2.1. Governing equations

We consider an inviscid compressible flow without heat conduction, modelled by the Euler equations

(2.1a)$$\begin{gather} \frac{{\rm D}\rho}{{\rm D}t} ={-}\rho\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{v}, \end{gather}$$

(2.1b)$$\begin{gather}\rho\frac{{\rm D}\boldsymbol{v}}{{\rm D}t}={-}\boldsymbol{\nabla}p, \end{gather}$$

(2.1c)$$\begin{gather}\frac{{\rm D}s}{{\rm D}t}=0, \end{gather}$$

where $\rho$ is density, $\boldsymbol {v}$ is the velocity vector, $p$ is pressure and $s$ is entropy. For linear acoustics, we suppose the variables $\rho$, $\boldsymbol {v}$ and $p$ to be composed of the base flow quantities $\rho _0$, $\boldsymbol {v}_0$ and $p_0$, and small unsteady perturbations $\rho '$, $\boldsymbol {v'}$ and $p'$, i.e.

(2.2)\begin{equation} (\rho, \boldsymbol{v}, p) =(\rho_0+\rho', \boldsymbol{v}_0+\boldsymbol{v}', p_0+ p'), \end{equation}

where the perturbations are functions of $x$, $y$ and $t$. Then, we substitute (2.2) into equations (2.1a) and (2.1b), and neglect nonlinear terms.

To obtain a closed linearized equation system, we consider a homentropic (homogeneous and isentropic) and unidirectional flow, i.e. $s_0={\rm const.}$ and $p_0={\rm const.}$, which indicates by the equation of state $\rho (p_0, s_0)$ that under a perfect gas assumption the density is a constant, i.e. $\rho _0={\rm const.}$ (Campos Reference Campos2007). A Taylor series expansion of the pressure about the reference thermodynamic state denoted by the subscript $0$ (neglecting higher-order derivatives) gives

(2.3)\begin{equation} p = p_0+p' = p(\rho_0+\rho',s_0) \approx p(\rho_0,s_0)+\left[\frac{\partial p} {\partial\rho}(\rho,s_0)\right]\rho' = p_0+c^2\rho', \end{equation}

where $c=\sqrt {(\partial p/\partial \rho )}$ is the speed of sound. In this way, we omit the entropy equation (2.1c) and apply the linearized relation for the speed of sound to close the LEE system, which reads

(2.4a)$$\begin{gather} \frac{\partial \rho'}{\partial t} +\boldsymbol{v}_0\boldsymbol{\cdot}\boldsymbol{\nabla}\rho' +\rho_0\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{v}' +\boldsymbol{v}'\boldsymbol{\cdot}\boldsymbol{\nabla}\rho_0 +\rho'\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{v}_0 = 0, \end{gather}$$

(2.4b)$$\begin{gather}\rho_0\left(\frac{\partial \boldsymbol{v'}}{\partial t} +\boldsymbol{v}_0\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{v'}\right) +\boldsymbol{\nabla}p' +\rho_0\boldsymbol{v'}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{v}_0 +\rho'\boldsymbol{v}_0\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{v}_0 = 0, \end{gather}$$

(2.4c)$$\begin{gather}p'= c^2\rho'. \end{gather}$$

For the proposed unidirectional two-dimensional parallel shear flow, an exponential velocity profile is employed to mimic a boundary layer flow, i.e.

(2.5a)$$\begin{gather} u_0(y)=U_{\infty}(1-{\rm e}^{-{y}/{\delta}}), \end{gather}$$

(2.5b)$$\begin{gather}v_0(y)=0, \end{gather}$$

where $u_0$ and $v_0$ stand for the mean flow velocity in the $x$- and $y$-directions, respectively, $U_\infty$ is the free-stream velocity and $\delta$ is the shear layer thickness, which is a multiplier of the hydrodynamic boundary layer thickness. An intuitive illustration of the base flow is shown in figure 1. Definitions of the incident angle $\phi$, the propagation angle $\varTheta$ and the reflection coefficient $R$ in figure 1 are given in §§ 2.2 and 2.4.

Figure 1. Illustration of an exponential boundary layer flow. An incident acoustic wave from the free stream gives rise to a reflected wave with angle $\varTheta$, characterized by the reflection coefficient $R$. The critical layer $y_c$ is marked by the dashed line. Further, a geometric relation of an acoustic wave with frequency $\tilde {\omega }$ in a medium at rest, between wavenumber vector $\tilde {\boldsymbol {k}}$ and streamwise wavenumber $\tilde {\alpha }$, is given by their angle $\phi$ (incident angle).

Inserting (2.5) into (2.4) and eliminating $p'$, we get a system, which, non-dimensionalized by $U_\infty$, $\delta$ and $\rho _0$, reads

(2.6a)$$\begin{gather} \frac{\partial{\rho}}{\partial {t}} +(1-{\rm e}^{-{y}})\frac{\partial{\rho}}{\partial{x}} +\left(\frac{\partial{u}}{\partial{x}}+\frac{\partial{v}}{\partial{y}}\right) = 0, \end{gather}$$

(2.6b)$$\begin{gather}\frac{\partial {u}}{\partial {t}} +(1-{\rm e}^{-{y}})\frac{\partial {u}}{\partial{x}} +{\rm e}^{-{y}}{v}+\frac{1}{M^2}\frac{\partial{\rho}}{\partial{x}}=0, \end{gather}$$

(2.6c)$$\begin{gather}\frac{\partial {v}}{\partial {t}} +(1-{\rm e}^{-{y}})\frac{\partial {v}}{\partial{x}} +\frac{1}{M^2}\frac{\partial {\rho}}{\partial{y}}=0, \end{gather}$$

where $M=U_{\infty }/c$ is the global or free-stream Mach number. For the sake of simplicity, we have dropped the primes in (2.6), but keep in mind that we analyse the perturbations, while $x$, $y$ and $t$ have been respectively normalized by $\delta$ and $\delta /U_\infty$.

It is worth noting that we have adopted a different dimensionless approach here than the traditional approach in the field of acoustics, i.e. using the free-stream velocity $U_\infty$ rather than the speed of sound $c$ to non-dimensionalize. The current approach agrees with the dimensionless variables introduced for the stability analysis in Zhang & Oberlack (Reference Zhang and Oberlack2021). We kept these definitions as it makes it more convenient to establish links between stability theory and acoustics, which is one of the key topics of the present work. Clearly, the parameters and results obtained by different dimensionless approaches are interchangeable.

Considering a two-dimensional perturbation, a normal-mode approach for density and velocity perturbations is applied. Similar to the classical stability theory, which involves Fourier decomposition in $x$ and $t$, we introduce

(2.7)\begin{equation} q(x,y,t) =\hat{q}(y){\rm e}^{{\rm i}(\alpha x-\omega t)}, \end{equation}

with $q\in (u,v,\rho )$, where the quantities $\hat {u}(y)$, $\hat {v}(y)$ and $\hat {\rho }(y)$ represent the amplitudes of the perturbations, $\alpha$ denotes the dimensionless streamwise wavenumber and $\omega$ stands for the dimensionless frequency.

Substituting the normal-mode ansatz (2.7) into the system of partial differential equations (2.6) results in a system of ordinary differential equations:

(2.8a)$$\begin{gather} \frac{{\rm d}\hat{v}}{{\rm d} y}+{\rm i}[-(\omega-\alpha+\alpha {\rm e}^{{-}y})\hat{\rho}+\alpha \hat{u}]=0, \end{gather}$$

(2.8b)$$\begin{gather}{\rm e}^{{-}y}\hat{v}+{\rm i}\left[\frac{\alpha}{M^2}\hat{\rho} -(\omega-\alpha+\alpha {\rm e}^{{-}y})\hat{u}\right]=0, \end{gather}$$

(2.8c)$$\begin{gather}\frac{1}{M^2}\frac{{\rm d}\hat{\rho}}{{\rm d}y}-{\rm i} (\omega-\alpha+\alpha {\rm e}^{{-}y})\hat{v}=0. \end{gather}$$

We rewrite equations (2.8) into an equivalent second-order differential equation with a single dependent variable $\hat {\rho }$. For this we express $\hat {v}$ by ${\rm d}\hat {\rho }/{{\rm d} y}$ through (2.8c), substitute the result for $\hat {v}$ in (2.8b) and thereafter express $\hat {u}$ in terms of $\hat {\rho }$ and ${\rm d}\hat {\rho }/{{\rm d}y}$. Finally, we substitute these results for $\hat {u}$ and $\hat {v}$ into (2.8a) to get the second-order ordinary differential equation for $\hat {\rho }$:

(2.9)\begin{equation} \frac{{\rm d}^2\hat{\rho}}{{{\rm d} y}^2} +\frac{2\alpha {\rm e}^{{-}y}}{\omega-\alpha+\alpha {\rm e}^{{-}y}} \frac{{\rm d}\hat{\rho}}{{\rm d} y} +[M^2(\omega-\alpha+\alpha {\rm e}^{{-}y})^2-\alpha^2]\hat{\rho}=0, \end{equation}

known as the PBE (Pridmore-Brown Reference Pridmore-Brown1958).

If a solution $\hat {\rho }$ to (2.9) is obtained, $\hat {u}$ and $\hat {v}$ can be expressed in terms of $\hat {\rho }$ as

(2.10)\begin{equation} \hat{u}={-}\frac{{\rm e}^{{-}y}}{M^2(\omega-\alpha+\alpha {\rm e}^{{-}y})^2} \frac{{\rm d}\hat{\rho}}{{\rm d} y} +\frac{\alpha}{M^2(\omega-\alpha+\alpha {\rm e}^{{-}y})}\hat{\rho} \end{equation}

and

(2.11)\begin{equation} \hat{v}=\frac{1}{{\rm i}M^2(\omega-\alpha+\alpha {\rm e}^{{-}y})} \frac{{\rm d}\hat{\rho}}{{\rm d} y}. \end{equation}

The PBE (2.9) not only constitutes an eigenvalue problem for modes to investigate stability problems, as was detailed in Zhang & Oberlack (Reference Zhang and Oberlack2021), but also describes acoustic wave propagation (non-resonant spectra) in boundary layer flows, which is the main topic of the present work.

In Zhang & Oberlack (Reference Zhang and Oberlack2021), an exact solution to (2.9) was derived in terms of the CHF denoted by ${\mbox {Hc}}(p_*,\alpha _*,\gamma _*,\delta _*,\sigma _*;z)$, where $p_*,\ \alpha _*,\ \gamma _*,\ \delta _*,\ \sigma _*$ stand for five parameters and $z$ is the independent variable (see e.g. Ronveaux & Arscott Reference Ronveaux and Arscott1995). The solution reads

(2.12)\begin{align} \hat{\rho}(y) &= C_1 \exp({{\rm i} M\alpha {\rm e}^{{-}y}+\sqrt{\theta}\,y}) {\mbox{Hc}}\left(p_*,\alpha_*,\gamma_*,\delta_*,\sigma_*;\frac{\alpha {\rm e}^{{-}y}}{\alpha-\omega}\right)\nonumber\\ &\quad +C_2 \exp({{\rm i} M\alpha {\rm e}^{{-}y}-\sqrt{\theta}\,y}) {\mbox{Hc}}\left(\tilde{p}_*,\tilde{\alpha}_*,\tilde{\gamma}_*,\tilde{\delta}_*, \tilde{\sigma}_*;\frac{\alpha {\rm e}^{{-}y}}{\alpha-\omega}\right), \end{align}

where

(2.13)\begin{equation} \theta={-}M^2(\alpha-\omega)^2+\alpha^2 \end{equation}

and the parameters are defined as follows:

(2.14)\begin{gather} \left. \begin{gathered} p_*=\tfrac{1}{2}{{\rm i}M(\alpha-\omega)}, \quad \alpha_*={\rm i} M (\alpha-\omega)-\tfrac{1}{2}-\sqrt{\theta}, \quad \gamma_*=1-2\sqrt{\theta},\\ \delta_*={-}2, \quad \sigma_*={\rm i}M(\alpha-\omega)-2M^2(\alpha-\omega)^2 -2\sqrt{\theta}[{\rm i}M(\alpha-\omega)+1], \end{gathered} \right\} \end{gather}

(2.15)\begin{gather} \left. \begin{gathered} \tilde{p}_*=\tfrac{1}{2}{{\rm i}M(\alpha-\omega)}, \quad \tilde{\alpha}_* ={\rm i}M(\alpha-\omega)-\tfrac{1}{2}+\sqrt{\theta}, \quad \tilde{\gamma}_*=1+2\sqrt{\theta}, \\ \tilde{\delta}_*={-}2, \quad \tilde{\sigma}_*={\rm i}M(\alpha-\omega) -2M^2(\alpha-\omega)^2+2\sqrt{\theta}[{\rm i}M(\alpha-\omega)+1]. \end{gathered} \right\} \end{gather}

In the following pages, for the sake of brevity, we omit the parameters of the $\mbox {Hc}$ function and express the $\mbox {Hc}$ function simply as $\mbox {Hc}(;z)$ and $\widetilde {\mbox {Hc}}(;z)$ with their respective parameters as in (2.14) and (2.15), and $z=\alpha \textrm {e}^{-y}/(\alpha -\omega )$. As, in this paper, we investigate boundary layer acoustics, only $\alpha \in \mathbb {R}$ and $\omega \in \mathbb {R}$ will be considered. Note that in stability problems one has $\omega \in \mathbb {C}$ for temporal instability or $\alpha \in \mathbb {C}$ for spatial instability.

2.2. Zone of silence

Assume that the incoming acoustic wave has unit amplitude and propagates at a constant speed of sound $c$ from the free stream, which is characterized by the dimensional quantities of frequency $\tilde {\omega }$ and horizontal wavenumber $\tilde {\alpha }$. The incident angle $\phi$, being a ‘secondary’ parameter, is defined as $\phi \in [0^\circ, 90^\circ ]$, which is the angle between the incident wave and the $x$-direction shown in figure 1. Here, first, we intend to establish a relationship between the incident angle $\phi$ and the frequency $\tilde {\omega }$ in order to facilitate $\phi$ as the main parameter to visually describe the incident acoustic wave. Using $\phi$ instead of $\tilde {\omega }$ reduces one parameter in analysing the critical cases shown in figure 2.

Figure 2. Borders of the zone of silence $\phi _{s_0}$ (thick blue solid line), $\phi _{s_1}$ (thick red solid line), $\phi _{s_2}$ (thick black solid line) in (2.24) and the critical value for the presence of the critical layer $\phi _c$ (black dashed line) in (2.26) as functions of the Mach number $M$. The grey region indicates the zone of silence. The vertically striped region is the propagation zone and the horizontally striped region represents the presence of the critical layer.

The horizontal wavenumber of an acoustic wave in a medium at rest is given by

(2.16)\begin{equation} \tilde{\alpha}=|\tilde{\boldsymbol{k}}|\cos(\phi)=\frac{\tilde{\omega}}{c}\cos(\phi). \end{equation}

In dimensionless form it gives

(2.17)\begin{equation} \omega=\frac{\alpha}{M\cos(\phi)}, \end{equation}

where a dimensionless wavenumber $\alpha =\tilde {\alpha }\delta$ has been used; or alternatively the wavelength $\tilde {\lambda }$ may be employed, $\tilde {\alpha }=2{\rm \pi} /\tilde {\lambda }$, so that we have

(2.18)\begin{equation} \alpha=\frac{2{\rm \pi}\delta }{\tilde{\lambda}}, \end{equation}

which implies a ratio between the shear layer thickness and the wavelength of the perturbation. This ratio indicates the range of $\alpha$ taken in § 3 from 0.1 to 10, implying that the thickness of the boundary layer varies from small to large relative to the wavelength scale.

Note that the incident angle $\phi$ in (2.17), which is an angle between the wavenumber vector $\tilde {\boldsymbol {k}}$ and the streamwise direction, is strictly speaking for a medium at rest, i.e. $U_\infty =0$, and hence $\phi$ is only an ‘auxiliary’ parameter used as a replacement to implicitly express the frequency. It is not the true propagation angle of the acoustic wave in the free stream. The ‘true’ angle of the incident wave propagating towards the boundary layer in the free stream, due to the velocity of the free stream $U_{\infty }$, is determined by the wavenumber of the acoustic wave in the $y$-direction. This wavenumber is given by the solution to the PBE in the free stream.

In the free stream $(y\to \infty )$, where any shear is absent, the PBE (2.9) simplifies to

(2.19)\begin{equation} \frac{{\rm d}^2\hat{\rho}}{{{\rm d} y}^2} +[M^2(\omega-\alpha)^2-\alpha^2]\hat{\rho}=0, \end{equation}

which has oscillatory (waveform) solutions when the term in (2.19) in square brackets is greater than zero, i.e. $\theta =-M^2(\alpha -\omega )^2+\alpha ^2<0$. The same result can also be derived directly by setting $y\to \infty$ in the exact solution (2.12), where $\mbox {Hc}(;0)=1$ and $\widetilde {\mbox {Hc}}(;0)=1$, and the solution reads

(2.20)\begin{equation} \hat{\rho}(y)=C_1 {\rm e}^{\sqrt{\theta}y}+C_2 {\rm e}^{-\sqrt{\theta}y}, \end{equation}

where the principal value of $\sqrt {\theta }$ is taken and the branch cut is along the negative real axis.

For $\theta <0$, the wavenumber in the $y$-direction in the free stream reads

(2.21)\begin{equation} \beta=\sqrt{-\theta}, \end{equation}

where the principal value of the square root is taken. It follows that the two solutions in (2.20) stand for an outgoing wave $(C_1 \textrm {e}^{\textrm {i}\beta y})$ and an incoming wave $(C_2 \textrm {e}^{-\textrm {i}\beta y})$, respectively, for $\theta <0$ and $\theta \in \mathbb {R}$. This can be intuitively observed by the shifts of the peaks and troughs of the waves in time by adding the time dependence in the normal mode (2.7).

It is important to note that this work is restricted to purely acoustic free-stream behavior, which means real wavenumbers $\beta \in \mathbb {R}$ and real values $\theta <0$. The case where $\theta \in \mathbb {C}$ or $\theta >0$ was treated by Zhang & Oberlack (Reference Zhang and Oberlack2021) and is not considered in the present work. For $\theta >0$, there is an exponential behavior of the solution (2.20) in the free stream. Hence a non-wave type of behavior is observed, which is explicitly excluded here. The angle of propagation with respect to the $x$-axis of an acoustic wave in the free stream is therefore defined as

(2.22)\begin{equation} \varTheta=\arctan\left(\frac{\beta}{\alpha}\right), \end{equation}

as shown in figure 1.

Next, two kinds of acoustic waves in boundary layer flows are distinguished as propagating waves and evanescent waves, for waves that can propagate in a vertical direction and waves that cannot. These two cases correspond to $\theta <0$ and $\theta >0$ in (2.13), respectively. Considering the relation (2.17) gives the critical case, i.e.

(2.23)\begin{equation} -\theta=\alpha^2\left[\left(\frac{1}{\cos(\phi)}-M\right)^2-1 \right]=0, \end{equation}

which gives two solutions $\phi _s$ corresponding to ‘zones of silence’ given by

(2.24) \begin{equation} \phi_{s}\in\left\{ \begin{array}{@{}ll} \left[0, \arccos\left(\dfrac{1}{M+1}\right)\right], & M\leq2,\\[12pt] \left[\arccos\left(\dfrac{1}{M-1}\right), \arccos\left(\dfrac{1}{M+1}\right)\right], & M>2.\\ \end{array} \right. \end{equation}

In (2.24), the lower border for $M\leq 2$ is denoted as $\phi _{s_0}$, the upper border for both cases as $\phi _{s_1}$ and the lower border for $M>2$ as $\phi _{s_2}$. An intuitive illustration is shown in figure 2, where the zone of silence is marked in grey. No acoustic waves exist for the areas defined in (2.24) or figure 2.

The introduction of $\phi$ instead of $\omega$ gives the critical case (2.24), which has only one independent variable $M$. It is possible to avoid the introduction of $\phi$, which, however, would imply that the critical case for $\omega$ involves two variables instead of one, i.e. $\alpha$ and $M$.

Equation (2.24) indicates that acoustic perturbations with $\phi _s$ as the incident angle do not exist in a waveform in the free stream, i.e. their amplitudes decay exponentially and tend to zero. Therefore, they are called evanescent waves. Whereas outside the $\phi _s$ region, acoustic perturbations in the free stream exist as oscillatory waves, i.e. they can propagate in the $y$-direction and therefore denote propagating waves. We define the region outside of $\phi _s$ as the ‘propagation zone’, which is represented by vertical stripes in figure 2. The introduction of the concept of the incident angle instead of the frequency physically defines the range of parameter $\phi$ for the existence of incident and outgoing waves at different Mach numbers.

2.3. Critical layer

As per its definition, the critical layer is a wall-parallel plane, where the phase velocity of the acoustic wave $\omega /\alpha$ is equal to the local base flow velocity $u_0(y)$, i.e. $\omega /\alpha = 1-\textrm {e}^{-y}$. Considering an incident acoustic wave in (2.17), the location of the critical layer is given by

(2.25)\begin{equation} y_{c}={-}{\rm ln}\left(1-\frac{1}{M\cos(\phi)}\right), \end{equation}

as shown in figure 1. It follows from (2.25) that, in order to ensure the existence of a critical layer, the argument of the logarithm has to be greater than zero and smaller than one, and, with this, the incident angle has to be less than a critical value given by

(2.26)\begin{equation} \phi_{c}=\arccos\left(\frac{1}{M}\right), \end{equation}

as displayed in figure 2. When the incident angle is greater than $\phi _c$, i.e. $\phi >\phi _c$, there is no critical layer in the shear flow. For the case $\phi =\phi _c$, the critical layer moves to infinity.

Figure 2 shows limiting curves of the zone of silence $\phi _s$ as well as the corresponding critical incident angle of the critical layer $\phi _c$ as a function of the Mach number. The thick red and thick black solid lines in figure 2 represent the borders $\phi _{s_1}$ and $\phi _{s_2}$ of the zone of silence according to (2.24). The grey region between them is the zone of silence. The critical incident angle for the presence of the critical layer is represented by the black dashed line. These critical angles, $\phi _{s_1}$, $\phi _{s_2}$ and $\phi _c$, provide a reference for the computation domain in § 3.

In § 3, we focus only on the propagation zone ($\theta <0$) where the critical layer is present ($\phi <\phi _c$) because these conditions ensure the presence of incident and reflected waves in the free stream and the occurrence of over-reflection. To satisfy these two conditions simultaneously, the chequered region shown in figure 2 is concerned. Considering the transformation of $\phi$ corresponding to the chequered region to $\omega$, it is equivalent to $\omega /\alpha \in [1/M,1-1/M]$. A detailed study of the critical layer leading to over-reflection is given in § 2.5.

2.4. Reflection coefficient

In this section, we derive the reflection coefficient $R$, which is the amplitude ratio of the reflected to the incident wave. This will be based on the exact solution (2.12) to the PBE (2.9) and the boundary conditions. We consider first the boundary condition at the wall, i.e. $y=0$. There, the condition is obtained through the impermeability condition, wherein, in the present paper, we adopt the simplest case, i.e. a rigid (inelastic) wall. Thus, the normal component of the velocity perturbation at the wall vanishes, i.e.

(2.27)\begin{equation} \hat{v}(0)=0. \end{equation}

Using the boundary condition (2.27) together with (2.11) and (2.12), where the derivative of the density perturbation reads

(2.28)\begin{align} \frac{{\rm d}\hat{\rho}}{{\rm d} y} &= C_1\left[(-{\rm i}M\alpha {\rm e}^{{-}y}+\sqrt{\theta}) {\mbox{Hc}}\left(;\frac{\alpha {\rm e}^{{-}y}}{\alpha-\omega}\right) -\frac{\alpha {\rm e}^{{-}y}}{\alpha-\omega}{\mbox{Hc}}' \left(;\frac{\alpha {\rm e}^{{-}y}}{\alpha-\omega}\right)\right]\nonumber\\ &\quad\times \exp({{\rm i}M\alpha {\rm e}^{{-}y}+\sqrt{\theta}y})\nonumber\\ &\quad+C_2\left[(-{\rm i}M\alpha {\rm e}^{{-}y}-\sqrt{\theta}) {\widetilde{\mbox{Hc}}}\left(;\frac{\alpha {\rm e}^{{-}y}}{\alpha-\omega}\right) -\frac{\alpha {\rm e}^{{-}y}}{\alpha-\omega}{\widetilde{\mbox{Hc}}}' \left(;\frac{\alpha {\rm e}^{{-}y}}{\alpha-\omega}\right)\right]\nonumber\\ &\quad\times \exp({{\rm i}M\alpha {\rm e}^{{-}y}-\sqrt{\theta}y}), \end{align}

we obtain

(2.29)\begin{align} v(0) &= \frac{-{\rm i}{\rm e}^{{\rm i}M\alpha}}{M^2\omega} \left\{C_1\left[(-{\rm i}M\alpha +\sqrt{\theta}) {\mbox{Hc}}\left(;\frac{\alpha }{\alpha-\omega}\right) -\frac{\alpha}{\alpha-\omega}{\mbox{Hc}}' \left(;\frac{\alpha }{\alpha-\omega}\right)\right]\right.\nonumber\\ &\qquad\left.+\,C_2\left[(-{\rm i}M\alpha -\sqrt{\theta}) {\widetilde{\mbox{Hc}}}\left(;\frac{\alpha}{\alpha-\omega}\right) -\frac{\alpha }{\alpha-\omega} {\widetilde{\mbox{Hc}}}' \left(;\frac{\alpha}{\alpha-\omega}\right)\right]\right\}=0. \end{align}

From (2.29), a relation between $C_1$ and $C_2$ is induced and reads

(2.30)\begin{equation} C_1={-}C_2\dfrac{(-{\rm i}M\alpha -\sqrt{\theta}) {\widetilde{\mbox{Hc}}}\left(;\dfrac{\alpha}{\alpha-\omega}\right) -\dfrac{\alpha }{\alpha-\omega}{\widetilde{\mbox{Hc}}}' \left(;\dfrac{\alpha}{\alpha-\omega}\right)} {(-{\rm i}M\alpha +\sqrt{\theta}) {\mbox{Hc}}\left(;\dfrac{\alpha}{\alpha-\omega}\right) -\dfrac{\alpha}{\alpha-\omega} {\mbox{Hc}}'\left(;\dfrac{\alpha}{\alpha-\omega}\right)}. \end{equation}

We assume that the amplitude of the incoming wave at $y\to \infty$, which is the second term of (2.12), is unity and thus, combining (2.12) with (2.30), the amplitude of the density perturbation reads

(2.31)\begin{align} \hat{\rho}(y) &=R \exp({{\rm i} M\alpha {\rm e}^{{-}y}+\sqrt{\theta}y}) {\mbox{Hc}}\left(;\frac{\alpha {\rm e}^{{-}y}}{\alpha-\omega}\right)\nonumber\\ &\quad +\exp({{\rm i} M\alpha {\rm e}^{{-}y}-\sqrt{\theta}y}) {\widetilde{\mbox{Hc}}}\left( ;\frac{\alpha {\rm e}^{{-}y}}{\alpha-\omega}\right), \end{align}

where $R$ is the reflection coefficient defined by

(2.32)\begin{equation} R={-}\dfrac{(-{\rm i}M\alpha -\sqrt{\theta}) {\widetilde{\mbox{Hc}}} \left(;\dfrac{\alpha}{\alpha-\omega}\right) -\dfrac{\alpha }{\alpha-\omega} {\widetilde{\mbox{Hc}}}'\left(;\dfrac{\alpha}{\alpha-\omega}\right)} {(-{\rm i}M\alpha +\sqrt{\theta}) {\mbox{Hc}}\left(;\dfrac{\alpha}{\alpha-\omega}\right) - \dfrac{\alpha}{\alpha-\omega} {\mbox{Hc}}'\left(;\dfrac{\alpha}{\alpha-\omega}\right)}. \end{equation}

In (2.32), there are three parameters, i.e. the wavenumber $\alpha$, the Mach number $M$ and the frequency $\omega$, where $\omega$ can be replaced by the incident angle $\phi$ according to (2.17).

In Campos & Kobayashi (Reference Campos and Kobayashi2013), a similar reflection coefficient is obtained for the PBE with a hyperbolic-tangent profile, which is used to mimic a boundary layer flow. Not only is an over-reflection of acoustic waves observed but the reflection coefficient has an unusual high peak at a certain frequency (see the case $M = 4.5$ in figure 6(a) in Campos & Kobayashi (Reference Campos and Kobayashi2013) and note that there is a mistake in that figure, i.e. figure 6(a) should be for $|R|$ and not for $|T|$). For this, no further explanation was given, but the values lie close to the eigenvalues we obtained for instability.

In the remaining part of the paper, we will analyse the reflection coefficient $R$ and we will observe various effects such as over-reflection or resonant over-reflection. Further, in § 2.5 we give a detailed analysis that for boundary layer flows the presence of a critical layer is intimately linked to the occurrence of over-reflection.

The explicit form of the reflection coefficient (2.32) points to a variety of physical phenomena. We recall that the temporal stability problems in Zhang & Oberlack (Reference Zhang and Oberlack2021) emerged from the eigenvalue equation derived therein, and which was obtained from the exact solution (2.12), in combination with boundary conditions of vanishing disturbances at infinity and zero wall-normal velocity. The former condition leads to $C_1=0$ in (2.12), and the latter condition gives $\textrm {d}\hat {\rho }/{\textrm {d} y}(0)=0$ in (2.28). An eigenvalue equation is therefore obtained. It is interesting to note that the same eigenvalue equation coincides with the current acoustic case, in which the numerator of $R$ is equal to zero. Acoustically, this indicates that no reflected waves are allowed to exist. But for unstable modes ($\omega _i>0$) in the temporal stability problem, setting $C_1=0$ implies the opposite scenario, i.e. that there are only outgoing waves and no incoming waves because the ‘outgoing wave’ must be redefined. Since the unstable modes show dispersive waves with amplitude decreasing to zero as $y$ tends to infinity, the group velocity (direction) is considered to be the true propagation velocity (direction) of the waves. In Zhang & Oberlack (Reference Zhang and Oberlack2021), it is demonstrated that the unstable wave with decreasing amplitude has a positive group velocity with a negative phase velocity. This leads to $C_1$ in the current context having an opposite physical meaning to that in the stability problem. This might give a physical insight: perturbations that acquire energy from the shear flow are not allowed to emit in the form of over-reflections, thereby manifesting themselves as the onset of a temporal–spatial instability, i.e. in a certain parameter range we obtain unstable eigenvalues $\omega \in \mathbb {C}$ for temporal instability or $\alpha \in \mathbb {C}$ for spatial instability.

To distinguish resonant over-reflection and hyper-reflection very clearly within the present work, let us consider the following setting. From an acoustic point of view, in the free stream, the first term of (2.31) represents the outgoing wave, while the second term represents the incoming wave. Taking the principal value of $\sqrt {\theta }$, setting $C_2=0$ means that a phenomenon occurs in which the incident wave is zero at an infinite distance from the wall while the reflected wave persists with a constant amplitude for $y\to \infty$. Considering the boundary condition at the wall, an eigenvalue problem is formulated, in which the eigenvalue equation is equivalent to setting the denominator in (2.32) to zero. Thus, in this case, the sought eigenvalues lead to an infinite $R$. To distinguish this case from the over-reflection induced by resonant frequencies in this paper, we define the above phenomenon of the reflection coefficient $R$ tending to infinity as ‘hyper-reflection’.

The focus of the present paper is to investigate over-reflection, and specifically over-reflection induced by the resonant frequencies in an instability context which we will presently name ‘resonant over-reflection’. Infinitely strong over-reflection, i.e. hyper-reflection as defined above, is not considered.

2.5. Analysis of the critical layer

In this section, we show how the critical layer is intimately linked to over-reflection. We further present an analytical relation between the amplitude of the density perturbation at the critical layer and the reflection coefficient $R$ of the boundary layer flow.

For this, a transformation of the independent variable $y$ is introduced in order to eliminate the first-order derivative in the PBE (2.9) and bring it to its normal form by introducing

\[ {\tilde y} = \int \exp \left(-\int \frac{2\alpha({\rm d}u_0(y)/{{\rm d}y})}{\omega-\alpha u_0(y)}{\rm d}y\right){{\rm d}}y \]

(see Olver et al. Reference Olver, Lozier, Boisvert and Clark2010).

In concrete terms and using $u_0(y)$, given by its dimensionless form $u_0(y)=1-\textrm {e}^{-y}$, we obtain

(2.33)\begin{equation} {\tilde y}=2\alpha(\alpha-\omega){\rm e}^{{-}y} -\tfrac{1}{2}{\alpha^2}{\rm e}^{{-}2y}+y(\alpha-\omega)^2. \end{equation}

With (2.33), the transformed PBE (2.9) takes the form

(2.34)\begin{equation} \frac{{\rm d}^2\hat\rho}{{\rm d}{\tilde y}^2}+\frac{[M^2(\omega-\alpha u_0)^2-\alpha^2]}{(\omega-\alpha u_0)^4}\hat\rho=0. \end{equation}

Next, (2.34) is multiplied by the conjugate solution ${{\hat \rho }^*}$ and integrated between the boundaries defined by ${\tilde y}_1$ and ${\tilde y}_2$. We limit ourselves to the imaginary part, which we refer to by $\mathrm {Im}$. This is motivated by a certain invariant of the transformed PBE, which will be explained in more detail below at (2.41). With the above we obtain

(2.35) \begin{equation} \left.\mathrm{Im}\left(\frac{{\rm d}\hat\rho}{{\rm d}{\tilde y}} {{\hat\rho}^*}\right)\right|_{{\tilde y}_1}^{{\tilde y}_2} =\int_{{\tilde y}_1}^{{\tilde y}_2}\mathrm{Im}\left(\frac{\alpha^2}{(\omega-\alpha u_0)^4}-\frac{M^2}{(\omega-\alpha u_0)^2}\right)|\hat\rho|^2\,{\rm d}{\tilde y}. \end{equation}

We will discuss both sides of (2.35) thoroughly in three parts in the following. First, the left-hand side of (2.35) is analysed with respect to its relation with the reflection coefficient $R$. Second, the left-hand side of (2.35) is proven to be associated with a jump of the above-mentioned invariant at the critical layer. In the presence of a critical layer, this invariant jumps directly at the critical layer, i.e. is constant above and below the critical layer. The step-like variation of this invariant is further related to the logarithmic singularity at the critical layer. These two facts indicate that the reflection coefficient is always larger than one in the presence of a critical layer. Third, we will discuss the right-hand side of (2.35) and evaluate the integrals in order to obtain an analytical relation between the reflection coefficient and the perturbed quantities at the critical layer.

To begin with the evaluation of the left-hand side of (2.35), it reveals a close link to the reflection coefficient $R$. For this, we recall the principal asymptotics of the acoustic solution in the limit $y\to \infty$, which is taken from (2.20). An incident wave with unit amplitude and a reflected wave with complex amplitude $R$ at $y\to \infty$ are assumed, which refers to ${\tilde y}_2$. At ${\tilde y}_1$, corresponding to the wall, i.e. $y=0$, the rigid-wall boundary condition (2.27) leads with (2.11) to a vanishing $y$ derivative of the density. To sum up, the boundary conditions read

(2.36a) \begin{equation} \hat\rho(y\to\infty)\sim R{\rm e}^{{\rm i}\beta y}+{\rm e}^{-{\rm i}\beta y}, \end{equation}

and

(2.36b) \begin{equation} \left.\frac{{\rm d}\hat\rho}{{\rm d}{\tilde y}}\right|_{{\tilde y}_1} =\left.\frac{{\rm d} y}{{\rm d}{\tilde y}}\frac{{\rm d}\hat\rho}{{\rm d} y}\right|_{y=0}=0. \end{equation}

Inserting (2.36a) and (2.36b) into the left-hand side of (2.35), it can be reshaped to

(2.37) \begin{equation} \left.\mathrm{Im}\left(\frac{{\rm d}\hat\rho}{{\rm d}{\tilde y}} {{\hat\rho}^*}\right)\right|_{{\tilde y}_1}^{{\tilde y}_2} =\left.\mathrm{Im}\left(\frac{{\rm d}\hat\rho}{{\rm d} y}\frac{{\rm d} y}{{\rm d} {\tilde y}}{{\hat\rho}^*}\right)\right|_{{\tilde y}_1}^{{\tilde y}_2} =\frac{\beta}{(\alpha-\omega)^2}(|R|^2-1). \end{equation}

We further rewrite the left-hand side of (2.37) below to obtain a relationship between the reflection coefficient, namely the right-hand side of (2.37), and the physical properties at the critical layer. Considering the $y$ derivative of the left-hand side of (2.37), we show that it vanishes outside the critical layer, i.e.

(2.38)\begin{equation} \frac{{\rm d}}{{\rm d} y}\left(\mathrm{Im}\left(\frac{{\rm d}\hat\rho}{{\rm d}{\tilde y}}{{\hat\rho}^*}\right)\right)=\mathrm{Im}\left(\frac{{\rm d}^2\hat\rho}{{\rm d}\tilde y^2}\hat\rho^*+\left|\frac{{\rm d}\hat\rho}{{\rm d}\tilde y}\right|^2\right)=\mathrm{Im}\left(\frac{{\rm d}^2\hat\rho}{{\rm d}{\tilde y^2}}{{\hat\rho}^*}\right), \end{equation}

which is, using (2.34), rewritten as

(2.39)\begin{equation} \frac{{\rm d}}{{\rm d} y}\left(\mathrm{Im}\left(\frac{{\rm d}\hat\rho}{{\rm d}{\tilde y}}{{\hat\rho}^*}\right)\right)=\mathrm{Im}\left(-\frac{[M^2(\omega-\alpha u_0)^2-\alpha^2]}{(\omega-\alpha u_0)^4}|\hat\rho|^2\right)=0\Big|_{y\neq y_c} \end{equation}

The above derivation only holds outside the critical layer, since at the critical layer the coefficient function in (2.34) is singular. From this, it follows that

(2.40)\begin{equation} I=\mathrm{Im}\left(\frac{{\rm d}\hat\rho}{{\rm d}{\tilde y}}{{\hat\rho}^*}\right), \end{equation}

is a constant, which undergoes a jump at the critical layer. Therefore, we call $I$ a quasi-invariant, since it is invariant only in limited domains. Thus, the boundaries between which the left-hand side of (2.37) is evaluated can also be chosen just above and below the critical layer ${\tilde y}_c$, which yields

(2.41) \begin{equation} I\Big|_{{\tilde y}_1}^{{\tilde y}_2} =\left.\mathrm{Im}\left(\frac{{\rm d}\hat\rho}{{\rm d} {\tilde y}}{{\hat\rho}^*}\right)\right|_{{\tilde y}_1}^{{\tilde y}_2}=\left.\mathrm{Im}\left(\frac{{\rm d}\hat\rho}{{\rm d} {\tilde y}}{{\hat\rho}^*}\right)\right|_{{\tilde y}_c^-}^{{\tilde y}_c^+}=\frac{\beta}{(\alpha-\omega)^2}(|R|^2-1).\end{equation}

Subsequently Appendix A will be used to evaluate the jump of the quasi-invariant at the critical layer. Transforming the left-hand side of (2.41) to the coordinate $\xi =\omega /\alpha -u_0(y)$, which is used to derive the Frobenius solution (A6) at the critical layer in Appendix A, yields

(2.42) \begin{equation} I\Big|_{{\tilde y}_1}^{{\tilde y}_2} =\left.\mathrm{Im}\left(\frac{{\rm d}\hat\rho}{{\rm d} {\tilde y}}{{\hat\rho}^*}\right)\right|_{{\tilde y}_c^-}^{{\tilde y}_c^+}=\left.\mathrm{Im}\left(\frac{{\rm d} y}{{\rm d}\tilde y}\frac{{\rm d}\xi}{{\rm d} y}\frac{{\rm d}\hat\rho}{{\rm d}\xi}\hat\rho^*\right)\right|_{0^+}^{0^-} ={-} u_0'(y_c)\left.\mathrm{Im}\left(\frac{1}{\xi^2}\frac{{\rm d}\hat\rho}{{\rm d}\xi}\hat\rho^*\right)\right|_{0^+}^{0^-}. \end{equation}

Taking the Fuchs–Frobenius solution (A6) into (2.42) leads to

(2.43) \begin{align} I\Big|_{{\tilde y}_1}^{{\tilde y}_2} &=\left.\mathrm{Im}\left(\frac{{\rm d}\hat\rho}{{\rm d} {\tilde y}}{{\hat\rho}^*}\right)\right|_{{\tilde y}_c^-}^{{\tilde y}_c^+} ={-} 3cu_0'(y_c)|B|^2 \, {\mathrm{Im}}({\rm ln}(\xi))\Big|^{0^-}_{0^+}\nonumber\\ &=-3cu_0'(y_c)|B|^2 \ln (0^-)={-}3{\rm \pi} c u_0'(y_c)|B|^2, \end{align}

where $B$ is a constant in the Fuchs–Frobenius series and $c$ is a rational function in $\alpha$ and $\omega$ as defined in (A7). The causal choice of the logarithmic branch cut as explained in Appendix B is important since it leads to $\mathrm {Im}(\textrm {{ln}}(0^-))={\rm \pi}$. Note that an non-causal choice of the branch cut would lead to the opposite sign.

Taking into account that $c$ is always negative in the presence of a critical layer, as shown in Appendix A by analysing (A8), we find that the right-hand side of (2.43) is always positive. Thus it can be concluded that the jump of $I=\hat \rho ^*\, \textrm {d}\hat \rho /\textrm {d}\tilde y$ over the critical layer is also always positive and thus, in considering (2.41), the reflection coefficient in the presence of a critical layer is always greater than one, i.e. $R>1$, since we obtain

(2.44)\begin{equation} \frac{\beta}{(\alpha-\omega)^2}(|R|^2-1)={-}3{\rm \pi} c\, u_0'(y_c)|B|^2. \end{equation}

In the absence of a critical layer, the right-hand side of (2.35) vanishes since the coefficients are real and the integral contains no singularity that would lead to an imaginary part, following the theory of generalized functions and distributions given in Galapon (Reference Galapon2016). Likewise, considering (2.37) with the knowledge that $I=\hat \rho ^*\, \textrm {d}\hat \rho /\textrm {d}\tilde y$ is invariant over the entire physical domain in the absence of a critical layer, we conclude that $R=1$ must hold without a critical layer. This means that without a critical layer there is no mechanism of damping or amplification of the reflected wave.

Thus, energy can only be transferred from the base flow to the acoustic wave, and the acoustic wave cannot be damped, since $R=1$ in the absence of a critical layer and $R>1$ in the presence of a critical layer.

In the last part of this section, we focus on the evaluation of the right-hand side of (2.35). For this, we evaluate both parts of (2.35) separately using the theory of distributions, which leads to

(2.45)\begin{align} & \int_{{\tilde y}_1}^{{\tilde y}_2}\mathrm{Im}\left(\frac{\alpha^2}{(\omega-\alpha u_0)^4}-\frac{M^2}{(\omega-\alpha u_0)^2}\right)|\hat\rho|^2\,{\rm d}{\tilde y} \nonumber\\ & \quad =\left.{\rm \pi}\left[\left(\frac{M^2}{\alpha}\frac{{\rm d}}{{\rm d}\xi}-\frac{\alpha}{6}\frac{{\rm d}^3}{{\rm d}\xi^3}\right) \frac{|\hat\rho|^2}{{{\rm d} u_0}/{{\rm d}{\tilde y}}}\right]\right|_{\xi=0}, \end{align}

where we have further introduced a coordinate based on the location of the critical layer, i.e.

(2.46)\begin{equation} \xi=\omega-\alpha u_0(y), \end{equation}

and therefore $\xi =0$ is the location of the critical layer. A detailed derivation of (2.45) is given in Appendix C.

In a final step, we use (2.35), where the left-hand side was replaced by (2.37) and the right-hand side by (2.45). Transforming this back to the initial variable $y$ leads to a relation between the amplitude of the reflected wave and the amplitude of density perturbation and its derivations at the critical layer, which reads

(2.47)\begin{equation} \frac{\beta}{(\alpha-\omega)^2}(|R|^2-1) =\left.{\rm \pi}\alpha^2(\alpha-\omega)^5\left[\frac{{\rm d}}{{\rm d} y} \left(\frac{|\hat\rho|^2}{{{\rm d}u_0}/{{\rm d}y}}\right) -2\frac{|\hat\rho|^2}{{{\rm d} u_0}/{{\rm d}y}}\right]\right|_{y=y_c}. \end{equation}

There is only information on the amount of the reflection coefficient here; the phase shift of the reflected wave compared to the incident one does not result from (2.47). (Since $R$ is complex, the phase shift of the reflected wave is included in the argument of $R$.)

Equation (2.47) is similar to the ‘unitarity condition’, introduced by Lapin (Reference Lapin2011) in the context of waves interacting with a jet; thus we call it the‘extended unitarity condition’ in terms of the current context of acoustics.

With the key results (2.44) and (2.47), we show that the over-reflection is directly caused by the critical layer. The critical layer causes a jump of the quasi-invariant $I$, which leads to the amplitude of the reflected wave being larger than that of the incident wave. An analytical relation between the reflection coefficient and the amplitude of density perturbation is given by the extended unitarity condition.

3.1. Non-resonant over-reflection

Figure 3 displays the numerical results for the over-reflection coefficient as a function of Mach number $M$ and incident angle $\phi$, where the wavenumber $\alpha$ is chosen in the range between $0.1$ and $1$. Therein we have also included contour lines, where the thick solid line corresponds to the border $\phi _{s_2}$ between the propagation zone and the zone of silence, which is shown in figure 2. The reflection coefficient in the latter region is either equal to one or has no real physical significance due to exponential non-oscillatory decay of the amplitude into the free stream.

Figure 3. Over-reflection coefficient $R$ defined by (2.32) as a function of $M$ and $\phi$, for a range of $\alpha$ from $0.1$ to $1$: (a) $\alpha =0.1$, (b) $\alpha =0.2$, (c) $\alpha =0.3$, (d) $\alpha =0.6$, (e) $\alpha =0.8$ and (f) $\alpha =1$. The thick black solid line defines the over-reflection border $\phi _{s_2}$ according to figure 2.

Observing figure 3(a–f), we note first that the over-reflection coefficient decreases with increasing wavenumbers. For small wavenumbers, as in figure 3(a–c), there are rather large values of the over-reflection coefficient. In particular, for $\alpha =0.1$, the maximum value even reaches approximately $R=3$, while the values of the over-reflection coefficient shown in figure 3(d–f) are relatively small. This result means that, for small wavenumbers, relatively strong over-reflections occur.

In addition, the large values of the over-reflection coefficient gather around the line of silence $\phi _{s_2}$ and go along with large gradients nearby. Particularly visible is that, at $\alpha =0.1$, a large gradient of variation in the over-reflection coefficient near the over-reflection border $\phi _{s_2}$ can be detected by observing the contours from $R=1.9$ to $R=2.7$, which become progressively narrower. And as the wavenumber $\alpha$ increases, in figure 3(b,c), the peak gradually moves away from the line of silence $\phi _{s_2}$. Meanwhile, the contours become sparse, which indicates that the drastic variation becomes flatter as $\alpha$ increases as in figure 3(d–f). This result suggests that over-reflections generated by small wavenumbers are more sensitive to variations in the Mach number than over-reflections generated by moderate wavenumbers.

3.2. Coincidence of resonantly over-reflected waves and unstable acoustic modes

In this section, we focus on the resonant over-reflection and its connection to unstable acoustic modes. We first depict a phenomenon in which an unusual peak of the over-reflection coefficient appears with increasing wavenumbers in figure 4. Subsequently, through figure 5 and figure 6, we establish links between the resonant frequencies and the peaks, i.e. links between unstable modes and resonant over-reflection.

Figure 4. Over-reflection coefficient $R$ defined by (2.32) as a function of $M$ and $\phi$, for a range of $\alpha$ from $1.1$ to $1.9$: (a) $\alpha =1.1$, (b) $\alpha =1.2$, (c) $\alpha =1.3$, (d) $\alpha =1.5$, (e) $\alpha =1.7$ and (f) $\alpha =1.9$. The thick black solid line defines the over-reflection border $\phi _{s_2}$ according to figure 2. The red solid line defines the local/global maximum. The red dashed line defines $R$ at resonant frequencies.

Figure 5. Over-reflection coefficient $R$ as a function of $\alpha$ and $\phi$ for different Mach numbers: (a) $M=4$, (b) $M=4.5$ and (c) $M=5$. In this parameter domain, only the first unstable mode induced resonant over-reflections, and their peaks are shown and marked by thick solid lines.

Figure 6. Coincidence of unstable modes and resonant over-reflection. (a) The first unstable acoustic mode of temporal instability for different Mach numbers, where the eigenvalue $\omega \in \mathbb {C}$, as a function of $\alpha$; here $\omega _r$ is the resonant frequency and $\omega _i$ is the growth rate. (b) The $\alpha$–$\omega _r$ plane in (a) is converted to the $\alpha$–$\phi$ plane according to (2.17). (c) Local maximum value of the resonant over-reflection in figure 5 and their projections on the $\alpha$–$\phi$ plane. The dashed lines in panels (b) and (c) represent the projection of the solid lines on the $\alpha$–$\phi$ plane.

Figure 4 depicts the variation of the over-reflection coefficient in a range of wavenumbers between $\alpha =1.1$ and $\alpha =1.9$, where the range of Mach numbers is still chosen to be between $M=2$ and $M=5$. Note that in figure 4 we have now swapped the coordinates $M$ and $\phi$ to better observe the key effect in this subsection, i.e. that at a wavenumber $\alpha = 1.1$, an unusual local peak appears near the incident angle $\phi \approx 75$ and Mach number $M\approx 4.5$. This local peak grows as the wavenumber increases and becomes a global peak finally at $\alpha \approx 1.3$. Thereafter, the peak continues to increase with wavenumber until $\alpha \approx 2$ in figure 7(a) and then starts decreasing again. These peaks occur within a very small range of the parameter $\phi$. The variation of the over-reflection coefficient in the remaining part is not affected by the peak and remains largely smooth. As the wavenumber increases, the remaining part slowly decreases.

Figure 7. Over-reflection coefficient $R$ defined by (2.32) as a function of $M$ and $\phi$, for a range of $\alpha$ from $2$ to $10$: (a) $\alpha =2$, (b) $\alpha =3$, (c) $\alpha =4$, (d) $\alpha =7$ and (e) $\alpha =10$. The thick black solid line defines the over-reflection border $\phi _{s_2}$ according to figure 2. The red solid line defines the resonant over-reflection induced by unstable modes.

In the following, the goal is to unravel the connection between resonant over-reflections and unstable modes. To achieve this, in figure 5(a–c) we show the results of the over-reflection coefficient as a function of the wavenumber $\alpha$ and the incident angle $\phi$, where the wavenumber ranges from $1$ to $10$ and the Mach numbers are fixed to $M=4$, $M=4.5$ and $M=5$. From figure 5, we note that the resonant over-reflection peaks appear as the wavenumber increases. To explain the peaks, we next establish the relationship between the resonant frequencies and the resonant over-reflections.

In Zhang & Oberlack (Reference Zhang and Oberlack2021), they deduce an eigenvalue problem for the temporal stability of the exponential boundary layer profile, and this is essentially based on solution (2.12). The imaginary part of the eigenvalue derived there, i.e. the growth rate $\omega _i$, is a function of the Mach number $M$, and the wavenumber $\alpha$ is of key importance here. In figure 6(a), we apply this result and show the curves of the growth rate $\omega _i$ and the resonant frequency $\omega _r$, i.e. the imaginary and real parts of the eigenvalue of temporal unstable modes, of the first temporal unstable modes as functions of the wavenumber $\alpha$ for fixed Mach numbers $M=4$, $M=4.5$ and $M=5$. In order to relate the unstable modes, i.e. the resonant frequencies $\omega _r$ and the growth rate $\omega _i$, to the corresponding over-reflection coefficient, we convert $\omega _r$ to the incident angle $\phi$ according to (2.17) in figure 6(b). At the same time, we extract the projection to the $\alpha$–$\phi$ plane of the maximum curve of the resonant peak, the thick black line, from figure 5 and place it in figure 6(c). Also, the dashed lines in figure 6(b) are projections of the main curves onto the $\alpha$–$\phi$ plane. By comparing figure 6(b) with figure 6(c), we find that the corresponding dashed lines highly coincide and the trends of the peaks are largely synchronized. We therefore conclude that the resonant frequency of unstable modes triggers the peak of the over-reflection coefficient. This suggests that an unusual over-reflection enhancement occurs when the frequency of the incident wave is close to the resonant frequency.

This conclusion is first verified by the results in figure 4 and explains well the appearance of the local and global peaks. If converted to frequency, then the incident angles corresponding to peaks are approximately equal to the real part of the resonant frequencies in the first unstable modes. Here we have to further indicate that the local/global maximum value of the over-reflection coefficient, i.e. the red solid line, and the over-reflection coefficient corresponding to the resonant frequency, i.e. the red dashed line, do not exactly coincide; however, they differ by at most two decimal digits of accuracy. This minor difference reaches its maximum around $\alpha \approx 1.7$ and then continuously decreases with increasing $\alpha$. As $\alpha \geq 2$, the difference becomes almost unrecognizable and therefore is not shown in figure 7.

Next, we apply our conclusion to higher unstable modes. In the stability analysis presented in Zhang & Oberlack (Reference Zhang and Oberlack2021), the higher modes gradually appear as the wavenumber increases. However, the first mode always remains the most unstable mode, i.e. the largest $\omega _i$. This feature is also observed in resonant over-reflections induced by unstable modes. Figure 7 illustrates the variation of the over-reflection coefficient for wavenumbers in a range from $\alpha =2$ to $\alpha =10$ and Mach number from $M=2$ to $M=5$. We find that, at $\alpha =3$, the second mode appears and induces a new local peak. At $\alpha =7$, the third mode appears. Although the third unstable mode is rather weak, it still induces a local peak. At the same time, the first mode always maintains its maximum peak.

With increasing wavenumber $\alpha$, i.e. figure 7(d,e), the effect of resonant frequencies on over-reflections causes a steep increase of $(R-1)$ of tens of times compared to the surrounding $(R-1)$ of non-resonant over-reflections. In regions where non-resonant over-reflections occur, the over-reflection coefficient is very close to one. Once the resonance frequency intervenes, extremely steep peaks appear. Another character that can be obtained by observing figure 7 is that the maximum value of the non-resonant over-reflection continues to move away from the $\phi _{s_2}$ line relative to the over-reflection of small and moderate wavenumbers. In figure 7(b–e), the over-reflection coefficient even forms a monotonic rise with Mach numbers. Meanwhile, $R$ in non-resonant over-reflection regions is very insensitive to variations in Mach numbers.

More specific data about resonant frequencies of higher unstable modes are given in Appendix D.

3.3. Eigenfunction

In this section, we will present and discuss the eigenfunction (2.31), which is intuitively separated into the first and second terms. The first term represents the eigenfunction for reflected waves and the second term stands for the eigenfunction for unitary incident waves from the free stream.

Figure 8 illustrates three representative patterns. Of these, figure 8(a–c) corresponds to the case where there is no critical layer and hence no over-reflection occurs ($R=1$). It can be observed that the amplitudes of the incident wave in figure 8(a) and the reflected wave in figure 8(b) are equal. Meanwhile, we observe from these panels the effect of shear layers on the direction of acoustic wave propagation. This effect is caused by the significant velocity gradient close to the wall. In addition, a slight increase in the amplitude of the acoustic wave near the wall is detected.

Figure 8. Eigenfunctions of the incident (a,d,g) and reflected (b,e,h) acoustic waves and equation (2.31) (c,f,i) for Mach number $M=4$ and wavenumber $\alpha =2$: (a–c) for frequency $\omega =2.88$ ($\phi \approx 80^{\circ }$) and $R=1$; (d–f) for frequency $\omega =0.58$ ($\phi \approx 30^{\circ }$) and $R=1.08$; and (g–i) for frequency $\omega =1.35$ ($\phi \approx 68^{\circ }$) and $R=1.04$. The black dashed line stands for the location of the critical layer $y_c$. The angle of propagation in the free stream $\varTheta$ according to (2.22) is shown for incident waves.

Figure 8(d–f) and figure 8(g–i) correspond to the case of over-reflections. Figure 8(g–i) coincides with the occurrence of the resonant over-reflection, but the over-reflection coefficient only produces local peaks in $R$. Thus, the over-reflection coefficient for figure 8(g–i) is less than the non-resonant over-reflection coefficient for figure 8(d–f). Observing the second and third rows, different patterns of over-reflection are detected. In the second row, particularly visible in figure 8(f), the amplitude of the acoustic wave near the wall is less than the amplitude of the acoustic wave in the free stream. In contrast, in the third row, i.e. figure 8(g–i), the amplitude of the acoustic wave near the wall is much greater than the amplitude of the acoustic wave in the free stream.

In addition, from figures 8(f) and 8(i), we find that the amplitude of acoustic waves is relatively small near the critical layer. This suggests that the critical layer has an attenuating effect on acoustic waves, in agreement with the conclusion proposed in Campos & Serrão (Reference Campos and Serrão1998). It is important to point out that the transformation between the second and third rows occurs continuously and smoothly with variations of $\omega$, i.e. their difference is not due to the resonant over-reflection. In the vicinity of $\omega =1.35$, i.e. the region where non-resonant over-reflections occur, e.g. $\omega =1.32$, the pattern of the eigenfunction remains similar to the third row. Conversely, the pattern of the eigenfunction of the resonant over-reflection always resembles that in the third row, i.e. the amplitude near the wall is much greater than that in the free stream. This suggests, from another point of view, that resonant over-reflection (instability) is a special case of over-reflections.

In figure 8(g–i), we note a pattern of the eigenfunction similar to that of the unstable modes in Zhang & Oberlack (Reference Zhang and Oberlack2021). Through figure 8(g–i) we clearly observe the acoustic waves trapped between the first relative sonic line and the wall, similar to the surface wave (mode) (see e.g. Rienstra & Hirschberg Reference Rienstra and Hirschberg2020). (For a detailed definition of the relative sonic line, see e.g. Knisely (Reference Knisely2018). Here it means that the relative Mach number equals minus one.) This indicates that complex reflections and refractions occur near the wall. In contrast to the non-resonant over-reflection in figure 8(d–f), on the one hand, the distance between the first sonic line and the wall is not sufficient to generate strong complex reflections and refractions, and, on the other hand, the attenuating effect of the critical layer covers this distance. Thus, as read from the pattern in the second row of figure 8, no related instability may exist. As the critical layer moves away from the wall, the surface wave (mode) becomes apparent, where complex reflections and refractions occur near the wall, which is an infallible sign of a related unstable mode at a particular frequency.

The above inference is confirmed by figure 9. Similar to the setting in figure 8, figure 9 shows the eigenfunction of acoustic waves but at the Mach number $M=5$, where figure 9(a–c) for $\alpha =2.2$ corresponds to the maximum value of the peak in figure 5(c). The enhancement of the reflected waves can be clearly observed by the colour gradient. Meanwhile, the amplitudes near the wall are much greater than in the remaining domain, where complex reflection and refraction occurs. This region is necessary to induce instability as was shown in Zhang & Oberlack (Reference Zhang and Oberlack2021). From figure 9(d–f) for $\alpha =0.1$, we observe similar acoustic wave propagation to the second pattern in figure 8. Small wavenumbers $\alpha$ do not allow for complex reflections and refractions that increase the amplitude near the wall. Therefore, despite the large over-reflection coefficient $R=2.42$, no resonant over-reflections could occur with the near-wall pattern in figure 9(d–f), i.e. no unstable modes exist related to these parameters.

Figure 9. Eigenfunctions of the incident (a,d) and reflected (b,e) acoustic waves and equation (2.31) (c,f) for Mach number $M=5$: (a–c) for wavenumber $\alpha =2.33$, frequency $\omega =1.34$ ($\phi \approx 70^{\circ }$) and $R=1.46$ corresponding to the global maximum value in figure 5(c); and (d–f) for wavenumber $\alpha =0.1$, frequency $\omega =0.078$ ($\phi \approx 75^{\circ }$) and $R=2.42$. The black dashed line stands for the location of the critical layer $y_c$. The angle of propagation in the free stream $\varTheta$ according to (2.22) is shown for incident waves.