3.1. Dispersion relations and eigenvectors

The perturbation field $\tilde {\boldsymbol{\varphi }} = (\tilde {\rho }, \tilde {u}_i, \tilde {p})^{{T}}$ takes a harmonic travelling-wave form:

(3.1) \begin{align} \tilde {\boldsymbol{\varphi }} = \hat {\boldsymbol{\varphi }} \exp \left [\mathrm{i}(k_{\kern-1pt j} x_{\kern-1pt j} - \omega t)\right ], \end{align}

where $\hat {\boldsymbol{\varphi }}$ is the complex amplitude, $k_{\kern-1pt j}$ are the wavenumber components and $\omega$ is the angular frequency. Substituting this ansatz into (2.7) yields the equations in the spectral domain:

(3.2a) \begin{align} (u_{\kern-1pt j} k_{\kern-1pt j} - \omega ) \hat {\rho } + \rho k_{\kern-1pt j} \hat {u}_j &= 0, \\[-12pt] \nonumber \end{align}

(3.2b) \begin{align} \rho (u_{\kern-1pt j} k_{\kern-1pt j} - \omega ) \hat {u}_i + k_i \hat {p} &= 0, \\[-12pt] \nonumber \end{align}

(3.2c) \begin{align} (u_{\kern-1pt j} k_{\kern-1pt j} - \omega ) \hat {p} + \gamma p k_{\kern-1pt j} \hat {u}_j &= 0. \\[10pt] \nonumber \end{align}

For non-trivial solutions, this homogeneous system requires the determinant of the coefficient matrix to vanish, yielding the characteristic polynomial:

(3.3) \begin{align} \rho ^3 (k_{\kern-1pt j} u_{\kern-1pt j} - \omega )^3 \left [(k_{\kern-1pt j} u_{\kern-1pt j} - \omega )^2 - c^2 k_{\kern-1pt j} k_{\kern-1pt j}\right ] = 0, \end{align}

where $c = \sqrt {\gamma p / \rho }$ is the sound speed. The eigenspace of each irreducible factor corresponds to a distinct type of wave: acoustic waves and entropy/vorticity waves.

Acoustic waves, governed by the quadratic factor in (3.3), satisfy the dispersion relation:

(3.4) \begin{align} (k_{\kern-1pt j} u_{\kern-1pt j} - \omega )^2 - c^2 k_{\kern-1pt j} k_{\kern-1pt j} = 0, \end{align}

with the corresponding eigenvector:

(3.5) \begin{align} \boldsymbol{e}_a = \left [ \rho ,\ \frac {k_1 c^2}{\omega - u_{\kern-1pt j} k_{\kern-1pt j}},\ \frac {k_2 c^2}{\omega - u_{\kern-1pt j} k_{\kern-1pt j}},\ \frac {k_3 c^2}{\omega - u_{\kern-1pt j} k_{\kern-1pt j}},\ \rho c^2 \right ]^{{T}}. \end{align}

The entropy and vorticity waves, associated with the cubic factor $(k_{\kern-1pt j} u_{\kern-1pt j} - \omega )^3$ in (3.3), satisfy the dispersion relation:

(3.6) \begin{align} k_{\kern-1pt j} u_{\kern-1pt j} - \omega = 0, \end{align}

corresponding to three independent eigenvectors. From a physical perspective, these can be further classified into irrotational entropy waves and rotational vorticity waves. Specifically, the entropy wave is characterised by a density perturbation with no velocity or pressure fluctuations:

(3.7) \begin{align} \boldsymbol{e}_e = [\rho , 0, 0, 0, 0]^{{T}}. \end{align}

For vorticity waves, the continuity equation (3.2a ) enforces the divergence-free constraint $k_{\kern-1pt j} \hat {u}_j = 0$ . Consequently, the two-dimensional vorticity subspace can be spanned by the following two linearly independent eigenvectors.

(i) When the wave vector components satisfy $(k_2, k_3) \neq (0, 0)$ , the eigenvectors are given by

(3.8) \begin{align} \begin{aligned} \boldsymbol{e}_{v_1} &= c \, \boldsymbol{q}^{{u}}_1, \quad \boldsymbol{q}_1 = \bigl [0,\; -k_2^2 - k_3^2,\; k_1 k_2,\; k_1 k_3,\; 0 \bigr ]^{{T}}, \\ \boldsymbol{e}_{v_2} &= c \, \boldsymbol{q}^{{u}}_2, \quad \boldsymbol{q}_2 = \bigl [0,\; 0,\; k_3,\; -k_2,\; 0 \bigr ]^{{T}}, \end{aligned} \end{align}

where $\boldsymbol{q}^{{u}} = \boldsymbol{q} / |\boldsymbol{q}|$ denotes the unit vector of $\boldsymbol{q}$ .

(ii) When $(k_2, k_3) = (0, 0)$ , the eigenvectors become of the form

(3.9) \begin{align} \begin{aligned} \boldsymbol{e}_{v_1} &= \bigl [0,\; 0,\; c,\; 0,\; 0 \bigr ]^{{T}}, \\ \boldsymbol{e}_{v_2} &= \bigl [0,\; 0,\; 0,\; c,\; 0 \bigr ]^{{T}}. \end{aligned} \end{align}

The eigenvectors $\boldsymbol{e}_{v_1}$ , $\boldsymbol{e}_{v_2}$ , defined earlier each represent a pure vorticity mode, characterised by vanishing density and pressure perturbations.

It is important to note that in our analysis, the dispersion relations are kept in their implicit form ((3.4) and (3.6)), rather than solving explicitly for the temporal modes, as was done by Mckenzie & Westphal (Reference Mckenzie and Westphal1968) and Huang & Wang (Reference Huang and Wang2019) (e.g. $\omega = u_{\kern-1pt j} k_{\kern-1pt j} \pm c |\boldsymbol{k}|$ ). This implicit formulation facilitates a unified treatment of both temporal and spatial modes in § 3.2. As will be shown shortly, the spatial analysis can capture evanescent wave behaviour that temporal analysis cannot.

3.2. Temporal and spatial analysis

Building on the implicit dispersion relations derived in § 3.1, we analyse the temporal and spatial modes associated with initial-value and boundary-value problems, respectively.

3.2.1. Temporal analysis

Temporal analysis takes real wavenumbers $k_{\kern-1pt j}$ and determines the frequency $\omega$ via the dispersion relations ((3.4), (3.6)), given by

(3.10) \begin{align} \begin{cases} \omega _{a_1},\omega _{a_2} = u_{\kern-1pt j} k_{\kern-1pt j} \pm c |\boldsymbol{k}| & \text{for acoustic waves}, \\ \omega _{ev} = u_{\kern-1pt j} k_{\kern-1pt j} & \text{for entropy and vorticity waves}, \end{cases} \end{align}

where $|\boldsymbol{k}| = \sqrt {k_{\kern-1pt j} k_{\kern-1pt j}}$ is the wavenumber magnitude, and the plus and minus signs denote the fast and slow acoustic modes, respectively.

For an initial perturbation $\tilde {\boldsymbol{\varphi }}(x,y,z,0)$ imposed at $t=0$ , its spectral representation in the wavenumber domain is obtained via the three-dimensional spatial Fourier transform:

(3.11) \begin{align} \hat {\boldsymbol{\varphi }}(k_1,k_2,k_3) = \mathcal{F}_{x,y,z}[\tilde {\boldsymbol{\varphi }}(x,y,z,0)], \end{align}

where $\hat {\boldsymbol{\varphi }}(k_1,k_2,k_3)$ is decomposed as a superposition of five linearly independent eigenvectors: two acoustic modes $\boldsymbol{e}_{a_1}$ , $\boldsymbol{e}_{a_2}$ (3.5), the entropy mode $\boldsymbol{e}_e$ (3.7) and the two vorticity modes $\boldsymbol{e}_{v_1}$ , $\boldsymbol{e}_{v_2}$ ((3.8), (3.9)). That is,

(3.12) \begin{align} \hat {\boldsymbol{\varphi }} = a_{a_1} \boldsymbol{e}_{a_1} + a_{a_2} \boldsymbol{e}_{a_2} + a_e \boldsymbol{e}_e + a_{v_1} \boldsymbol{e}_{v_1} + a_{v_2} \boldsymbol{e}_{v_2}, \end{align}

where $a_{a_1},a_{a_2},a_e,a_{v_1},a_{v_2}$ are the respective amplitudes of these modes.

The time-dependent solution is given by

(3.13) \begin{align} \begin{split} \tilde {\boldsymbol{\varphi }}(x,y,z,t) &= \iiint _{\mathbb{R}^3} \left ( a_{a_1} \boldsymbol{e}_{a_1} \mathrm{e}^{ -{i}\omega _{a_1} t} + a_{a_2} \boldsymbol{e}_{a_2} \mathrm{e}^{ -{i}\omega _{a_2} t} \right . \\ &\quad \left . +(a_{e} \boldsymbol{e}_{e} + a_{v_1} \boldsymbol{e}_{v_1} + a_{v_2} \boldsymbol{e}_{v_2}) \mathrm{e}^{ -{i}\omega _{ev} t} \right )\mathrm{e}^{ {i}k_{\kern-1pt j} x_{\kern-1pt j}}\, \mathrm{d} k_1 \,\mathrm{d} k_2 \,\mathrm{d} k_3. \end{split} \end{align}

3.2.2. Spatial analysis

In contrast to temporal analysis, spatial analysis addresses a boundary-value problem: given the real frequency $\omega$ and real tangential wavenumbers $k_2, k_3$ , the normal wavenumber $k_1$ is obtained by solving the dispersion relations (3.4) (for acoustic waves) and (3.6) (for entropy/vorticity waves). This yields

(3.14) \begin{align} \begin{cases} k_1^{a_1},k_1^{a_2} = \dfrac {\varOmega u_1 \pm c\sqrt {D}}{u_1^2 - c^2} & \text{for acoustic waves}, \\ k_1^{ev} = \dfrac {\varOmega }{u_1} & \text{for entropy-vorticity waves}, \end{cases} \end{align}

where

(3.15) \begin{align} \varOmega = \omega - u_2 k_2 - u_3 k_3, \quad D = \varOmega ^2 + \big(u_1^2 - c^2 \big) \big(k_2^2 + k_3^3 \big). \end{align}

Since the acoustic dispersion relation (3.4) is quadratic in $k_1$ , the discriminant $D$ determines the nature of the acoustic modes: $D \gt 0$ yields two real roots corresponding to propagating waves, while $D \lt 0$ gives complex conjugate roots corresponding to evanescent waves. In supersonic upstream flow ( $u_1 \gt c$ ), $D$ is always positive, ensuring two propagating acoustic modes. In contrast, in subsonic downstream flow ( $u_1 \lt c$ ), $D$ can be either positive or negative, depending on the specific values of $\omega , k_2$ and $k_3$ . The parameter $\varOmega$ , as will be shown later in § 5.1, denotes a Doppler-shifted frequency when an oblique shock is transformed to an equivalent normal shock.

For boundary-value problems involving a disturbance at $x=0$ , the spectral representation is obtained via the Fourier transform in $y, z, t$ :

(3.16) \begin{align} \hat {\boldsymbol{\varphi }}(k_2,k_3,\omega ) = \mathcal{F}_{y,z,t}[\tilde {\boldsymbol{\varphi }}(0,y,z,t)]. \end{align}

Using the same five-mode decomposition as in (3.12):

(3.17) \begin{align} \hat {\boldsymbol{\varphi }} = a_{a_1} \boldsymbol{e}_{a_1} + a_{a_2} \boldsymbol{e}_{a_2} + a_e \boldsymbol{e}_e + a_{v_1} \boldsymbol{e}_{v_1} + a_{v_2} \boldsymbol{e}_{v_2}, \end{align}

where $k_1$ in the eigenvectors ( $\boldsymbol{e}_{a_1}$ , $\boldsymbol{e}_{a_2}$ , $\boldsymbol{e}_{e}$ , $\boldsymbol{e}_{v_1}$ , $\boldsymbol{e}_{v_2}$ ) is determined by (3.14). Disturbance evolution in the x-direction is then given by

(3.18) \begin{align} \begin{split} \tilde {\boldsymbol{\varphi }}(x,y,z,t) &= \iiint _{\mathbb{R}^3} \left ( a_{a_1} \boldsymbol{e}_{a_1} \mathrm{e}^{ {i}k_1^{a_1} x} + a_{a_2} \boldsymbol{e}_{a_2} \mathrm{e}^{ {i}k_1^{a_2} x} \right . \\ &\quad \left . +(a_{e} \boldsymbol{e}_{e} + a_{v_1} \boldsymbol{e}_{v_1} + a_{v_2} \boldsymbol{e}_{v_2}) \mathrm{e}^{ {i}k_1^{ev} x} \right )\mathrm{e}^{ {i}(k_2 y + k_3 z- \omega t)} \,\mathrm{d} k_2\, \mathrm{d} k_3\, \mathrm{d} \omega . \end{split} \end{align}

3.3. Group velocity and propagation direction

The sign of the group velocity relative to the shock commonly serves to distinguish incident waves (impinging on the shock) from generated waves (resulting from the shock–disturbance interaction). Waves with group velocities directed towards the shock are classified as incident waves and must be prescribed as boundary conditions; those with group velocities directed away from the shock are regarded as generated waves and emerge as part of the solution. However, this approach becomes ambiguous when dealing with evanescent modes, which possess complex group velocities. In this section, we examine this difficulty and propose an alternative criterion to systematically handle evanescent modes.

For entropy and vorticity waves, the group velocity,

(3.19) \begin{align} g^{ev}_i = \partial \omega /\partial k_i = u_i ,\end{align}

equals the base flow velocity, indicating that these waves are convected by the mean flow. For acoustic waves, the group velocity is derived from the dispersion relation (3.4):

(3.20) \begin{align} g^a_i = \frac {\partial \omega }{\partial k_i} = u_i + \frac {c^2 k_i}{\omega - u_{\kern-1pt j} k_{\kern-1pt j}}. \end{align}

For temporal modes, the acoustic group velocity follows the form $g^{a\pm }_i = u_i \pm c k_i/|\boldsymbol{k}|$ , with + and – denoting fast and slow acoustic waves, respectively. For spatial modes, by applying (3.20), the shock-normal component of the group velocity for the two acoustic spatial modes is

(3.21) \begin{align} g^{a_1}_1, g^{a_2}_1 = \frac {u_1^2 - c^2}{u_1 \pm \frac {\varOmega c}{\sqrt {D}}}, \end{align}

where $\varOmega$ and $D$ are defined in (3.15).

For propagating modes ( $D \gt 0$ ), the upstream flow ( $u_1 \gt c$ ) yields $g^{a_1}_1 \gt 0$ and $g^{a_2}_1 \gt 0$ from (3.15), corresponding to two incident waves. Conversely, in the downstream flow ( $u_1 \lt c$ ), (3.15) gives $g^{a_1}_1 \gt 0$ and $g^{a_2}_1 \lt 0$ , which correspond to one outgoing and one incident wave, respectively.

For evanescent modes ( $D \lt 0$ ), the interpretation becomes problematic because the group velocity is complex (Gerasik & Stastna Reference Gerasik and Stastna2010). Previous work by Huang & Wang (Reference Huang and Wang2019) associated evanescent modes with a positive real group velocity, interpreted as downstream orientation. Our analysis, however, reveals that the real part of the evanescent mode group velocity is actually negative (i.e. it points upstream). We demonstrate this by considering an evanescent mode with $k_1 = \textrm {Re}(k_1) + {i} {\mathrm{Im}}(k_1)$ , combining (3.14) and (3.21), and evaluating its real part, which yields

(3.22) \begin{align} \textrm {Re} \big(g^{a_1,a_2}_1 \big) = \frac {u_1 D}{\varOmega ^2 + \big(k_2^2 + k_3^2 \big) u_1^2}. \end{align}

Since $D \lt 0$ and the denominator is always positive, both evanescent waves satisfy $\textrm {Re}(g^{a_1,a_2}_1) \lt 0$ , confirming the upstream orientation. Thus, for evanescent modes, the sign of the shock-normal group velocity is not applicable to evanescent waves.

This motivates an alternative classification based on the spatial decay direction. To illustrate this point, we consider a simple example based on the two-dimensional acoustic wave equation:

(3.23) \begin{align} \frac {\partial ^2 p}{\partial t^2} = c^2 \left ( \frac {\partial ^2 p}{\partial x^2} + \frac {\partial ^2 p}{\partial y^2} \right ), \end{align}

where $c$ is the sound speed. Imposing the boundary condition $p(0,y,t) = \cos (3y - ct)$ at $x=0$ yields the general solution:

(3.24) \begin{align} p(x,y,t) = \begin{cases} \mathrm{e}^{3x} \cos (3y - ct) & \text{for } x \lt 0, \\[5pt]\mathrm{e}^{-3x} \cos (3y - ct) & \text{for } x \gt 0. \end{cases} \end{align}

This result shows that the evanescent wave admits two physically admissible solutions: one decaying exponentially in the $-x$ direction and the other decaying in the $+x$ direction, as illustrated in figure 2.

Figure 2.

Illustration of evanescent wave behaviour at $x=0$ . (a) Numerical solution demonstrating one decaying exponentially in the $-x$ direction and the other decaying in the $+x$ direction. (b) Comparison between theoretical solution and numerical results.

Thus, in the disturbance-shock problem, we classify evanescent modes according to their spatial decay direction relative to the shock interface. Since evanescent modes are confined to the post-shock region, the mode decaying in the $+x$ direction (away from the shock) is identified as the generated wave ( $a_1$ ), whereas the mode decaying in the $-x$ direction (towards the shock) represents the incident wave ( $a_2$ ). The directional properties of both propagating acoustic modes and evanescent modes are summarised in table 1.

Table 1.

Wave characteristics for acoustic and evanescent modes in uniform flows, where the discriminant $D$ is defined in (3.15). $^*$ Classification is based on spatial decay direction.

Table 1 Long description

A table comparing wave characteristics in uniform flows. The table has three columns: Region, Wave mode, and k1 solution. It has six rows, including the header row. The regions are Upstream (L) and Downstream (R). For Upstream (L), the wave modes are Acoustic, Entropy-vorticity, and Acoustic (if DR > 0). For Downstream (R), the wave modes are Evanescent (if DR < 0) and Entropy-vorticity. The k1 solutions are provided for each wave mode. Row 1: Upstream (L), Acoustic, k_{1L}^{a1,a2} = (Ω_L u_{1L} ± c_L sqrt(D_L)) / (u_{1L}^2 - c_L^2). Row 2: Upstream (L), Entropy-vorticity, k_{1L}^{ev} = Ω_L / u_{1L}. Row 3: Upstream (L), Acoustic (if D_R > 0), k_{1R}^{a1,a2} = (Ω_R u_{1R} ± c_R sqrt(D_R)) / (u_{1R}^2 - c_R^2). Row 4: Downstream (R), Evanescent (if D_R < 0), k_{1R}^{a1,a2} = (Ω_R u_{1R} ± i c_R sqrt(|D_R|)) / (u_{1R}^2 - c_R^2). Row 5: Downstream (R), Entropy-vorticity, k_{1R}^{ev} = (Ω_R) / (u_{1R}).

3.4. Energy characteristics of evanescent waves

To better understand the physical nature of evanescent waves, we analyse the disturbance energy using the disturbance energy equation. This equation can be derived from the linearised Euler equation (2.7) and takes the form

(3.25) \begin{align} \frac {\partial \tilde E}{\partial t} + \frac {\partial }{\partial x_{\kern-1pt j}} \big ( \tilde E u_{\kern-1pt j} + \tilde {p} \tilde {u}_j \big ) = 0, \end{align}

where the disturbance energy density $\tilde E$ is defined as

(3.26) \begin{align} \tilde E = \frac {\rho }{2} \tilde {u}_i \tilde {u}_i + \frac {\tilde {p}^2}{2\rho c^2}. \end{align}

For temporally periodic disturbances, we define the time-averaged energy flux as

(3.27) \begin{align} J_j = \frac {1}{T} \int _{0}^{T} \big ( \tilde E u_{\kern-1pt j} + \tilde {p} \tilde {u}_j \big )\, \mathrm{d}t, \end{align}

where $T = 2\pi /\omega$ is the period. Averaging (3.25) over one period eliminates the time-derivative term and directly yields $\partial J_j/\partial x_{\kern-1pt j} = 0$ , indicating that $J_j$ is conserved. Substituting the disturbance quantities for an evanescent wave yields

(3.28) \begin{align} J_1 &= 0, \\[-12pt] \nonumber \end{align}

(3.29) \begin{align} J_2 &= \frac {1}{2} \rho c^2 \frac {c^2 \left ( \varOmega k_2 + \big(k_2^2 + k_3^2 \big) u_2 \right )}{\varOmega ^2 + \big(k_2^2 + k_3^2 \big) u_1^2}, \\[-12pt] \nonumber \end{align}

(3.30) \begin{align} J_3 &= \frac {1}{2} \rho c^2 \frac {c^2 \left ( \varOmega k_3 + \big(k_2^2 + k_3^2 \big) u_3 \right )}{\varOmega ^2 + \big(k_2^2 + k_3^2\big) u_1^2}. \\[10pt] \nonumber \end{align}

Thus, the shock-normal component $J_1$ vanishes, whereas the tangential components $J_2$ and $J_3$ are non-zero. This implies that the time-averaged energy flux of evanescent waves is directed parallel to the shock, not normal to it. In this sense, evanescent waves differ fundamentally from damped waves, whose energy dissipates along the direction of propagation.

Physically, this parallel energy transport mirrors that of optical evanescent waves. In the limiting case of zero fluid velocity, our problem is equivalent to that of light travelling across an interface, governed by the wave equation. When total internal reflection occurs, an optical evanescent wave is generated in the rarer medium, whose time-averaged energy flux is directed parallel to the interface.

4.1. Wavenumber relationship

Small disturbances across the shock interface must satisfy continuity conditions that reflect the conservation of oscillatory properties. Specifically, the frequency $\omega$ and the tangential wavenumbers $(k_2,k_3)$ must remain continuous, since they represent intrinsic oscillation characteristics that cannot be changed abruptly at the interface. Consequently, the matching conditions at the shock may be stated as

(4.1) \begin{align} \omega _L = \omega _R, \quad k_{2L} = k_{2R}, \quad k_{3L} = k_{3R}, \end{align}

where subscripts $L$ and $R$ denote upstream and downstream quantities, respectively.

Following the spatial framework established in § 3.2.2, we prescribe the real-valued parameter triplet $(\omega , k_2, k_3)$ and obtain $k_1$ from the dispersion relation (3.14). This approach simultaneously resolves all wave modes in the upstream and downstream flows. Table 2 summarises the spatial mode solutions for disturbances across the shock interface. In the upstream region, all waves propagate with real $k_1$ . In the downstream region, entropy-vorticity waves remain propagating, while acoustic waves are propagating for $D_R\gt 0$ , but become evanescent (with complex-conjugate $k_1$ and exponential decay) for $D_R\lt 0$ . This spatial analysis framework naturally accommodates both propagating and evanescent disturbances within a unified mathematical treatment, offering a consistent basis for analysing shock–disturbance interactions.

Table 2.

Shock-normal wavenumber $k_1$ in terms of the prescribed parameters $(\omega , k_2, k_3)$ , with $\varOmega = \omega - u_2 k_2 - u_3 k_3$ and $D = \varOmega ^2 + (u_1^2 - c^2)(k_2^2 + k_3^2)$ (see (3.15)).

Table 2. Long description

The table presents spatial mode solutions for disturbances across a shock interface. It has six rows and five columns. The column headers are a sub R 1, a sub R e, a sub R v 1, a sub R v 2, and \eta hat. The row labels are L a 1, L a 2, L e, L v 1, L v 2, and R a 2. The values in the table are as follows: Row 1: L a 1, 0.7258, 0.06854, -0.1042, 0, 0.2698i. Row 2: L a 2, 0.03719, 0.2407, -0.3660, 0, -0.008232i. Row 3: L e, 0.1783, 0.9554, -0.5517, 0, 0.5369i. Row 4: L v 1, -0.1261, 0.03151, 0.9714, 0, -0.3797i. Row 5: L v 2, 0, 0, 0, 0.7698, 0. Row 6: R a 2, 0.005234, -0.2513, 0.3821, 0, -0.9596i.

To relate the orientations of wave vectors across the shock, we define the wave vector angle $\theta$ as the angle between the wave vector $\boldsymbol{k}$ and the shock-normal direction ( $x_1$ -axis), i.e.

(4.2) \begin{align} \tan \theta = \sqrt {k_2^2 + k_3^2}\,/\,k_1. \end{align}

Since $k_2$ and $k_3$ are equal on both sides of the shock, the wave vector angles are directly related through $k_1$ . For propagating waves where $k_1$ is real-valued, this angular relationship takes the form:

(4.3) \begin{align} \frac {\tan \theta _L}{\tan \theta _R} = \frac {k_{1R}}{k_{1L}}. \end{align}

For evanescent waves with complex-valued $k_{1R}$ , we take the real part $\textrm {Re}(k_{1R})$ in the definition of the wave-vector angle, i.e. $\tan \theta = \sqrt {k_2^2 + k_3^2}\,/\,\textrm {Re}(k_{1R})$ .

4.2. Amplitude relationship

To determine the amplitude relationship of disturbances across the shock, the disturbance-induced oscillation of the shock surface itself must be included in the analysis. Mathematically, this moving interface can be described by an implicit function $ f(x, y, z, t) = 0$ . Considering that the unperturbed shock remains stationary at $x = 0$ , the position of the perturbed shock is given by

(4.4) \begin{align} f(x, y, z, t) = x - \eta (y, z, t) = 0, \end{align}

where $\eta (y, z, t)$ represents the small displacement of the shock surface from its unperturbed position.

Since the standard Rankine–Hugoniot relations assume a stationary shock, they cannot be applied directly when the shock is in motion. Previous studies, such as those of Mckenzie & Westphal (Reference Mckenzie and Westphal1968) and Huang & Wang (Reference Huang and Wang2019), employed coordinate transformations to render the shock stationary in a moving reference frame. In contrast, this study adopts a more direct approach by using generalised Rankine–Hugoniot relations derived from the weak solution conditions of the conservation equations in the stationary frame. Based on the governing conservation laws (2.1), these generalised relations are expressed as

(4.5) \begin{align} \frac {\partial f}{\partial t} [\![ \boldsymbol{U} ]\!] + \frac {\partial f}{\partial x_{\kern-1pt j}} [\![ \boldsymbol{F}_{\kern-1pt j} ]\!] = 0, \end{align}

where $[\![ \boldsymbol{\cdot }]\!] = (\boldsymbol{\cdot })_L - (\boldsymbol{\cdot })_R$ denotes the jump from the upstream to the downstream side of the shock. The conservative variables $\boldsymbol{U}$ and fluxes $\boldsymbol{F}_{\kern-1pt j}$ are defined previously in (2.2) and (2.3).

Using the shock-interface description (4.4) in the weak-solution condition (4.5) yields

(4.6) \begin{align} [\![ \boldsymbol{F}_1 ]\!] - \frac {\partial \eta }{\partial t} [\![ \boldsymbol{U} ]\!] - \frac {\partial \eta }{\partial y} [\![ \boldsymbol{F}_2 ]\!] - \frac {\partial \eta }{\partial z} [\![ \boldsymbol{F}_3 ]\!] = 0, \end{align}

where $\boldsymbol{F}_1$ , $\boldsymbol{F}_2$ and $\boldsymbol{F}_3$ represent the flux vectors in the $x$ -, $y$ - and $z$ -directions, respectively. To analyse small-amplitude perturbations, the flux $\boldsymbol{F}_1$ is linearised up to first order with respect to small perturbations $\boldsymbol{\tilde {\varphi }} = (\tilde {\rho }, \tilde {u}, \tilde {v}, \tilde {w}, \tilde {p})^{{T}}$ :

(4.7) \begin{align} \boldsymbol{F}_1 = {\unicode{x1D643}} \boldsymbol{\tilde {\varphi }}. \end{align}

Here, ${\unicode{x1D643}} = {\partial \boldsymbol{F}_1}/{\partial \boldsymbol{\varphi }}$ is the Jacobian matrix, explicitly written as

(4.8) \begin{align} {\unicode{x1D643}} = \begin{bmatrix} u & \rho & 0 & 0 & 0 \\ u^2 & 2\rho u & 0 & 0 & 1 \\ uv & \rho v & \rho u & 0 & 0 \\ uw & \rho w & 0 & \rho u & 0 \\ u \left ( E - \frac {p}{(\gamma - 1)\rho } \right ) & \rho E + p + \rho u^2 & \rho uv & \rho uw & \frac {\gamma u}{\gamma - 1} \\ \end{bmatrix}. \end{align}

To derive explicit amplitude relationships, the perturbations at $x=0$ are expressed in Fourier form as $\boldsymbol{\tilde {\varphi }} = \hat {\boldsymbol{\varphi }} \mathrm{e}^{\mathrm{i} (k_2 y + k_3 z - \omega t)}$ , while the shock deformation is expressed as

(4.9) \begin{align} \eta (y, z, t) = \hat {\eta } \mathrm{e}^{\mathrm{i} (k_2 y + k_3 z - \omega t)}, \end{align}

where $\hat {\eta }$ is the deformation amplitude. Note that the shock deformation shares the same frequency and transverse wavenumbers as the incident waves. Substituting these Fourier representations into the generalised Rankine–Hugoniot relation (4.6), along with the linearised flux (4.7), yields the matching condition:

(4.10) \begin{align} [\![ {\unicode{x1D643}} \hat {\boldsymbol{\varphi }} ]\!] + \hat {\eta }[\![ \boldsymbol{\varGamma } ]\!]=0, \end{align}

where $\boldsymbol{\varGamma }= \mathrm{i} \omega \boldsymbol{U} - \mathrm{i} k_2 \boldsymbol{F}_2 - \mathrm{i} k_3 \boldsymbol{F}_3$ .

To determine the amplitudes of individual wave modes, the perturbations on each side of the shock are decomposed into a superposition of five linearly independent modes as given in (3.17). Substituting this decomposition into the matching condition (4.10) leads to a linear algebraic system:

(4.11) \begin{align} {\unicode{x1D648}} \boldsymbol{X} = {\unicode{x1D649}} \boldsymbol{Y}, \end{align}

where the system components are defined as

(4.12) \begin{align} \begin{aligned} \boldsymbol{X} &= [ a_{Ra_1}, a_{Re}, a_{Rv_1}, a_{Rv_2}, \hat {\eta } ]^{{T}}, \\ \boldsymbol{Y} &= [ a_{La_1}, a_{La_2}, a_{Le}, a_{Lv_1}, a_{Lv_2},a_{Ra_2} ]^{{T}}, \\ {\unicode{x1D648}} &= [ {\unicode{x1D643}}_R \boldsymbol{e}_{Ra_1}, {\unicode{x1D643}}_R \boldsymbol{e}_{Re}, {\unicode{x1D643}}_R \boldsymbol{e}_{Rv_1}, {\unicode{x1D643}}_R \boldsymbol{e}_{Rv_2}, -[\![ \boldsymbol{\varGamma } ]\!] ],\\ {\unicode{x1D649}} &= [{\unicode{x1D643}}_L \boldsymbol{e}_{La_1},{\unicode{x1D643}}_L \boldsymbol{e}_{La_2},{\unicode{x1D643}}_L \boldsymbol{e}_{Le},{\unicode{x1D643}}_L \boldsymbol{e}_{Lv_1},{\unicode{x1D643}}_L \boldsymbol{e}_{Lv_2}, -{\unicode{x1D643}}_R \boldsymbol{e}_{Ra_2}]. \end{aligned} \end{align}

For a given wave triplet ( $\omega , k_2, k_3$ ), once the incident wave amplitudes are specified, this linear system uniquely determines the corresponding generated wave amplitudes $(a_{Ra_1}, a_{Re}, a_{Rv_1}, a_{Rv_2})$ and shock deformation $\hat {\eta }$ . The complete solution in physical space can then be obtained by inverse Fourier transform using (3.18) given in § 3.2.2.

It is instructive to examine (4.11) from a linear-algebraic viewpoint. The system describes a linear mapping from the six-dimensional input vector $\boldsymbol{Y}$ to the five-dimensional output vector $\boldsymbol{X}$ through $\boldsymbol{X}={\unicode{x1D648}}^{-1}{\unicode{x1D649}}\boldsymbol{Y}$ . The dimensional mismatch implies that the mapping is surjective, but not injective; consequently, different input vectors $\boldsymbol{Y}$ can produce the same output $\boldsymbol{X}$ . Physically, the non-trivial solution space of the homogeneous equation ${\unicode{x1D649}}\boldsymbol{Y}=0$ indicates that the downstream incident acoustic mode $a_{Ra_2}$ is not independent: it can be expressed as a linear combination of the upstream incident modes. Any vector $\boldsymbol{Y}$ in this solution space yields the same (trivial) output $\boldsymbol{X}=0$ in the matching condition.

Therefore, we can eliminate $a_{Ra_2}$ from the formulation and restrict attention to the upstream incident amplitude vector $\boldsymbol{A}_L = [a_{La_1}, a_{La_2}, a_{Le}, a_{Lv_1}, a_{Lv_2}]^{{T}}$ . The modified mapping relation becomes

(4.13) \begin{align} \boldsymbol{X} = {\unicode{x1D648}}^{-1}\tilde {{\unicode{x1D649}}}\boldsymbol{A}_L, \end{align}

where $\tilde {{\unicode{x1D649}}}$ denotes the leading $5 \times 5$ submatrix of $\unicode{x1D649}$ obtained by removing its last column. Since this reduced system establishes a bijection between $\boldsymbol{A}_L$ and $\boldsymbol{X}$ , the inverse mapping is uniquely defined. This property enables the solution of the inverse problem: given a desired disturbance response $\boldsymbol{X}$ (the amplitudes of generated waves and the shock deformation), one can uniquely determine the required upstream incident wave vector via

(4.14) \begin{align} \boldsymbol{A}_L = \tilde {{\unicode{x1D649}}}^{-1}{\unicode{x1D648}}\boldsymbol{X}. \end{align}

This inverse formulation provides a direct method to tailor a specific downstream disturbance field by designing the appropriate combination of upstream incident waves. Note that eliminating $a_{Ra_2}$ is merely a convenient choice to fix one degree of freedom; any other way of fixing a degree of freedom, such as prescribing another incident wave or imposing a relation among modes, would also yield a bijective mapping and uniquely determine all incident waves, whether upstream or downstream.