_{\textit{mn}}^-$ to zero, and keep only the sound waves propagating in the positive direction of the $z$ -axis. Now we can extract the incident wave $\psi _i$ from (2.28) as

(2.29) \begin{equation} \psi _i(\xi ,\eta ,z) = \left \{\!\! \begin{array}{ll} 0, &\xi \gt \xi _0, \\[4pt] A_{\textit{mn}}^+\,\varPsi _{\textit{mn}}(\xi ,\eta )\exp \left(\textrm {i}\omega \mu _{\textit{mn}}^+ z\right), &0\lt \xi \lt \xi _0. \end{array} \right . \end{equation}

2.4. Scattered wave

For the acoustic field scattered by the incident wave at the duct lip, we also assume

(2.30) \begin{equation} \psi _s(\xi ,\eta ,z,t) = \psi _s(\xi ,\eta ,z)\exp (-\textrm {i}\omega t), \end{equation}

and define the half-range Fourier transforms

(2.31) \begin{align} \beta _-(\xi ,\eta ,u) &= \int _{-\infty }^{0}\psi _s(\xi ,\eta ,z)\exp (-\textrm {i}\omega u z)\,\textrm {d}z, \end{align}

(2.32) \begin{align} \beta _+(\xi ,\eta ,u) &= \int _{0}^{+\infty }\psi _s(\xi ,\eta ,z)\exp (-\textrm {i}\omega u z)\,\textrm {d}z, \end{align}

where $u$ is the dimensionless axial wavenumber. The conventional Fourier transform is then obtained by simply taking

(2.33) \begin{equation} \beta (\xi ,\eta ,u) = \beta _-(\xi ,\eta ,u) + \beta _+(\xi ,\eta ,u). \end{equation}

The scattered wave (2.30) is also governed by (2.5). Substituting (2.30) into (2.5) and performing the Fourier transform (2.33), we obtain

(2.34) \begin{equation} \frac {2}{\cosh 2\xi - \cos 2\eta } \left (\frac {\partial ^2 \beta }{\partial \xi ^2}+\frac {\partial ^2 \beta }{\partial \eta ^2}\right ) + \omega ^2 \lambda ^2 \beta = 0, \end{equation}

where

(2.35) \begin{equation} \lambda ^2 = (1-uM)^2-u^2, \end{equation}

for $\lambda = \lambda _- \lambda _+$ with $\lambda _\pm =[1-u(M\pm 1)]^{1/2}$ . By taking the principal branches of the square root function, it could be derived that $\lambda _\pm$ go from $u_\pm$ to $\pm \infty$ , respectively, where $u_\pm = 1/(M\pm 1)$ .

We treat the axial wavenumber $u$ as a parameter, and then perform the variable separation

(2.36) \begin{equation} \beta (\xi ,\eta ,u) = \beta _\xi (\xi ,u)\, \beta _\eta (\eta ,u). \end{equation}

The applicability of variable separation is related not only to the coordinate system but also to the differential operators involved. Several related studies (Lowson & Baskaran Reference Lowson and Baskaran1975; Peake & Cooper Reference Peake and Cooper2001; Oliveira & Gil Reference Oliveira and Gil2014) have shown that the separation of spatial variables in the wave equation under elliptic coordinates is feasible. Boyer, Kalnins & Miller (Reference Boyer, Kalnins and Miller1976) provided a more detailed mathematical explanation of this issue. Substitution of (2.36) into (2.34) yields the form of the angular Mathieu equation

(2.37) \begin{equation} \frac {\textrm {d}^2\beta _\eta }{\textrm {d}\eta ^2} +(A - 2Q\cos 2\eta ) \beta _\eta = 0, \end{equation}

and the radial Mathieu equation

(2.38) \begin{equation} \frac {\textrm {d}^2\beta _\xi }{\textrm {d}\xi ^2} -(A - 2Q\cosh 2\xi ) \beta _\xi = 0, \end{equation}

where $Q = \omega ^2 \lambda ^2 /4$ is a function of $u$ , and $A$ denotes the eigenvalues of the corresponding Mathieu functions, like $a$ in (2.13) and (2.14). Similar to the discussion in § 2.3, the solution of (2.37) and (2.38) with arbitrary parametric coefficients $B_{0,\kappa }(u)$ and $B_{1,\kappa }(u)$ is

(2.39) \begin{equation} \beta (\xi ,\eta ,u) = \left \{\!\! \begin{array}{ll} \sum _{\kappa =0}^{+\infty } B_{0,\kappa }(u)\,{\textit{Mc}}_\kappa ^{(3)}(\xi ,Q)\, {\textit{ce}}_\kappa (\eta ,Q), &\xi \gt \xi _0, \\[6pt] \sum _{\kappa =0}^{+\infty } B_{1,\kappa }(u)\,{\textit{Mc}}_\kappa ^{(1)}(\xi ,Q)\, {\textit{ce}}_\kappa (\eta ,Q), &0\lt \xi \lt \xi _0, \end{array} \right . \end{equation}

where ${\textit{Mc}}_\kappa ^{(k)}$ is the $k$ th-kind radial Mathieu function of even form with non-negative integer order $\kappa$ , and ${\textit{ce}}_\kappa$ is the angular Mathieu function of even form with the same order $\kappa$ . In a practical implementation, the above infinite series would be truncated to a finite dimension.

Physically, (2.39) shows that an incident wave will be scattered into various different radial and angular Mathieu modes from the open end of an elliptic duct. The corresponding amplitudes of each mode can be rigorously determined based on the theoretical solutions that are to be developed below. In (2.39), we make an assumption that the scattered field resulting from an incident elliptic mode of even form is solely described by even-form Mathieu functions. This assumption holds due to the orthogonality property of angular Mathieu functions (McLachlan Reference McLachlan1947). However, if the incident wave consists of both even-form and odd-form elliptic modes, then the scattered field should be expressed as a linear combination of both even-form and odd-form Mathieu functions.

From (2.39), we find that $\beta$ can be expressed as the infinite series of the angular Mathieu functions of integer order, i.e.

(2.40) \begin{equation} \beta (\xi ,\eta ,u) = \sum _{\kappa =0}^{+\infty }\hat {\beta }_\kappa (\xi ,u)\,{\textit{ce}}_\kappa (\eta ,Q). \end{equation}

By comparing (2.39) and (2.40), we express the coefficients $\hat {\beta }_\kappa$ in the form of matrix operation as

(2.41) \begin{equation} \hat {\beta }_\kappa (\xi ,u) = \left \{\!\! \begin{array}{ll} B_{0,\kappa }(u)\,{\textit{Mc}}_\kappa ^{(3)}(\xi ,Q), &\xi \gt \xi _0, \\[6pt] B_{1,\kappa }(u)\,{\textit{Mc}}_\kappa ^{(1)}(\xi ,Q), &0\lt \xi \lt \xi _0. \end{array}\right . \end{equation}

2.5. Boundary conditions

There are three possible forms of the uniform mean flow: completely stationary, in the positive $z$ -axis direction, or in the negative $z$ -axis direction. Accordingly, the semi-infinite elliptic duct would be an ordinary duct, an outlet duct, or an inlet duct. It is important to note that the vortex sheet shown in figure 3 truly exists only in the second scenario, where the semi-infinite elliptic duct is used as an outlet duct. For the first and third scenarios, where the semi-infinite elliptic duct is used as an ordinary duct or an inlet duct, the vortex sheet is merely used to establish the necessary boundary conditions between the inner and outer sides.

The radial displacement of the sound particles on the vortex sheet is defined as

(2.42) \begin{equation} w(\eta ,z,t) = w(\eta , z)\exp (-\textrm {i}\omega t). \end{equation}

Omitting the time harmonic factor $\exp (-\textrm {i}\omega t)$ , particle displacement $w(\eta ,z)$ on the vortex sheet is related to the particle velocity as

(2.43) \begin{align} \left (-\textrm {i}\omega + M\frac {\partial }{\partial z}\right )w(\eta ,z) &= \frac {\partial }{\partial \xi }\psi _s\left(\xi _0^+, \eta , z\right), \quad \xi \gt \xi _0, \quad z\gt 0, \end{align}

(2.44) \begin{align} \left (-\textrm {i}\omega +M\frac {\partial }{\partial z}\right )w(\eta ,z) &= \frac {\partial }{\partial \xi }\psi _s\left(\xi _0^-, \eta , z\right), \quad 0\lt \xi \lt \xi _0, \quad z\gt 0, \end{align}

where the two limiting variables $\xi _0^+$ and $\xi _0^-$ approach $\xi _0$ from the positive and negative directions, respectively.

Readers may notice that (2.43) and (2.44) appear to be physically incorrect. The right-hand sides do not represent dimensionless particle velocity since the radial coordinate base $\partial _\xi$ is not a unitary base. To address this issue arising from coordinate transformations, we must note that in (2.42), (2.43) and (2.44), the particle displacement and velocity are expressed by their corresponding covariant components based on the local contravariant bases $(\partial _\xi ,\partial _\eta ,\partial _\zeta )$ . Due to the invariant nature of vector fields under changes in coordinate bases, these equations remain valid.

The continuity of the acoustic pressure across the vortex sheet yields

(2.45) \begin{equation} \left (\!-\textrm {i}\omega + M\frac {\partial }{\partial z}\right )\psi _s\left(\xi _0^+,\eta ,z\right) = \left (\!-\textrm {i}\omega + M\frac {\partial }{\partial z}\right )\! \left[\psi _s\left(\xi _0^-,\eta ,z\right)+\psi _i\left(\xi _0^-,\eta ,z\right)\right], \!\quad z\gt 0. \end{equation}

The boundary condition of the rigid duct wall is

(2.46) \begin{equation} \frac {\partial }{\partial \xi } \psi _s\left(\xi _0^-, \eta , z\right) = \frac {\partial }{\partial \xi } \psi _s\left(\xi _0^+, \eta , z\right) = 0, \quad z\leqslant 0. \end{equation}

3.1. Wiener–Hopf equation

First, introduce the half-range Fourier transforms

(3.1) \begin{align} F_+(\eta ,u) &= \int _{0}^{+\infty } w(\eta ,z)\exp (-\textrm {i}\omega u z)\, \textrm {d}z, \end{align}

(3.2) \begin{align} G_-(\eta ,u) &= \int _{-\infty }^{0} \left (-\textrm {i}\omega + M\frac {\partial }{\partial z}\right )\left [\psi _s\left(\xi _0^+,\eta ,z\right)-\psi _s\left(\xi _0^-,\eta ,z\right)\right ] \exp (-\textrm {i}\omega uz)\, {\rm d}z, \end{align}

(3.3) \begin{align} G_+(\eta ,u) &= \int _{0}^{+\infty }\left (-\textrm {i}\omega + M\frac {\partial }{\partial z}\right )\left [\psi _s\left(\xi _0^+,\eta ,z\right)-\psi _s\left(\xi _0^-,\eta ,z\right)\right ] \exp (-\textrm {i}\omega uz)\,{\rm d}z, \end{align}

and the conventional Fourier transform

(3.4) \begin{equation} G(\eta ,u) = G_-(\eta ,u) + G_+(\eta ,u). \end{equation}

Note that

(3.5) \begin{equation} \psi _s\left(\xi _0^+,\eta ,z\right)-\psi _s\left(\xi _0^-,\eta ,z\right) = \psi _i(\xi _0,\eta ,z),\quad z\gt 0. \end{equation}

Therefore, by performing (3.3) on the incident wave (2.29), we obtain

(3.6) \begin{equation} G_+(\eta ,u) = A_{\textit{mn}}^+ \frac {1-\mu _{\textit{mn}}^+ M}{\mu _{\textit{mn}}^+ - u} {\textit{Mc}}_{m}^{(1)} (\xi _0,q_{\textit{mn}})\, {\textit{ce}}_m (\eta ,q_{\textit{mn}}). \end{equation}

Now we turn to the key part of this theoretical development. The implementation of the Wiener–Hopf method requires decoupling of the angular variable $\eta$ . To achieve this, we put forward an assumption, i.e. the distribution of almost all continuous and integrable periodic functions along the angular variable $\eta$ , e.g. $f(\eta )$ , can be decomposed in a method that imitates the Fourier series expansion. The basis function of this decomposition will be the angular Mathieu functions of integer orders with the same parameter, say $Q$ . Then $f(\eta )$ can be written in the series form

(3.7) \begin{equation} f(\eta ) = \sum _{\kappa =0}^{+\infty } \hat {f}_{c,\kappa }(\eta )\, {\textit{ce}}_\kappa (\eta ,Q) + \sum _{\kappa =1}^{+\infty } \hat {f}_{s,\kappa }(\eta )\, {\textit{se}}_\kappa (\eta ,Q). \end{equation}

The rationale behind our assumption is quite straightforward. The angular Mathieu functions of integer orders with the same parameter $Q$ can be expressed by Fourier series as (A1)–(A4). And for the $\kappa$ th-order angular Mathieu function ${\textit{ce}}_\kappa (\eta ,Q)$ or ${\textit{se}}_\kappa (\eta ,Q)$ , the complex amplitudes of its corresponding Fourier series form one of the linear independent eigenvectors derived from the recurrence relations (see Appendix A for more details), which means that in a Hilbert space, the angular Mathieu functions of integer orders with the same parameter $Q$ can also form a set of linearly independent basis functions. Actually, we can further prove the orthogonality conditions as follows (McLachlan Reference McLachlan1947):

(3.8) \begin{align} \int _{0}^{2\unicode{x03C0} }{\textit{ce}}_m(\eta ,Q)\,{\textit{ce}}_p(\eta ,Q)\,\textrm {d}\eta &= 0, \quad m\neq p, \quad m,p\ \text{non-negative integers}, \end{align}

(3.9) \begin{align} \int _{0}^{2\unicode{x03C0} }{\textit{se}}_m(\eta ,Q)\,{\textit{se}}_p(\eta ,Q)\,\textrm {d}\eta &= 0, \quad m\neq p, \quad m,p\ \text{positive integers}, \end{align}

(3.10) \begin{align} \int _{0}^{2\unicode{x03C0} }{\textit{ce}}_m(\eta ,Q)\,{\textit{se}}_p(\eta ,Q)\,\textrm {d}\eta &= 0, \quad m\neq p, \quad m,p\ \text{positive integers}. \end{align}

Equation (3.10) also suggests that considering only even modes does not lead to a loss of generality, since elliptic modes of even and odd form are linearly independent.

At this stage, it is important to clarify that the functions that we intend to decompose are not arbitrary. They all arise from an even-form elliptic mode as indicated by (2.29) within the context of this study. Utilising the orthogonality condition (3.10), we assert that only even-form Mathieu functions need be employed as basis functions for this decomposition, such as in the case of $G_+(\eta ,u)$ :

(3.11) \begin{equation} G_+(\eta ,u) = \sum _{\kappa =0}^{+\infty } \hat {G}_{+,\kappa }(u)\, {\textit{ce}}_\kappa (\eta ,Q). \end{equation}

Similarly, we have the series expansions of $G_-(\eta ,u)$ and $F_+(\eta ,u)$ as

(3.12) \begin{align} G_-(\eta ,u) &= \sum _{\kappa =0}^{+\infty } \hat {G}_{-,\kappa }(u)\, {\textit{ce}}_\kappa (\eta ,Q), \end{align}

(3.13) \begin{align} F_+(\eta ,u) &= \sum _{\kappa =0}^{+\infty } \hat {F}_{+,\kappa }(u)\, {\textit{ce}}_\kappa (\eta ,Q). \end{align}

Different from (3.11), where $G_+(\eta ,u)$ is already known in (3.6), $G_-(\eta ,u)$ and $F_+(\eta ,u)$ are unknown functions that need to be solved, and we can do so only by solving $\hat {F}_{+,\kappa }(u)$ first using the Wiener–Hopf method in the remaining part of this subsection.

By applying a Fourier transform along the axial direction $z$ to (2.43) and (2.44), we obtain

(3.14) \begin{align} -\textrm {i}\omega (1-uM)\,F_+(\eta ,u) &= \frac {\partial }{\partial \xi }\beta (\xi _0^+,\eta ,u), \quad \xi \gt \xi _0, \end{align}

(3.15) \begin{align} -\textrm {i}\omega (1-uM)\,F_+(\eta ,u) &= \frac {\partial }{\partial \xi }\beta (\xi _0^-,\eta ,u), \quad 0\lt \xi \lt \xi _0. \end{align}

Next, substitute the series (2.40) and (3.13) into (3.14) and (3.15). By comparing the coefficients, we obtain

(3.16) \begin{align} -\textrm {i}\omega (1-uM)\,\hat {F}_{+,\kappa }(u) &= {{\textit{Mc}}_\kappa ^{(3)'}}(\xi _0,Q)\, B_{0,\kappa }(u), \quad \xi \gt \xi _0, \end{align}

(3.17) \begin{align} -\textrm {i}\omega (1-uM)\,\hat {F}_{+,\kappa }(u) &= {{\textit{Mc}}_\kappa ^{(1)'}}(\xi _0,Q)\, B_{1,\kappa }(u), \quad 0\lt \xi \lt \xi _0. \end{align}

The continuity conditions across the vortex sheet (2.45) and the boundary condition of the rigid wall (2.46) yield

(3.18) \begin{equation} \hat {G}_{-,\kappa }(u) + \hat {G}_{+,\kappa }(u) = \textrm {i}\omega (1-uM) [{\textit{Mc}}_\kappa ^{(1)}(\xi _0,Q)\, B_{1,\kappa }(u) -{\textit{Mc}}_\kappa ^{(3)}(\xi _0,Q)\, B_{0,\kappa }(u)]. \end{equation}

Combining (3.16), (3.17) and (3.18), we obtain the Wiener–Hopf equation

(3.19) \begin{equation} \hat {G}_{-,\kappa }(u) + \hat {G}_{+,\kappa }(u) = K_\kappa (u)\, \hat {F}_{+,\kappa }(u), \end{equation}

where the Wiener–Hopf kernel is

(3.20) \begin{eqnarray} K_\kappa (u) = \omega ^2(1-uM)^2\left [\frac {{\textit{Mc}}_\kappa ^{(3)}(\xi _0,Q)}{{{\textit{Mc}}_\kappa ^{(3)'}}(\xi _0,Q)} - \frac {{\textit{Mc}}_\kappa ^{(1)}(\xi _0,Q)}{{{\textit{Mc}}_\kappa ^{(1)'}}(\xi _0,Q)}\right ]\!. \end{eqnarray}

3.2. Analytic strip and integration path

A unique solution requires an appropriate radiation condition that guarantees sound absorption at infinity. The Sommerfeld condition, which ensures that scattered waves radiate only outwards from known sources, is applied to maintain the causality of the initial value problem. Jones & Morgan (Reference Jones and Morgan1972) showed the proper approach by adding a positive imaginary component to the angular frequency $\omega$ . To mathematically describe the behaviour of acoustic waves radiating away to infinity, we introduce a complex angular frequency

(3.21) \begin{equation} \omega = |\omega |\exp (\textrm {i}\varepsilon ),\quad 0\leqslant \varepsilon \leqslant \frac {\unicode{x03C0} }{2}, \end{equation}

with $\varepsilon$ close to $\unicode{x03C0} /2$ , all waves decaying exponentially. Then we let $\varepsilon$ go to zero, the far-field radiation condition is correctly adopted, and the solution is obtained at a nearly real angular frequency. This process is vital for us to understand the behaviour of the Wiener–Hopf kernel function in the complex $u$ -plane, and gives us important insights into how to construct the analytical strip and integration path.

Figure 6.

Schematics of the analytic strip and the integration path: $(a)$ when $\varepsilon$ is sufficiently large, the two complex half-planes $R_\pm$ define the overlapped strip $S$ with $M\lt 0$ ; $(b)$ as $\varepsilon$ goes to 0, the analytic strip degenerates into a slightly deformed integration path that bypasses all possible acoustic zeros, poles and the aerodynamic zero $u_0$ .

To determine the analytical strip, it is necessary to discuss the regularity of $\beta (\xi ,\eta ,u)$ . As shown in figure 6 $(a)$ , since $Q=\omega ^2\lambda ^2/4$ , the regularity conditions of the radial Mathieu functions require that $u$ can only take the value within the analytic strip defined by the half-planes $R_-$ and $R_+$ , where the half-planes $R_\pm$ are defined, respectively, as

(3.22) \begin{equation} \pm \textrm {Im}(u-u_\pm )\lt \mp \tan (\varepsilon )\,\text{Re}(u-u_\pm ). \end{equation}

As $\varepsilon \rightarrow 0$ , the integration path is slightly deformed around the aerodynamic double zero $u_0=1/M$ and the acoustic zeros and poles. Finally, as shown in figure 6 $(b)$ , the analytic strip $S$ degenerates into the deformed integration path $\mathit \varGamma$ that goes from $-\infty +0\textrm {i}$ to $+\infty -0\textrm {i}$ and crosses the real axis at $u_c$ with

(3.23) \begin{equation} u_c = \frac {u_-+u_+}{2} = \frac {M}{M^2-1}. \end{equation}

For static cases with stationary flows, $M=u_c=0$ . For outlet cases, $M\gt 0$ , hence $u_c\lt 0$ . For inlet cases, $M\lt 0$ , hence $u_c\gt 0$ .

3.3. Kernel factorisation

Kernel factorisation is the key step in the implementation of the Wiener–Hopf technique. To clarify this procedure, we demonstrate the factorisation of $K_\kappa (u)$ . Detailed descriptions can be found in Munt (Reference Munt1977) and Rienstra (Reference Rienstra1984).

Before factorisng $K_\kappa (u)$ , we note that there are obvious aerodynamic double zeros associated with the Mach number of the uniform mean flow when $M\neq 0$ . These can be straightforwardly derived from (3.20) as

(3.24) \begin{equation} u_0 = \frac {1}{M}, \quad M\neq 0. \end{equation}

It would be beneficial to remove the possible double zeros prior to the factorisation of (3.20), which yields

(3.25) \begin{equation} K_\kappa (u) = \tilde {K}_\kappa (u)\,(u-u_0)^b, \end{equation}

where $b$ is an enumerable integer depending on the axial mean flow. For a static test case ( $M=0$ ), the aerodynamic zeros do not exist, hence $b=0$ . Otherwise, for the case with a uniform flow ( $M\neq 0$ ), we set $b=2$ .

Assume that $K_\kappa (u)$ is factorised into the form

(3.26) \begin{equation} K_\kappa (u) = \frac {K_{+,\kappa }(u)}{K_{-,\kappa }(u)}(u-u_0)^b, \end{equation}

with $K_{\pm ,\kappa }$ regular and non-zero on $R_\pm$ . Using Cauchy’s integration gives

(3.27) \begin{equation} \log K_{\pm ,\kappa }(u) = -\frac {1}{2\unicode{x03C0} \textrm {i}} \int _C \frac {\log [K_\kappa (\zeta )/(u-u_0)^b]}{\zeta -u}\,\textrm {d}\zeta . \end{equation}

As $\varepsilon \rightarrow 0$ , the integration path $C$ goes from $-\infty +i\varepsilon$ to $+\infty -i\varepsilon$ and crosses the real axis at $u_c=(u_-+u_+)/2$ . Since the integration path is symmetric about $u_c$ on the complex $u$ -plane, performing variable substitution and integrating along the new path gives

(3.28) \begin{align} \log K_{\pm ,\kappa }(u) &= -\frac {1}{2\unicode{x03C0} {\rm i}} \int _{0}^{+\infty -\textrm {i}\varepsilon }\left \{ \frac {\log [K_\kappa (u_c-v)/(u-u_0)^b]}{u_c-v-u} \right .\nonumber\\&\quad \left . +\, \frac {\log [K_\kappa (u_c+v)/(u-u_0)^b]}{u_c+v-u} \right \}\textrm {d}v, \end{align}

with

(3.29) \begin{equation} v(\mu ) = \left \{\!\! \begin{array}{ll} \mu + \textrm {i}\left (\frac {\Delta u - \varepsilon }{\Delta u^2}\mu ^2 -\mu \right ), & 0\leqslant \mu \lt \Delta u, \\ \mu - \textrm {i}\varepsilon , & \mu \geqslant \Delta u, \end{array} \right . \end{equation}

where $\Delta u = (u_+ - u_-)/2$ . For $K_{+,\kappa }(u)$ , if $u$ is above the integration contour, then we add the contributions of the pole $\zeta = u$ . In contrast, for $K_{-,\kappa }$ , a similar operation needs to be done when $u$ is below the integration contour.

3.4. General solutions

Substituting (3.26) into (3.19) gives

(3.30) \begin{equation} K_{+,\kappa }(u)\, \hat {F}_{+,\kappa }(u)\, (u-u_0)^b = K_{-,\kappa }(u)\,\hat {G}_{-,\kappa }(u) + K_{-,\kappa }(u)\,\hat {G}_{+,\kappa }(u). \end{equation}

By taking out the acoustic poles, we obtain

(3.31) \begin{align} &K_{+,\kappa }(u)\, \hat {F}_{+,\kappa }(u)\, (u-u_0)^b - K_{-,\kappa }\left(\mu _{\textit{mn}}^+\right)\, \hat {G}_{+,\kappa }(u) =K_{-,\kappa }(u)\,\hat {G}_{-,\kappa }(u) \nonumber \\ &\quad + \hat {G}_{+,\kappa }(u)\left[K_{-,\kappa }(u)-K_{-,\kappa }\left(\mu _{\textit{mn}}^+\right)\right]. \end{align}

The left- and right-hand sides of (3.31) are regular on the half-planes $R_\pm$ , respectively. According to analytic continuation in the complex analysis theory, (3.31) defines an entire function $E_\kappa (u)$ that is regular on the whole complex $u$ -plane.

3.4.1. Static cases

In the static flow case, there is no need to introduce the Kutta condition. It is necessary to discuss the behaviour of $E_\kappa (u)$ as $|u|\rightarrow \infty$ . The expansions of Mathieu functions in series of cross-products of Bessel functions (Appendix A) provide insightful details regarding $K_{\kappa }(u)$ , and ultimately lead to the inference that $K_{\pm ,\kappa }(u)\sim \textit {O}(u^{\mp 1/2})$ as $|u|\rightarrow \infty$ . For a static case ( $M=0$ and $b=0$ ), it is straightforward to assert $\psi _s(\xi _0,z)\sim \textit {O}(z)$ as $z\rightarrow 0$ . Following Noble (Reference Noble1958), this condition results in the conclusion that $F_{+,\kappa }(u)\sim \textit {O}(u^{-2})$ and $G_{-,\kappa }(u)\sim \textit {O}(u^{-1})$ as $|u|\rightarrow \infty$ , derived from (2.6), (3.1) and (3.19). Hence $E_\kappa (u)\sim \textit {O}(u^{-1/2})$ as $|u|\rightarrow \infty$ . Following Liouville’s theorem, the left-hand side of (3.31) is identically zero, i.e.

(3.32) \begin{equation} K_{+,\kappa }(u)\, \hat {F}_{+,\kappa }(u) - K_{-,\kappa }\left(\mu _{\textit{mn}}^+\right)\, \hat {G}_{+,\kappa }(u) = 0. \end{equation}

Hence

(3.33) \begin{equation} \hat {F}_{+,\kappa }(u) = K_{+,\kappa }^{-1}(u)\, K_{-,\kappa }\left(\mu _{\textit{mn}}^+\right)\, \hat {G}_{+,\kappa }(u). \end{equation}

3.4.2. Outlet cases

When the elliptic duct is used as a nozzle (given that we are only addressing uniform flow, this nozzle holds limited practical applicability), the full Kutta condition is applied at the duct lip, ensuring that all vortices shed exclusively from the lip. Note that the problem addressed in this study involves only uniform flow, unlike the work of Gabard & Astley (Reference Gabard and Astley2006), where instability waves form at the duct lip due to the velocity difference between the jet and the ambient flow, and propagate downstream. Therefore, flow instability and its effects are not considered in this study. For an outlet problem ( $M\gt 0$ and $b=2$ ), the velocity potential is a simple zero at the duct lip, which means $\psi _s(\xi _0,z)\sim \textit {O}(z^{3/2})$ as $z\rightarrow 0$ . Similarly, we have $F_{+,\kappa }(u)\sim \textit {O}(u^{-5/2})$ and $G_{-,\kappa }(u)\sim \textit {O}(u^{-3/2})$ as $|u|\rightarrow \infty$ . Hence $E_\kappa (u)\sim \textit {O}(u^{-1})$ as $|u|\rightarrow \infty$ , and (3.31) turns out to be

(3.34) \begin{equation} K_{+,\kappa }(u)\, \hat {F}_{+,\kappa }(u)\, (u-u_0)^2 - K_{-,\kappa } \big(\mu _{\textit{mn}}^+ \big)\, \hat {G}_{+,\kappa }(u) = 0. \end{equation}

Then

(3.35) \begin{equation} \hat {F}_{+,\kappa }(u) = \frac {K_{+,\kappa }^{-1}(u)\, K_{-,\kappa } \big(\mu _{\textit{mn}}^+ \big)\, \hat {G}_{+,\kappa }(u)}{(u-u_0)^2}. \end{equation}

3.4.3. Inlet cases

When the elliptic duct serves as an inlet, vortex shedding at the lip is considered non-physical, making the application of the Kutta condition unnecessary. For an inlet problem set-up ( $M\lt 0$ and $b=2$ ), the velocity potential is singular at the lip, which translates to $\psi _s(\xi _0,z)\sim \textit {O}(z^{1/2})$ as $z\rightarrow 0$ . Therefore, we can derive that $F_{+,\kappa }(u)\sim \textit {O}(u^{-3/2})$ and $G_{-,\kappa }(u)\sim \textit {O}(u^{-1/2})$ as $|u|\rightarrow \infty$ . Then $E_\kappa (u)\sim \textit {O}(1)$ as $|u|\rightarrow \infty$ , which implies that $E_\kappa (u)\equiv E_\kappa (u_0)$ . We obtain from (3.31) that

(3.36) \begin{equation} K_{+,\kappa }(u)\, \hat {F}_{+,\kappa }(u)\, (u-u_0)^2 - K_{-,\kappa }\left(\mu _{\textit{mn}}^+\right)\, \hat {G}_{+,\kappa }(u) = - K_{-,\kappa }\left(\mu _{\textit{mn}}^+\right)\, \hat {G}_{+,\kappa }(u_0). \end{equation}

Then

(3.37) \begin{equation} \hat {F}_{+,\kappa }(u) = \frac {K_{+,\kappa }^{-1}(u)\, K_{-,\kappa }\left(\mu _{\textit{mn}}^+\right)\, [\hat {G}_{+,\kappa }(u) - \hat {G}_{+,\kappa }(u_0)]}{(u-u_0)^2}. \end{equation}

3.4.4. Inverse Fourier transforms

On substituting $\hat {F}_{+,\kappa }(u)$ into (3.16) and (3.17), we get $B_{0,\kappa }(u)$ , $B_{1,\kappa }(u)$ solved respectively as

(3.38) \begin{align} B_{0,\kappa }(u) &= -\textrm {i}\omega (1-uM) \frac {\hat {F}_{+,\kappa }(u)}{{{\textit{Mc}}_\kappa ^{(3)'}}(\xi _0,Q)},\quad \xi \gt 0, \end{align}

(3.39) \begin{align} B_{1,\kappa }(u) &= -\textrm {i}\omega (1-uM) \frac {\hat {F}_{+,\kappa }(u)}{{{\textit{Mc}}_\kappa ^{(1)'}}(\xi _0,Q)},\quad 0\lt \xi \lt \xi _0. \end{align}

Therefore, the substitution of (3.38) and (3.39) into (2.41) gives

(3.40) \begin{align} \hat {\beta }_\kappa (\xi ,u) &= -\textrm {i}\omega (1-uM) \frac {{\textit{Mc}}_\kappa ^{(3)}(\xi ,Q)}{{{\textit{Mc}}_\kappa ^{(3)'}}(\xi _0,Q)} \hat {F}_{+,\kappa }(u),\quad \xi \gt \xi _0, \end{align}

(3.41) \begin{align} \hat {\beta }_\kappa (\xi ,u) &= -\textrm {i}\omega (1-uM) \frac {{\textit{Mc}}_\kappa ^{(1)}(\xi ,Q)}{{{\textit{Mc}}_\kappa ^{(1)'}}(\xi _0,Q)} \hat {F}_{+,\kappa }(u),\quad 0\lt \xi \lt \xi _0. \end{align}

Then the scattered field is calculated by performing the inverse Fourier transforms

(3.42) \begin{align} \psi _s(\xi ,\eta ,z) &= \frac {\omega }{2\unicode{x03C0} }\int _{{\mathit \varGamma }} \left (\sum _{0}^{\infty }\hat {\beta }_\kappa (\xi ,u)\,{\textit{ce}}_\kappa (\eta ,Q)\right ) \exp (\textrm {i}\omega u z)\,\textrm {d}u, \end{align}

(3.43) \begin{align} p_s(\xi ,\eta ,z) &= \frac {\textrm {i}\omega ^2}{2\unicode{x03C0} }\int _{{\mathit \varGamma }} \left (\sum _{0}^{\infty }\hat {\beta }_\kappa (\xi ,u)\,{\textit{ce}}_\kappa (\eta ,Q)\right )(1-uM) \exp (\textrm {i}\omega u z)\,\textrm {d}u, \end{align}

where the integration path $\mathit \varGamma$ is a straight line in the analytic strip $S$ (see figure 6). The path $\mathit \varGamma$ can be expressed as

(3.44) \begin{equation} {\mathit \varGamma } = \{z\in \mathbb{C}\mid z = x\exp (-\textrm {i}\varepsilon ), \, x\in \mathbb{R}\}, \end{equation}

where $z\in \mathbb{C}$ represents points in the complex plane, and $x\in \mathbb{R}$ represents the real parameter that spans all real numbers. Substituting (3.40) and (3.41) into (3.42) and (3.43), we obtain the velocity potential field as

(3.45) \begin{align} \psi _s(\xi ,\eta ,z) &= \frac {\omega ^2}{2\unicode{x03C0} \textrm {i}}\int _{{\mathit \varGamma }} \left (\sum _{\kappa =0}^{+\infty } \frac {{\textit{Mc}}_\kappa ^{(3)}(\xi ,Q)\,{\textit{ce}}_\kappa (\eta ,Q)\,\hat {F}_{+,\kappa }(u)}{{{\textit{Mc}}_\kappa ^{(3)'}}(\xi _0,Q)}\right ) \nonumber \\ &\quad\times(1-uM)\exp (\textrm {i}\omega u z)\,\textrm {d}u, \quad \xi \gt \xi _0, \\[-12pt] \nonumber \end{align}

(3.46) \begin{align} \psi _s(\xi ,\eta ,z) &= \frac {\omega ^2}{2\unicode{x03C0} \textrm {i}}\int _{{\mathit \varGamma }} \left (\sum _{\kappa =0}^{+\infty } \frac {{\textit{Mc}}_\kappa ^{(1)}(\xi ,Q)\,{\textit{ce}}_\kappa (\eta ,Q)\,\hat {F}_{+,\kappa }(u)}{{{\textit{Mc}}_\kappa ^{(1)'}}(\xi _0,Q)}\right ) \nonumber \\ &\quad\times(1-uM)\exp (\textrm {i}\omega u z)\,\textrm {d}u, \quad 0\lt \xi \lt \xi _0, \end{align}

and the sound pressure field as

(3.47) \begin{align} p_s(\xi ,\eta ,z) &= \frac {\omega ^3}{2\unicode{x03C0} }\int _{{\mathit \varGamma }} \left (\sum _{\kappa =0}^{+\infty } \frac {{\textit{Mc}}_\kappa ^{(3)}(\xi ,Q)\,{\textit{ce}}_\kappa (\eta ,Q)\,\hat {F}_{+,\kappa }(u)}{{{\textit{Mc}}_\kappa ^{(3)'}}(\xi _0,Q)}\right ) \nonumber \\ &\quad\times(1-uM)^2\exp (\textrm {i}\omega u z)\,\textrm {d}u, \quad \xi \gt \xi _0, \\[-12pt] \nonumber \end{align}

(3.48) \begin{align} p_s(\xi ,\eta ,z) &= \frac {\omega ^3}{2\unicode{x03C0} }\int _{{\mathit \varGamma }} \left (\sum _{\kappa =0}^{+\infty } \frac {{\textit{Mc}}_\kappa ^{(1)}(\xi ,Q)\,{\textit{ce}}_\kappa (\eta ,Q)\,\hat {F}_{+,\kappa }(u)}{{{\textit{Mc}}_\kappa ^{(1)'}}(\xi _0,Q)}\right ) \nonumber \\ &\quad\times(1-uM)^2\exp (\textrm {i}\omega u z)\,\textrm {d}u, \quad 0\lt \xi \lt \xi _0. \end{align}

3.5. Reflection coefficients

It is of great practical value to assess the acoustic fields scattering back into the elliptic duct. The modal decomposition of the scattered sound pressure field inside the elliptic duct yields

(3.49) \begin{align} p_s(\xi ,\eta ,z,t) &= \sum _{\tilde {m}=0}^{+\infty }\sum _{\tilde {n}=0}^{+\infty } \left[C_{\tilde {m}\tilde {n}}^- \exp \left(\textrm {i}\omega \mu _{\tilde {m}\tilde {n}}^- z\right)+C_{\tilde {m}\tilde {n}}^+ \exp\left (\textrm {i}\omega \mu _{\tilde {m}\tilde {n}}^+ z\right)\right]\, \varPsi _{\tilde {m}\tilde {n}}(\xi ,\eta ) \nonumber \\ &\quad\times\exp (-\textrm {i}\omega t), \quad 0\lt \xi \lt \xi _0, \end{align}

where $C_{\tilde {m}\tilde {n}}^\pm$ are the complex amplitudes. The reflection coefficients of one single incident elliptic mode $(m,n)$ are then defined as

(3.50) \begin{equation} R_{\tilde {m}\tilde {n}} = \frac {C_{\tilde {m}\tilde {n}}^-}{\textrm {i}\omega A_{\textit{mn}}^+\left(1-\mu _{\textit{mn}}^+ M\right)}. \end{equation}

Following Rienstra (Reference Rienstra1984) and Gabard & Astley (Reference Gabard and Astley2006), $C_{\tilde {m}\tilde {n}}^-$ can be evaluated by calculating the residues of the integrand in (3.48) at the corresponding acoustic poles below the integration path $\mathit \varGamma$ . After comparing the results with the decomposition (3.49), we have

(3.51) \begin{equation} C_{\tilde {m}\tilde {n}}^- = \frac {\omega \left(1-\mu _{\tilde {m}\tilde {n}}^- M\right)^2\, \hat {F}_{+,\tilde {m}}\left(\mu _{\tilde {m}\tilde {n}}^-\right)}{\unicode{x03C0} \left[\mu _{\tilde {m}\tilde {n}}^-(M^2-1)-M\right]\,\mathit{\varLambda }'(q_{\tilde {m}\tilde {n}})}, \end{equation}

where

(3.52) \begin{equation} \mathit{\varLambda }'(q_{\tilde {m}\tilde {n}}) = \left [\frac {\textrm {d}}{\textrm {d} Q}{{\textit{Mc}}_{\tilde {m}}^{(1)'}}(\xi _0,Q)\right ]_{Q=q_{\tilde {m}\tilde {n}}}. \end{equation}