2.1. The first condition of TMs: eigenmode condition

Lifshitz & Pitaevskii (Reference Lifshitz and Pitaevskii1981) derived a general travelling-wave formulation of the eigenmode condition of continuous systems. For the present problem sketched in figure 1, i.e. eigenmodes in a 2-D acoustic waveguide containing cavities or plates in the resonant region from $x=0$ to $x=L$, the travelling-wave interpretation is that an eigenmode may be regarded as resulting from the superposition of travelling waves in the $\pm x$ directions reflected by the region boundaries at $x=0$ and $x=L$. Since the eigenfunction is one valued, such a travelling-wave decomposition leads to the characteristic condition of the eigenmode, that is, the travelling waves remain unchanged after a loop or round trip of propagation and reflection in the resonant region. For cases with a sufficiently long resonant region, i.e. $L \gg 1$, the influences of $k^{\mp }_e$ cut-off DMs (evanescent waves) generated by the scattering of $k^{\pm }_p$ cut-on DMs (propagating waves) at one end on the wave scattering at the opposite end and on the loop characteristics can be neglected. The one-valued or the loop closure principle for each eigenmode is then formulated in the asymptotic form (Lifshitz & Pitaevskii Reference Lifshitz and Pitaevskii1981)

(2.1)\begin{equation} R_0(\omega)R_L(\omega) \mathrm{exp} \big\lbrace -\mathrm{i} \big[ k^+_p (\omega)- k^-_p (\omega) \big] L\big\rbrace=1, \end{equation}

where $R_0$ and $R_L$ are the reflection coefficients at region boundaries, $k^{\pm }_p$ are wavenumbers and the frequency $\omega$ can be either real or complex. Note that, for the simplicity of (2.1), it is assumed that only a single propagating wave in the $\pm x$ directions is involved in the eigenmode. The loop closure principle includes both magnitude and phase constraints

(2.2)$$\begin{gather} |R_0R_L| \mathrm{exp} \big\lbrace \big[ \mathrm{Im} (k^+_p) - \mathrm{Im} (k^-_p) \big] L\big\rbrace = 1, \end{gather}$$

(2.3)$$\begin{gather}\mathrm{arg} \left( R_0R_L \right) + \big[ - \mathrm{Re} (k^+_p) + \mathrm{Re} (k^-_p) \big] L =-2m{\rm \pi}, \end{gather}$$

where $m$ is an integer.

Figure 1. Travelling-wave components of an eigenfunction in an acoustic resonator–waveguide system: $k^{\pm }_p$ cut-on DMs (propagating waves) in the resonant region, $k^{\pm }_r$ cut-on DMs in the waveguide that can radiate energy to infinity if not decoupled and $k^{\pm }_e$ cut-off DMs (evanescent waves).

The travelling-wave formulation provides physical insights into the eigenmodes of continuous systems. It reveals that waves travelling in opposite directions and wave reflections at the two ends of the resonant region are four indispensable ingredients of an eigenmode, because the loop must be closed. For acoustics, it also reveals the connection between the system eigenmodes being real or complex and the wave reflection characteristics, illustrated as follows. Assume the resonant region in figure 1 is a 2-D hard-walled duct segment of length $L$ and height $H=1$. Acoustic DMs (Rienstra & Hirschberg Reference Rienstra and Hirschberg2021) in the $\pm x$ directions are in the form of

(2.4)\begin{equation} p = P(y)\exp(-\mathrm{i} k x)\exp(\mathrm{i} \omega t ), \end{equation}

where $p$ is the pressure disturbance, $P(y)$ is the modal profile. The dispersion relation of DMs is

(2.5)\begin{equation} k_n^{{\pm}} ={\pm} \sqrt{\omega^2-\left[ (n-1){\rm \pi} \right] ^2}, \end{equation}

where $n=1,2,3\ldots$ and $n=1$ for the plane waves, $k_n^{\pm }$ is the wavenumber of the $n$th DM in the $\pm x$ directions. All variables are normalized by the sound speed, fluid density and the duct height. As the frequency varies from $\omega /{\rm \pi} =1.5-6\mathrm {i}$ to $1.5+6\mathrm {i}$, (2.5) is plotted in figure 2 for the first eight DMs in the $\pm x$ directions, of which the first two DMs are cut-on (propagating) and the others are cut-off (evanescent). First, consider in figure 1 the duct segment being closed at $x=0$ and $x=L$ by straight walls, so that $R_0=R_L=1$ for each DM. Each pair of $k_n^+$ and $k_n^-$ cut-on DMs can satisfy (2.2) and form an eigenmode at a real-valued frequency selected by (2.3). Second, consider an open system as in figure 1, so $|R_0| \leq 1$ and $|R_L| \leq 1$. A pair of $k_n^+$ and $k_n^-$ cut-on DMs can also satisfy (2.2) and underpin an eigenmode at a real frequency selected by (2.3) if $|R_0|=|R_L|=1$ for that pair of DMs. If $|R_0R_L|<1$ for a pair of $k_n^+$ and $k_n^-$ cut-on DMs, then a real-frequency eigenmode cannot be formed by that pair. Nevertheless, a positive $\mathrm {Im}(\omega )$ leads to $\mathrm {exp} \lbrace [\mathrm {Im} (k_n^+) - \mathrm {Im} (k_n^-) ] L\rbrace >1$ for cut-on DMs (see figure 2), thus (2.2) can be satisfied at a complex frequency selected by (2.2) and (2.3). So, it is total reflection or not rather than an acoustic system being geometrically closed or open that decides eigenmodes of the system being real or complex. A pair of $k_n^+$ and $k_n^-$ cut-off DMs always leads to $\mathrm {exp} \lbrace [ \mathrm {Im} (k_n^+) - \mathrm {Im} (k_n^-) ] L\rbrace <1$ at either real or complex frequencies (see figure 2), thus, without the participation of propagating waves, pairs of $k_n^+$ and $k_n^-$ evanescent waves alone cannot satisfy (2.2) and therefore cannot form an eigenmode.

Figure 2. Dispersion relation in a 2-D hard-walled acoustic waveguide of height $H=1$. (a) Complex frequency and (b) wavenumbers of the first eight $\pm x$ DMs.

2.2. The second condition of TMs: wave-trapping condition

There are spatial and temporal approaches of defining some special eigenmodes in an open system as TMs. In the spatial approach, the most commonly used definition is zero radiation. Specifically, the eigenfunction of a TM has no travelling-wave components that can radiate to infinity. In resonator–waveguide systems, this occurs in two different situations. One is that the system eigenmodes occur in a frequency range where all DMs in the infinite waveguide are cut-off, which can happen in waveguides with the Dirichlet boundary condition (Pagneux Reference Pagneux2013). The other situation is that there are cut-on DMs in the waveguide, but they do not participate in (or they are decoupled from) some eigenmodes of the system. In other words, the amplitudes or mode coefficients of $k^{\pm }_r$ cut-on DMs in figure 1 are zero if the eigenfunctions are decomposed into DMs

(2.6)\begin{equation} C^{{\pm}}_r=0. \end{equation}

In contrast, system eigenmodes with $C^{\pm }_r \neq 0$ are LMs. Trapped modes in the second situation are called embedded TMs or BICs (Hsu et al. Reference Hsu, Zhen, Stone, Joannopoulos and Soljac̆ić2016), since there are cut-on DMs in the waveguide when the TMs occur. All acoustic TMs in this study are embedded TMs, because the Neumann boundary condition leads to the cut-on plane wave in the waveguide at all frequencies.

In the situation of $L \gg 1$, if one neglects the influences of $k^{\mp }_e$ cut-off DMs generated by the scattering of $k^{\pm }_p$ cut-on DMs at one end on the wave scattering at the opposite end, the following two equivalents to the zero-radiation condition (2.6) can be easily obtained: the zero transmission from $k^{\pm }_p$ cut-on DMs inside the resonant region respectively to the outgoing $k^{\pm }_r$ cut-on DMs in the infinite waveguide

(2.7)\begin{equation} T_L = T_ 0 =0, \end{equation}

and the total reflection of $k^{\pm }_p$ cut-on DMs at the region boundaries

(2.8)\begin{equation} \left| R_L \right|=\left| R_0 \right|=1. \end{equation}

The temporal approach of defining TMs is that the free oscillation associated with an eigenmode of an open system does not decay in time, i.e. the modal frequency is real

(2.9)\begin{equation} \mathrm{Im}(\omega_{TM})=0. \end{equation}

In contrast, $\mathrm {Im}(\omega _{LM})>0$ under the $\exp (\mathrm {i} \omega t )$ convention.

One can obtain the equivalence between (2.6) and (2.9) from an obvious physical statement in this context (no other gains and losses) that spatially zero energy radiation is equal to temporally neutral free oscillations. Here, the equivalence of these four conditions that define some special eigenmodes of an open system as TMs is discussed based on the sketch of figure 1, where we assume a single cut-on DM in the $\pm x$ directions. In cases where multiple cut-on DMs are involved, the equivalence between zero radiation, zero transmission, total reflection and real modal frequency still holds in the physical sense, although (2.6), (2.7) and (2.8) need to be accordingly generalized (see § 4).

2.3. Computational model

The present resonator–waveguide systems are divided into homogeneous duct segments. The DMs are given by

(2.10)\begin{equation} k_n^{{\pm}} ={\pm} \sqrt{ \omega^2-\left[ (n-1){\rm \pi} /H_j\right] ^2}, \end{equation}

for wavenumbers and

(2.11)\begin{equation} P_n(y) = \mathrm{cos} \left[ (n-1){\rm \pi} y/H_j \right]\!, \end{equation}

for modal profiles, where $H_j$ is duct height and $j$ is the index of ducts. The DMs are normalized so that $|P_n(y) |_{max}=1$. The eigenfunction of a system eigenmode is expanded in terms of DMs in each duct with coefficients $C_{n,j}^{\pm }$ to be determined

(2.12)\begin{equation} p_j(x,y) = \sum_{n=1}^{N_j} C_{n,j}^{{\pm}} P_{n,j}(y) \exp(-\mathrm{i} k_{n,j}^{{\pm}} x), \end{equation}

where DMs in the $\pm x$ directions in the resonant region and outgoing DMs in the two semi-infinite waveguides are involved in the summation.

A loop matrix taking into account cut-on and cut-off DMs in the resonant region is defined (Doaré Reference Doaré2001; Gallaire & Chomaz Reference Gallaire and Chomaz2004; Doaré & de Langre Reference Doaré and de Langre2006; de Lasson et al. Reference de Lasson, Kristensen, Mørk and Gregersen2014; Dai Reference Dai2021a)

(2.13)\begin{equation} \boldsymbol{\mathsf{M}}_{lp} = \boldsymbol{\mathsf{R}}_{l}\boldsymbol{\mathsf{P}}_{l}\boldsymbol{\mathsf{R}}_r\boldsymbol{\mathsf{P}}_{r}. \end{equation}

In the calculations, $N$ less attenuated DMs in the $\pm x$ directions in the $H=1$ infinite waveguide are used and the number of less attenuated DMs used in duct segment(s) in the resonant region is accordingly $N_j=NH_j/H$. So, the total number of DMs in the $\pm x$ directions in the resonant region is $N_t=\sum N_j$. Here, $\boldsymbol{\mathsf{P}}_l (N_t \times N_t)$ and $\boldsymbol{\mathsf{P}}_r (N_t \times N_t)$ are propagation matrices of the left-running and right-running DMs, describing a wave travelling inside the resonant region; $\boldsymbol{\mathsf{P}}_{l,r}$ are diagonal matrices with the elements on the diagonal being $\exp ( \pm \mathrm {i} k_n^{\mp } L )$; $\boldsymbol{\mathsf{R}}_l (N_t \times N_t)$ and $\boldsymbol{\mathsf{R}}_r (N_t \times N_t)$ are the left and right reflection matrices, describing wave reflection at region boundaries; $\boldsymbol{\mathsf{R}}_{l,r}$ are extracted from the interface scattering matrices. Using the interface between the resonant region and the right semi-infinite waveguide as an example, the scattering matrix is obtained by numerically matching DMs on both sides of the interface (Kooijman et al. Reference Kooijman, Testud, Aurégan and Hirschberg2008; Kooijman, Hirschberg & Aurégan Reference Kooijman, Hirschberg and Aurégan2010; Dai & Aurégan Reference Dai and Aurégan2018). The continuity of $p$ and $\partial p/ \partial x$ at the interface and $\partial p/ \partial x=0$ on the vertical wall lead to the scattering matrix that links incoming waves to outgoing waves from the interface

(2.14)\begin{equation} \left(\begin{array}{@{}c@{}} \boldsymbol{C}_r^+ \\ \boldsymbol{C}_l^- \end{array}\right)= \boldsymbol{\mathsf{S}}\left(\begin{array}{@{}c@{}} \boldsymbol{C}_l^+ \\ \boldsymbol{C}_r^- \end{array}\right)\!, \end{equation}

where vectors $\boldsymbol {C}^{\pm }_l$ (respectively $\boldsymbol {C}^{\pm }_r$) contain coefficients of DMs in the $\pm x$ directions on the left and right sides of the interface

(2.15) \begin{equation} \boldsymbol{\mathsf{S}}= \Big(\begin{array}{@{}cc@{}} \boldsymbol{\mathsf{T}}^+ & \boldsymbol{\mathsf{R}}^- \\ \boldsymbol{\mathsf{R}}^+ & \boldsymbol{\mathsf{T}}^- \end{array}\Big)\!, \end{equation}

where $\boldsymbol{\mathsf{T}}^{\pm }$ and $\boldsymbol{\mathsf{R}}^{\pm }$ are transmission and reflection matrices in the $\pm x$ directions and $\boldsymbol{\mathsf{R}}^+=\boldsymbol{\mathsf{R}}_r$.

The loop closure principle (2.1) requires that an eigenvalue of $\boldsymbol{\mathsf{M}}_{lp}$ is unity

(2.16a,b)\begin{equation} \boldsymbol{\mathsf{M}}_{lp} \boldsymbol{C}_{lp}= k_{lp} \boldsymbol{C}_{lp} \quad \mbox{and}\quad k_{lp} =1. \end{equation}

Since (2.16a,b) is the condition of all eigenmodes, including both TMs and LMs, the temporal definition (2.9) is used in the calculations to distinguish TMs from LMs. For common TMs, the real-valued frequency $\omega$ is optimized with a given TM length $L$, for that an eigenvalue of $\boldsymbol{\mathsf{M}}_{lp}$ equals unity. For accidental TMs, both $L$ and real-valued $\omega$ are optimized. One optimization process solves one TM: the optimized $\omega$ is the eigenfrequency and the eigenvector $\boldsymbol {C}_{lp}$ corresponding to $k_{lp} =1$ contains mode coefficients of DMs, the linear superposition of which gives the eigenfunction. The eigenfunction is normalized so that $|p(x,y)|_{max}=1$. In the calculations, the iteration stops when the error between the target $k_{lp}$ and unity is less than $10^{-12}$, the number of $k_n^{\pm }$ DMs in the waveguide with $H=1$ varies from $N=400$ to 600 (for a round number of $N_j=NH_j/H$) and the relative errors of $\omega$ and $L$ are of the order of $10^{-5}$–$10^{-6}$ as $N$ is doubled in the checked cases: figures 4, 8 and 10.

3.1. Symmetry-protected TMs

The first example of symmetry-protected TMs is in a waveguide with a $y$-symmetric cavity, as displayed in figure 3. First, the mode coefficients of DMs in figure 3(d,h) indicate the split between $y$-antisymmetric and $y$-symmetric DMs in the $y$-symmetric system. The $y$-antisymmetric TMs are composed of only $y$-antisymmetric DMs. The only one outgoing cut-on DM ($k_1^+$) in $d2$ has a zero mode coefficient, indicating zero radiation. Second, only one pair of cut-on DMs ($k_2^{\pm }$) in the resonant region ($d1$) is involved, as shown in figure 3(b,f). The wave reflection at region boundaries is total reflection, i.e. $| R_0 |=| R_L |=1$. Third, the zero mode coefficient of $k_1^+$ DM in $d2$ also indicates that the transmission from $k_2^{\pm }$ DMs in the resonant region to the outgoing $k_1^{\pm }$ DMs in the waveguide is zero, i.e. $T_0 = T_ L =0$. So, all the three spatial features of TMs, i.e. (2.6), (2.7) and (2.8), are observed.

Figure 3. Trapped modes in a waveguide with a $y$-symmetric cavity of depth $D=2$: (a,e) ${\rm Re}(\,p)$ of short and long TMs ($L=0.3$, $\omega /{\rm \pi} =0.6667$; $L=10$, $\omega /{\rm \pi} =0.5045$); (b,f) ${\rm Re}(\,p)$ of $k_2^{\pm }$ cut-on DMs; (c,g) ${\rm Re}(\,p)$ of $k_4^{\pm }$ cut-off DMs; (d,h) modulus of mode coefficients of $k_n^+$ DMs in $d1$ and $d2$ in the TM eigenfunctions.

Note that, for all TMs in this paper, the number of cut-on DMs in each duct segment is known from the TM frequencies given in figure captions. The cut-on frequency of $k_n^{\pm }$ DMs is $\omega _{{cut\text -on}}/{\rm \pi} = (n-1)/H_j$.

Let us compare the short and long TMs in figure 3. In the short TM, $k_4^{\pm }$ evanescent DMs in $d1$ generated at $x=0$ or $x=L$ have a non-negligible amplitude at the opposite end, as shown in figure 3(c). On the other hand, $k_4^{\pm }$ DMs in the long TM have a quite small amplitude, as shown in figure 3(g). Especially, after a long-distance exponential decrease $\exp ( \mp \mathrm {i} k_4^{\pm } L) = 5.33 \times 10^{-20}$, $k_4^{\pm }$ DMs have a vanishingly small amplitude at the destinations. So, the long TM is visually clean and simple. It is essentially one pair of $k_2^{\pm }$ cut-on DMs in the resonant region with total reflection at region boundaries. The second difference between the short and long TMs is that $k_2^{\pm }$ DMs in $d1$ show different phase changes in total reflection: figure 3(b) is close to a hard-wall reflection (the phase change is zero and $\partial p/ \partial x =0$ at the reflection interface) whereas figure 3(f) is close to a pressure-release reflection (the phase change is ${\rm \pi}$ and $p=0$ at the reflection interface).

The second example of symmetry-protected TMs is in a waveguide containing a plate placed on the centreline, as displayed in figure 4. The mode coefficients of DMs in $d3$ indicate the split between $y$-antisymmetric and $y$-symmetric DMs. The amplitudes of all $y$-symmetric DMs in $d3$, including the only one cut-on DM ($k_1^+$), are zero. The magnitudes of $k_n^+$ DMs in respectively $d1$ and $d2$ are the same, as shown in figure 4(d,h), whereas the phases of $k_n^+$ DMs in $d1$ and $d2$ are the same for $y$-antisymmetric DMs but opposite for $y$-symmetric DMs (this phase information is not shown in mode coefficients, but it can be seen in figure 4b,c,f,g), which ensures the $y$-antisymmetric eigenfunction in the resonant region. In both $d1$ and $d2$, there is only one pair of cut-on DMs ($k_1^{\pm }$), and they have total reflection at the ends, as shown in figure 4(b,f). Visually, the short TM is complicated by evanescent waves, but the long TM is clean and simple, essentially two standing waves oscillating in $d1$ and $d2$ with equal amplitude and opposite phases. These two standing waves are spatially separated by the plate, however, they are coupled.

Figure 4. Trapped modes in a waveguide with a plate of thickness $T=0.1$ on the centreline: (a,e) ${\rm Re}(\,p)$ of short and long TMs ($L=0.3$, $\omega /{\rm \pi} =0.9574$; $L=10$, $\omega /{\rm \pi} =0.0950$); (b,f) ${\rm Re}(\,p)$ of $k_1^{\pm }$ cut-on DMs; (c,g) ${\rm Re}(\,p)$ of $k_2^{\pm }$ cut-off DMs; (d,h) modulus of mode coefficients of $k_n^+$ DMs in $d1$, $d2$ and $d3$ in the TM eigenfunctions.

Two necessary conditions for $y$-antisymmetric TMs in figures 3 and 4 are the $y$-symmetry of the system and $0<\omega _{TM}/{\rm \pi} <1$. However, these two conditions do not guarantee TMs because they do not guarantee eigenmodes (the first condition of TMs). Reducing the depth of the cavity shown in figure 3 so that $D< H=1$ leads to the orifice–waveguide system shown in figure 5. In the frequency range $0<\omega /{\rm \pi} <1$, one cannot expect $y$-antisymmetric eigenmodes of this system for any $L>0$, neither TMs nor LMs, because no $y$-antisymmetric DMs in $d1$ are cut-on. The only possible eigenmodes of the system in this frequency range are $y$-symmetric (underpinned by $k_1^{\pm }$ plane waves in $d1$) and damped by radiation into infinity. One of those $y$-symmetric LMs is shown in figure 5. In § 4, we will understand that the orifice–waveguide system does not support TMs in any frequency range.

Figure 5. Leaky mode in a waveguide with a $y$-symmetric orifice. Orifice height $D=0.2$, orifice length $L=10$ and $\omega /{\rm \pi} =0.0984 + 0.0127 \mathrm {i}$: (a) $|p|$; (b) modulus of mode coefficients of $k_n^+$ DMs in $d1$ and $d2$ in the LM eigenfunction.

3.2. Invisibility-protected TMs

An example of invisibility-protected TMs (Evans et al. Reference Evans, Linton and Ursell1993; Davies & Parnovski Reference Davies and Parnovski1998; Groves Reference Groves1998; Linton & McIver Reference Linton and McIver1998; Linton et al. Reference Linton, McIver, McIver, Ratcliffe and Zhang2002) is given in figure 6, where a 2-D waveguide contains a zero-thickness plate parallel to the guide walls. In both short and long TMs, the coefficients of DMs in $d3$ indicate the split of DMs, but in a way that is different from the $y$-symmetric split shown in figure 4(d,h). Here, the zero amplitudes of $k_n^+$ ($n=1,4,7\dots$) DMs in $d3$ indicate that this particular group of DMs in the guide are not involved in the TMs. Those DMs all have a node of vertical velocity at the plate position $y=1/3$, so they are not scattered by the plate. One may also say that the zero-thickness plate is invisible to those DMs in the guide. Since $k_2^{\pm }$ DMs do not have a node of vertical velocity within the guide, $k_2^{\pm }$ DMs are always scattered by a zero-thickness plate placed at any $y$-position in the guide. Therefore, the invisibility mechanism cannot protect an eigenmode from the radiation by $k_2^{\pm }$ DMs in the guide, and TMs protected by the invisibility mechanism alone exist only in the frequency range $0<\omega /{\rm \pi} <1$ where $k_2^{\pm }$ DMs in the guide are evanescent waves. The eigenfunction of the short TM in figure 6(a) is rather complex, whereas the long TM in figure 6(e) is simple and characterized by two standing waves in $d1$ and $d2$ formed by $k_1^{\pm }$ plane waves.

Figure 6. Trapped modes in a waveguide with a zero-thickness plate at $y=1/3$: (a,e) ${\rm Re}(\,p)$ of short and long TMs ($L=0.3$, $\omega /{\rm \pi} =0.9870$; $L=10$, $\omega /{\rm \pi} =0.0961$); (b,f) ${\rm Re}(\,p)$ of $k_1^{\pm }$ cut-on DMs; (c,g) ${\rm Re}(\,p)$ of $k_2^{\pm }$ cut-off DMs; (d,h) modulus of mode coefficients of $k_n^+$ DMs in $d1$, $d2$ and $d3$ in the TM eigenfunctions.

3.3. Symmetry–periodicity-protected TMs

Consider a 2-D waveguide containing $N$ identical plates. If the plates are placed at the same $x$-position and at $y$-positions so that the heights of the parallel duct segments separated by the plates satisfy $H_1=H_{N+1}= H_j/2$ where $j=2, 3 \ldots N$, then an $N$-period section of an infinite array of equally spaced plates in the $y$-direction is formed. Figure 7 displays symmetry–periodicity-protected TMs (Utsunomiya & Eatock Taylor Reference Utsunomiya and Eatock Taylor1999; Porter & Evans Reference Porter and Evans1999; Linton & McIver Reference Linton and McIver2002) in an acoustic waveguide with two plates of thickness $T=1/8$. In this case, two patterns of TMs exist, i.e. the $y$-antisymmetric pattern shown in figure 7(a,i) and the $y$-symmetric pattern shown in figure 7(e,m). The mode coefficients of DMs in $d4$ shown in the third column of figure 7 indicate the decoupling of TMs of each pattern from a particular group of outgoing DMs in the waveguide. The $y$-antisymmetric TMs are decoupled from the outgoing $k_1^{\pm }$ but not $k_2^{\pm }$ DMs in the waveguide, thus the TM frequency range is $0<\omega /{\rm \pi} <1$. The $y$-symmetric TMs are decoupled from the outgoing $k_1^{\pm }$ and $k_2^{\pm }$ but not $k_3^{\pm }$ DMs in the waveguide, thus the TM frequency range is $0<\omega /{\rm \pi} <2$. In the frequency range all the TMs, i.e. $0<\omega /{\rm \pi} <2$, only $k_1^{\pm }$ plane waves are cut-on in the duct segments in the resonant region. The long TMs shown in figure 7(i,m) are simple and characterized by multiple standing waves in $d$1, $d2$ and $d3$ formed by $k_1^{\pm }$ plane waves.

Figure 7. Trapped modes in a waveguide with two plates of thickness $T=1/8$ which form a 2-period section of a vertically symmetric–periodic system: (a,i) ${\rm Re}(\,p)$ of short and long $y$-antisymmetric TMs ($L=0.3$, $\omega /{\rm \pi} =0.9737$; $L=10$, $\omega /{\rm \pi} =0.0957$); (e,m) ${\rm Re}(\,p)$ of short and long $y$-symmetric TMs ($L=0.3$, $\omega /{\rm \pi} =1.5830$; $L=10$, $\omega /{\rm \pi} =0.0973$); (b,f,j,n) ${\rm Re}(\,p)$ of $k_1^{\pm }$ cut-on DMs; (c,g,k,o) ${\rm Re}(\,p)$ of $k_2^{\pm }$ cut-off DMs; (d,h,l,p) modulus of mode coefficients of $k_n^+$ DMs in $d1$, $d2$ and $d4$ in the TM eigenfunctions.

Symmetry–periodicity-protected TMs in an acoustic waveguide with three and four plates are shown in figure 8. In these two cases, there are respectively three and four patterns of TMs. In each pattern, the decoupled outgoing DMs in the waveguide, indicated by their mode coefficients, reveal the upper limit of TM frequency: $\omega /{\rm \pi} <3$ (figure 8f) and $\omega /{\rm \pi} <4$ (figure 8n) in the three-plate and four-plate cases respectively. As in the two-plate case, only $k_1^{\pm }$ DMs are cut-on in the duct segments in the resonant region. The long TMs in figure 8 are also very simple and characterized by multiple standing waves formed by $k_1^{\pm }$ plane waves. Symmetry–periodicity-protected TMs in any high-frequency range (thus with any large number of cut-on DMs in the waveguide) can be obtained by increasing the periods $N$.

Figure 8. Trapped modes in a waveguide with three plates ($T=1/12$) and four plates ($T=1/16$) which form a 3- and 4-period section of a vertically symmetric–periodic system: (a–c) ${\rm Re}(\,p)$ of long TMs ($L=10$; $\omega /{\rm \pi} =0.09569$, 0.09777 and 0.09817) belonging to three different patterns and (d–f) modulus of mode coefficients of $k_n^+$ DMs in $d1$, $d2$ and $d5$ in the TM eigenfunctions; (g–j) ${\rm Re}(\,p)$ of long TMs ($L=10$; $\omega /{\rm \pi} =0.09564$, 0.09782, 0.09846 and 0.09862) belonging to four different patterns and (k–n) modulus of mode coefficients of $k_n^+$ DMs in $d1$, $d2$, $d3$ and $d6$ in the TM eigenfunctions.

3.4. Accidental TMs

Figure 9 presents a simple and valuable example of accidental TMs, found previously by Xiong (Reference Xiong2016) in an $x$-symmetric $y$-asymmetric cavity–waveguide system. Those TMs were later calculated and described in Dai (Reference Dai2021a), here, we re-calculate and collect them for the purpose of the comparison of various types of TMs. The mode coefficients of DMs in $d2$ shown in the third column of figure 9 indicate that the single outgoing cut-on DM ($k_{1}^+$) has a zero amplitude, i.e. zero radiation. Note that the number of decoupled DMs in the waveguide in this case is different from that in the common TMs described above: only one outgoing DM in the $\pm x$ directions in the waveguide is decoupled in this case, whereas a group of an infinite number of outgoing DMs (only one or a few of them are cut-on, however) are decoupled in the cases above. The second column of figure 9 shows that the TMs are characterized by two standing waves respectively formed by $k_{1}^{\pm }$ and $k_{2}^{\pm }$ DMs in the resonant region with total reflection at region boundaries. Different phase changes in total reflection can be observed: $k_{1}^{\pm }$ DMs seem always have a hard-wall reflection as shown in figure 9(b,g,l); for $k_{2}^{\pm }$ DMs, the total reflection is close to a pressure-release reflection in figure 9(h) but close to a hard-wall reflection in figure 9(m). Also note that the standing-wave characteristics in this case are different from those in the common TMs. First, two standing waves coexist in the cavity segment in this case, whereas there is no coexistence of standing waves in a single duct segment in the cases above. Second, cut-on DMs with two different wavelengths are involved in the resonant region in this case, whereas all cut-on DMs in the resonant region have the same wavelength in cases above.

Figure 9. Accidental TMs in a waveguide with a cavity with depth $D=1.5$: (a,f,k) ${\rm Re}(\,p)$ of three TMs ($L=2.9025$ and $\omega /{\rm \pi} =0.6891$; $L=8.9333$ and $\omega /{\rm \pi} =0.6716$; $L=7.2356$ and $\omega /{\rm \pi} =0.9674$); (b,g,l) ${\rm Re}(\,p)$ of $k_1^{\pm }$ cut-on DMs; (c,h,m) ${\rm Re}(\,p)$ of $k_2^{\pm }$ cut-on DMs; (d,i,n) ${\rm Re}(\,p)$ of $k_3^{\pm }$ cut-off DMs; (e,j,o) modulus of mode coefficients of $k_n^+$ DMs in $d1$ and $d2$ in the TM eigenfunctions. Note that, in figures 9, 10 and 11, the calculated coefficients of the outgoing DMs decoupled by the accidental TM mechanism are of the order $10^{-8}$, which reduces as the number of DMs used in the calculations is increased.

Figure 10 displays accidental TMs in an open system with rotational symmetry. Those TMs are also characterized by two standing waves respectively formed by $k_{1}^{\pm }$ and $k_{2}^{\pm }$ DMs in the resonant region. The TMs and their standing-wave components are either point antisymmetric in figure 10(a–c) or point symmetric in figure 10(e–g).

Figure 10. Accidental TMs in a system with rotational symmetry: (a,e) ${\rm Re}(\,p)$ of two TMs ($L=1.3945$ and $\omega /{\rm \pi} =0.7171$; $L=5.6263$ and $\omega /{\rm \pi} =0.7109$); (b,f) ${\rm Re}(\,p)$ of $k_1^{\pm }$ cut-on DMs; (c,g) ${\rm Re}(\,p)$ of $k_2^{\pm }$ cut-on DMs; (d,h) modulus of mode coefficients of $k_n^+$ DMs in $d1$ and $d2$ in the TM eigenfunctions.

Figure 11 presents another example of accidental TMs, in which the travelling-wave components are more complex than those in figures 9 and 10. Those TMs were found previously by Duan et al. (Reference Duan, Koch, Linton and McIver2007) in a waveguide with a zero-thickness plate placed on the centreline of the guide. The accidental TM shown in figure 11(a), which is one of the Neumann TMs in the frequency range $2<\omega /{\rm \pi} <3$ in figure 2 of Duan et al. (Reference Duan, Koch, Linton and McIver2007), might be understood as a result of co-protection. Over the frequency range $2<\omega /{\rm \pi} <3$, $k_n^{\pm }$ ($n=1,2,3$) DMs are cut-on in the waveguide $d3$; over the frequency range $2<\omega /{\rm \pi} <4$, $k_n^{\pm }$ ($n=1,2$) DMs are cut-on in $d1$ and $d2$. First, the $y$-symmetry of the system decouples $y$-antisymmetric TMs from the outgoing $k_n^{\pm }$ ($n=1,3$) DMs in the waveguide (the same as figure 4). Second, the coupled $k_1^{\pm }$ and $k_2^{\pm }$ standing waves in $d1$ and $d2$ decouple the TMs from the outgoing $k_{2}^{\pm }$ DMs in the waveguide. Note that, under the $y$-antisymmetric constraint for the DMs in $d1$ and $d2$, only the ratio between $k_1^{\pm }$ and $k_2^{\pm }$ standing waves in one of the two ducts is relevant to the decoupling of the outgoing $k_{2}^{\pm }$ DMs in the waveguide, which is essentially the same situation as that in figures 9 and 10. Accidental TMs due to such co-protection with the plate having a finite thickness $T$ are shown in figure 11(f,k). With a finite-thickness plate, the frequency range of those accidental TMs is not $2<\omega /{\rm \pi} <3$ but $2H/( H-T )<\omega /{\rm \pi} <3$, where the lower limit is required by the condition that there are two cut-on DMs in $d1$ and $d2$. So, if $2H/( H-T )>3$, i.e. $T>H/3$, those accidental TMs do not occur.

Figure 11. Accidental TMs in a waveguide with zero-thickness and finite-thickness ($T=0.1$) plates on the centreline: (a,f,k) ${\rm Re}(\,p)$ of TMs ($L=1.2779$, $\omega /{\rm \pi} =2.1139$; $L=1.0819$, $\omega /{\rm \pi} =2.3416$; $L=2.0480$, $\omega /{\rm \pi} =2.9987$); (b,g,l) ${\rm Re}(\,p)$ of $k_1^{\pm }$ cut-on DMs; (c,h,m) ${\rm Re}(\,p)$ of $k_2^{\pm }$ cut-on DMs; (d,i,n) ${\rm Re}(\,p)$ of $k_3^{\pm }$ cut-off DMs; (e,j,o) modulus of mode coefficients of $k_n^+$ DMs in $d1$, $d2$ and $d3$ in the TM eigenfunctions.

In all TMs above except figure 3, the number of cut-on DM pairs in the resonant region that participate in the eigenfunctions is one (or more than one, as in figure 8a,b,g) larger than the number of outgoing cut-on DMs in the waveguide, which gives just the right amount of freedom (or sufficient freedom) to cancel the radiation (Linton & McIver Reference Linton and McIver2007). However, for example, if the plate in figures 4 and 11 is moved slightly off the centreline, the number of cut-on DM pairs in the resonant region can remain unchanged in a slightly changed frequency range but the TMs disappear. Therefore, we need to understand why in those scenarios the cut-on DM pairs in the resonant region cannot underpin a TM by mutual cancellation in radiation (see the phase compensation in (4.46) thanks to $k_1^{\pm } \neq\, k_2^{\pm }$).

4.1. Symmetry-protected TMs

Figure 12(a) corresponds to the TMs in figure 3. For $D=2H$, they happen under necessary conditions that only $k_{1}^{\pm }$ DMs are cut-on in $d2$ and only $k_1^{\pm }$ and $k_2^{\pm }$ DMs are cut-on in $d1$. For a larger $D$, the second condition is that only one pair of $y$-antisymmetric $k_n^{\pm }$ DMs is cut-on in $d1$. In the decomposed eigenfunction, the only incident wave on the interface is $k_2^+$ DM in $d1$. Then, the $y$-symmetry of the system ensures

(4.4)\begin{equation} \left| B_1^+ \right| =0,\end{equation}

and

(4.5)\begin{equation} \left| C_1^- \right| =0.\end{equation}

With (4.4) and (4.5), the expression of energy conservation (4.3) gives

(4.6)\begin{equation} \left| \frac{C_2^-}{C_2^+} \right|=1,\end{equation}

meaning that $k_2^+$ DM is totally reflected. Total reflection of $k_2^-$ DM and zero radiation to infinity also happen at the left end. Therefore, the loop magnitude constraint and the loop zero-radiation condition are satisfied by a $k_2^{\pm }$-loop of DMs in $d1$. For an $x$-symmetric system, the loop phase constraint (2.3) that selects the TM frequency for a given $L$ can be reformed

(4.7)\begin{equation} \mathrm{arg}\bigg( \frac{C_2^-}{C_2^+} \bigg) - k_2^+L =-m {\rm \pi},\end{equation}

where $m$ is an integer. Equation (4.7) means that the phase change of half of the loop is an integer multiple of ${\rm \pi}$. Even and odd values of $m$ respectively correspond to $x$-symmetric and $x$-antisymmetric eigenmodes.

The interface scattering matrix, which relates the outgoing waves to the incoming waves on the interface, is numerically calculated by matching DMs in $d1$ and $d2$ (the number of DMs used in the matching is the same as that in the calculations in § 3). From the interface scattering matrix, the value of $C_2^-/C_2^+$ is extracted, as shown in figure 13(a,b). The calculated modulus agrees with (4.6). The phase change varies continuously with frequency in such a way: $\mathrm {arg}( C_2^-/C_2^+ ) = {\rm \pi}$ and $\mathrm {arg}( C_2^-/C_2^+ ) \approx 0$ respectively at the lower and upper limits of the frequency range. An analysis of the phase change in total reflection at waveguide discontinuities is given in Appendix A. With the calculated $\mathrm {arg}( C_2^-/C_2^+ )$, (4.7) gives the $L$–$\omega$ relation in figure 13(c). From the comparison for $m=0$ and $m=1$ branches, we can see that the lines obtained from (4.7) predict TMs well, except when $L$ is small.

Figure 13. (a,b) Modulus and argument of $C_2^-/C_2^+$ in the interface scattering sketched in figure 12(a). (c) Solid and dashed lines: $L$–$\omega$ relation from (4.7) with $m$ being even and odd. Symbols: TMs; squares: TMs shown in figure 3.

Figure 12(b) corresponds to the TMs in figure 4, which happen under necessary conditions that only $k_{1}^{\pm }$ DMs are cut-on in $d3$ and only $k_{1}^{\pm }$ DMs are cut-on in $d1$ and $d2$. The scattering relation of cut-on DMs is written as

(4.8) \begin{equation} \left(\begin{array}{@{}c@{}} C_1^- \\ C_2^- \\ B_1^+ \end{array}\right) = \left[\begin{array}{@{}cc@{}} R_{11} & R_{12} \\ R_{21} & R_{22} \\ T_{11} & T_{12} \end{array}\right] \left(\begin{array}{@{}c@{}} C_1^+ \\ C_2^+ \end{array}\right)\!. \end{equation}

The $y$-symmetry of the system ensures

(4.9a–c)\begin{equation} T_{11} =T_{12},\quad R_{11} = R_{22}, \quad R_{12} = R_{21}. \end{equation}

Therefore, if the incident waves satisfy

(4.10)\begin{equation} C_1^+=- C_2^+, \end{equation}

then

(4.11)\begin{equation} B_1^+=T_{11}C_1^++ T_{12}C_2^+=0, \end{equation}

understood as meaning that $k_1^+$ DM in $d1$ and $k_1^+$ DM in $d2$ cancel each other in transmission to $k_1^+$ DM in $d3$. Due to (4.9b,c), the incidence combination (4.10) also leads to

(4.12)\begin{equation} \frac{C_1^-}{C_1^+} =R_{11} + R_{12} \frac{C_2^+}{C_1^+} =R_{21} \frac{C_1^+}{C_2^+} + R_{22}= \frac{C_2^-}{C_2^+}. \end{equation}

With (4.11) and (4.12), the expression of energy conservation (4.3) gives

(4.13)\begin{equation} \left| \frac{C_1^-}{C_1^+} \right| = \left| \frac{C_2^-}{C_2^+} \right| =1. \end{equation}

Firstly, (4.13) indicates that the two $k_1^+$ DMs in respectively $d1$ and $d2$ both have total reflection at the interface, i.e. the magnitudes of the DMs are unvarying in reflection. At real frequencies, the magnitudes of $k_1^{\pm }$ cut-on DMs are also unvarying in propagation. Secondly, (4.12) indicates that the phase changes of the two $k_1^+$ DMs are the same in reflection. The phase changes of the two $k_1^+$ DMs in propagation from the left to the right end (or of the two $k_1^-$ DMs in the reverse direction) are also the same for any $L$, owing to the same wavenumber. Therefore, once the amplitude ratio between the two $k_1^{\pm }$ loops in respectively $d1$ and $d2$ is prescribed as in (4.10), the loop magnitude constraint and the loop zero-radiation condition are satisfied. Although two loops are involved, $k_1^{\pm }$ loops in $d1$ and $d2$ share the same phase constraint owing to the equal wavenumber and the equal phase change in boundary reflection

(4.14)\begin{equation} \mathrm{arg}\left( \frac{C_j^-}{C_j^+} \right) - k_1^+L =-m {\rm \pi},\end{equation}

where $m$ is an integer, $j=1$ or 2. Even and odd values of $m$ respectively correspond to $x$-symmetric and $x$-antisymmetric TMs.

The interface scattering matrix is numerically calculated by matching DMs in $d1$, $d2$ and $d3$, from which the value of $C_1^-/C_1^+$ (or $C_2^-/C_2^+$) is extracted, as shown in figure 14(a,b). With the calculated $\mathrm {arg}( C_1^-/C_1^+ )$, (4.14) gives the $L$–$\omega$ relation in figure 14(c). From the comparison for $m=0$ and $m=1$ branches, i.e. Parker's $\beta$- and $\alpha$-modes (Parker Reference Parker1966), we can see that the lines from (4.14) predict TMs well.

Figure 14. (a,b) Modulus and argument of $C_1^-/C_1^+$ in the interface scattering sketched in figure 12(b) with the incidence combination (4.10). (c) Solid and dashed lines: $L$–$\omega$ relation from (4.14) with $m$ being even and odd. Symbols: TMs; squares: TMs shown in figure 4.

4.2. Invisibility-protected TMs

Figure 12(c) corresponds to the TMs in figure 6, which happen under necessary conditions that only $k_{1}^{\pm }$ DMs are cut-on in $d3$ and only $k_{1}^{\pm }$ DMs are cut-on in $d1$ and $d2$. The scattering relation of cut-on DMs is written as

(4.15)\begin{equation} \left(\begin{array}{@{}c@{}} C_1^- \\ C_2^- \\ B_1^+ \end{array}\right) = \left[\begin{array}{@{}cc@{}} R_{11} & R_{12} \\ R_{21} & R_{22} \\ T_{11} & T_{12} \end{array}\right] \left(\begin{array}{@{}c@{}} C_1^+ \\ C_2^+ \end{array}\right)\!. \end{equation}

Because the interface scattering matrix is an invariant to the combination of incident waves, the values of or the relations between some elements of the scattering matrix can be extracted from particular incidence combinations. Imagine $k_1^-$ DM incoming from $d3$, then the zero-thickness plate does not scatter the plane wave. So, $k_1^-$ DM propagates in the $-x$-direction in the waveguide as if the zero-thickness plate was not present. Reversing the direction of wave propagation gives a second solution (also with no evanescent waves being scattered)

(4.16)\begin{equation} \left. \begin{array}{l@{}} C_1^+=C_2^+=B_1^+,\\ C_1^-=C_2^-=0. \end{array}\right\} \end{equation}

Such a solution leads to

(4.17)\begin{equation} \left. \begin{array}{l@{}} T_{11} + T_{12}=1,\\ R_{11} + R_{12}=0,\\ R_{21} + R_{22}=0. \end{array}\right\} \end{equation}

Tracing the total transmission and zero reflection of mass fluxes incoming from $d1$ and $d2$ (mass fluxes can be easily traced here because there are no evanescent waves being scattered in this particular situation) gives

(4.18)\begin{equation} \left. \begin{array}{l@{}} T_{11} /T_{12}=H_1/H_2, \\ H_1 R_{11}+H_2 R_{21}=0, \\ H_1 R_{12}+H_2 R_{22} =0, \end{array}\right\} \end{equation}

where $H_1$ and $H_2$ are the heights of $d1$ and $d2$.

Now, we consider the wave scattering sketched in 12(c). According to (4.18), if the incident waves satisfy

(4.19)\begin{equation} \frac{C_1^+ }{C_2^+} =- \frac{H_2}{H_1}, \end{equation}

then

(4.20)\begin{equation} B_1^+=T_{11}C_1^++ T_{12}C_2^+=0. \end{equation}

Equations (4.18) and (4.19) also lead to

(4.21)\begin{equation} \frac{C_1^-}{C_1^+} =R_{11} + R_{12} \frac{C_2^+}{C_1^+} =R_{21} \frac{C_1^+}{C_2^+} + R_{22}= \frac{C_2^-}{C_2^+}. \end{equation}

We can see that this case is very similar to the second case in § 4.1, except for the different amplitude ratios of $k_1^{\pm }$ loops prescribed in (4.10) and (4.19). For this case, the same equations as (4.13) and (4.14) can be obtained. We skip them but plot the $C_1^-/C_1^+$ (or $C_2^-/C_2^+$) as a function of frequency and the $L$–$\omega$ relation in figure 15.

Figure 15. (a,b) Modulus and argument of $C_1^-/C_1^+$ in the interface scattering sketched in figure 12(c) with the incidence combination (4.19). (c) Solid and dashed lines: $L$–$\omega$ relation from the loop phase constraint (the same as (4.14)). Symbols: TMs; squares: TMs shown in figure 6.

4.3. Symmetry–periodicity-protected TMs

Figure 12(d) corresponds to the TMs in figure 7, which happen under necessary conditions that only $k_{1}^{\pm }$ DMs are cut-on in $d4$ in the $y$-antisymmetric pattern of TMs whereas only $k_1^{\pm }$ and $k_2^{\pm }$ DMs are cut-on in $d4$ in the $y$-symmetric pattern of TMs, and in both patterns only $k_{1}^{\pm }$ DMs are cut-on in $d1$, $d2$ and $d3$. The scattering relation of cut-on DMs is written as

(4.22)\begin{equation} \left(\begin{array}{@{}c@{}} C_1^- \\ C_2^- \\ C_3^- \\ B_1^+ \\ B_2^+ \end{array}\right) = \left[\begin{array}{@{}ccc@{}} R_{11} & R_{12} & R_{13} \\ R_{21} & R_{22} & R_{23} \\ R_{31} & R_{32} & R_{33} \\ T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \end{array}\right] \left(\begin{array}{@{}c@{}} C_1^+ \\ C_2^+ \\ C_3^+ \end{array}\right)\!. \end{equation}

The $y$-symmetry of the system ensures the following relations of transmission and reflection:

(4.23a–c)$$\begin{gather} T_{11}=T_{13}, \quad T_{21}=-T_{23}, \quad T_{22}=0, \end{gather}$$

(4.24a–d)$$\begin{gather}R_{11}=R_{33}, \quad R_{13}=R_{31}, \quad R_{12}=R_{32}, \quad R_{21}=R_{23}. \end{gather}$$

Consider an incidence combination $C_1^+ = -C_2^+ = C_3^+$, the resulting field is symmetric about the waveguide centreline $y=H/2$. Therefore, placing a zero-thickness infinite-length plate at $y=H/2$ does not change the field. The plate divides the system into two identical sub-systems. In each $y$-symmetric sub-system, the scattering of the $y$-antisymmetric incidence is the same as analysed in § 4.1. Therefore

(4.25)\begin{equation} \frac{C_1^-}{C_1^+} = \frac{C_2^-}{C_2^+} = \frac{C_3^-}{C_3^+}, \end{equation}

and

(4.26)\begin{equation} \left|\frac{C_1^-}{C_1^+}\right| = \left|\frac{C_2^-}{C_2^+}\right| = \left|\frac{C_3^-}{C_3^+}\right| =1. \end{equation}

Also, $B_1^+=0$ under the incidence combination $C_1^+ = -C_2^+ = C_3^+$ gives

(4.27)\begin{equation} T_{11}- T_{12}+T_{13}=0. \end{equation}

Now, we consider the wave scattering in long TMs sketched in 12(d). According to the transmission relations in (4.23a–c) and (4.27), if the incident waves satisfy

(4.28)\begin{equation} \left. \begin{array}{l@{}} C_1^++ 2 C_2^++ C_3^+=0, \\ C_1^+- C_3^+=0, \end{array}\right\} \end{equation}

then the net transmissions to both $k_{1}^+$ and $k_{2}^+$ DMs in $d4$ are zero

(4.29)\begin{equation} \left. \begin{array}{l@{}} B_1^+=T_{11}C_1^++ T_{12}C_2^++ T_{13}C_3^+=0, \\ B_2^+=T_{21}C_1^++ T_{22}C_2^++ T_{23}C_3^+=0. \end{array}\right\} \end{equation}

The only solution of (4.28) for the ratios between DM coefficients is

(4.30)\begin{equation} C_1^+= -C_2^+= C_3^+, \end{equation}

which corresponds to the reflection relations in (4.25) and (4.26) and the $y$-symmetric patten of TMs. To have zero net transmission to only $k_{1}^+$ DM in $d4$, another solution, associated with a $y$-antisymmetric incidence and the $y$-antisymmetric patten of TMs, is

(4.31a,b)\begin{equation} C_1^+= -C_3^+, \quad C_2^+=0. \end{equation}

For the incidence combination (4.31a,b), (4.24a,b) leads to

(4.32)\begin{equation} \frac{C_1^-}{C_1^+} =R_{11} + R_{13} \frac{C_3^+}{C_1^+} = R_{31} \frac{C_1^+}{C_3^+} + R_{33} = \frac{C_3^-}{C_3^+}. \end{equation}

Then, with $B_1^+=0$ and (4.32), if $k_2^+$ DM in $d4$ is cut-off, (4.3) gives

(4.33)\begin{equation} \left|\frac{C_1^-}{C_1^+}\right| = \left|\frac{C_3^-}{C_3^+}\right| =1. \end{equation}

This case is also similar to the second case in § 4.1 in the sense that, once $k_1^{\pm }$ loops in $d$1, $d2$ and $d3$ have the amplitude ratio as prescribed in (4.30) or (4.31a,b), the loop magnitude constraint and the loop zero-radiation condition are satisfied. For each TM pattern, $k_1^{\pm }$ loops in $d1$, $d2$ and $d3$ share the same phase constraint owing to the equal wavenumber and the equal phase change in boundary reflection

(4.34)\begin{equation} \mathrm{arg}\bigg( \frac{C_j^-}{C_j^+} \bigg) - k_1^+L=-m {\rm \pi},\end{equation}

where $m$ is an integer, $j=1$ or 3 for the $y$-antisymmetric pattern and $j=1$ or 2 or 3 for the $y$-symmetric pattern. The numerically calculated reflection coefficients and the $L$–$\omega$ relation from (4.34) are plotted in figure 16. The $L$–$\omega$ relation is verified by the TMs shown in figure 7 and the TMs calculated with $L=5$.

Figure 16. (a,b) Modulus and argument of $C_1^-/C_1^+$ in the interface scattering sketched in figure 12(d). Thin and thick lines: $y$-symmetric and $y$-antisymmetric pattens with the incidence combinations (4.30) and (4.31a,b). (c) Solid and dashed lines: $L$–$\omega$ relation from (4.34) with $m$ being even and odd. Symbols: TMs; squares: TMs shown in figure 7.

4.4. Accidental TMs

Figure 12(e) corresponds to the accidental TMs in figure 9, which happen under necessary conditions that only $k_{1}^{\pm }$ DMs are cut-on in $d2$ and only $k_1^{\pm }$ and $k_2^{\pm }$ DMs are cut-on in $d1$. The scattering relation of cut-on DMs is written as

(4.35)\begin{equation} \left(\begin{array}{@{}c@{}} C_1^- \\ C_2^- \\ B_1^+ \end{array}\right) = \left[\begin{array}{@{}cc@{}} R_{11} & R_{12} \\ R_{21} & R_{22} \\ T_{11} & T_{12} \end{array}\right] \left(\begin{array}{@{}c@{}} C_1^+ \\ C_2^+ \end{array}\right)\!. \end{equation}

If the incident waves satisfy

(4.36)\begin{equation} \frac{C_1^+}{C_2^+} =- \frac{T_{12}}{T_{11}}, \end{equation}

then

(4.37)\begin{equation} B_1^+=0. \end{equation}

The incident combination (4.36) also leads to

(4.38)\begin{equation} \left. \begin{array}{l@{}} \displaystyle \dfrac{C_1^-}{C_1^+} = R_{11} - R_{12}\dfrac{T_{11}}{T_{12}}, \\[12pt] \displaystyle \dfrac{C_2^-}{C_2^+} = R_{22} - R_{21}\dfrac{T_{12}}{T_{11}}. \end{array}\right\} \end{equation}

With (4.37), the expression of energy conservation (4.3) gives

(4.39)\begin{equation} k_1^+ \big( \big| C_1^+ \big| ^2 - \big| C_1^- \big| ^2 \big) +k_2^+ \big( \big| C_2^+ \big| ^2 - \big| C_2^- \big| ^2 \big) =0, \end{equation}

which can be written equivalently as

(4.40)\begin{equation} k_1^+ \big( \big| T_{12} \big| ^2 - \big| R_{11}T_{12}- R_{12}T_{11} \big| ^2 \big) +k_2^+ \big( \big| T_{11} \big| ^2 - \big| R_{22}T_{11}- R_{21}T_{12} \big| ^2 \big) =0. \end{equation}

For the incidences of only one of $k_1^+$ and $k_2^+$ DMs in $d1$, (4.3) gives

(4.41)\begin{equation} \left. \begin{array}{l@{}} \displaystyle k_1^+= k_1^+ \left| R_{11} \right| ^2 + k_2^+ \left| R_{21} \right| ^2 + k_r^+ \left| T_{11} \right| ^2, \\ \displaystyle k_2^+= k_1^+ \left| R_{12} \right| ^2 + k_2^+ \left| R_{22} \right| ^2 + k_r^+ \left| T_{12} \right| ^2, \end{array}\right\} \end{equation}

and for the incidence combinations $C_1^+/C_2^+ = - R_{22}/R_{21}$ and $C_1^+/C_2^+ = - R_{12}/R_{11}$, (4.3) gives

(4.42) \begin{equation} \left. \begin{array}{l@{}} \displaystyle k_1^+ \big( \big| R_{22} \big| ^2 - \big| R_{11}R_{22}- R_{12}R_{21} \big| ^2 \big) +k_2^+ \big| R_{21} \big| ^2 = k_r^+ \big| R_{22}T_{11}- R_{21}T_{12} \big| ^2, \\ \displaystyle k_1^+ \big| R_{12} \big| ^2 + k_2^+ \big( \big| R_{11} \big| ^2 - \big| R_{11}R_{22}- R_{12}R_{21} \big| ^2 \big) = k_r^+ \left| R_{11}T_{12}- R_{12}T_{11} \right| ^2. \end{array}\right\} \end{equation}

Note that $k_r^+$ denotes the wavenumber of the involved cut-on DM in $d2$. In the case in figure 9 $k_r^+=k_1^+$, whereas in figure 22(a,b) $k_r^+$ is unequal to the incident wavenumbers in $d1$. Combining (4.41) and (4.42), and the following reciprocity relation for wave reflection at the waveguide discontinuity (Cho Reference Cho1980):

(4.43)\begin{equation} k_1^+ R_{12} = k_2^+ R_{21}, \end{equation}

gives

(4.44) \begin{equation} k_1^+ \big( \big| T_{12} \big| ^2 - \big| R_{11}T_{12}- R_{12}T_{11} \big| ^2 \big) = k_2^+ \big( \big| T_{11} \big| ^2 - \big| R_{22}T_{11}- R_{21}T_{12} \big| ^2 \big)\!. \end{equation}

Relations (4.40) and (4.44) lead to

(4.45)\begin{equation} \left|\frac{C_1^-}{C_1^+}\right| = \left|\frac{C_2^-}{C_2^+}\right| =1. \end{equation}

So we have proved total reflection of each DM, i.e. no energy interchange between $k_{1}^{\pm }$ DMs and $k_{2}^{\pm }$ DMs, in the two-wave coupled total reflection.

The amplitude ratio of $k_1^+$ and $k_2^+$ DMs (4.36) ensures the zero net transmission (4.37) and the total reflection of each of $k_1^+$ and $k_2^+$ DMs (4.45) at the right end. Due to the total reflection of each of $k_1^+$ and $k_2^+$ DMs and the real wavenumbers of cut-on DMs at real frequencies, the modulus of the amplitude ratio of $k_1^-$ and $k_2^-$ DMs at the left end is the same as that given by (4.36), So, if the following two loop phase constraints are satisfied:

(4.46) \begin{equation} \left. \begin{array}{l@{}} \displaystyle \mathrm{arg}\bigg( \dfrac{C_1^-}{C_1^+} \bigg) - k_1^+L =- m_1 {\rm \pi}, \\[12pt] \displaystyle \mathrm{arg}\bigg( \dfrac{C_2^-}{C_2^+} \bigg) - k_2^+L =- m_2 {\rm \pi}, \end{array}\right\} \end{equation}

then zero radiation at both ends and of the two loops is ensured, where $m_1$ and $m_2$ are integers and they both are even or odd for the $x$-symmetric system in figure 9 so that $k_1^{\pm }$ and $k_2^{\pm }$ loops have the same $x$-symmetry. The difference between the phase changes of the two loops in reflection is compensated by the propagation, thanks to $k_1^{\pm } \neq\, k_2^{\pm }$. With total reflection at both ends, the magnitude constraint of the two loops is also satisfied, at real frequencies.

Now, the difference between the mechanisms of common and accidental TMs can be understood by comparing how the loop zero-radiation condition is satisfied in the two situations. In common TMs that involve multiple propagating-wave loops, loop zero radiation is ensured alone by the particular amplitude ratio between the loops and it is independent of the single loop phase constraint that selects $\omega _{TM}$ as a continuous function of $L$, due to the geometrical properties of the systems. On the other hand, loop zero radiation in accidental TMs depends on not only the amplitude ratio between the propagating-wave loops but also the loop phase constraints and there are two phase constraints. Only at the crossing points of the two phase constraints can zero-radiation loops be ensured, which is the reason for the discrete appearance of accidental TMs in the $L$–$\omega$ two-parameter space.

To illustrate (4.46), the incidence combination $C_1^+ /C_2^+$ in (4.36) and the reflection coefficients $C_1^- /C_1^+$ and $C_2^- /C_2^+$ in (4.38) are calculated by modal matching at the interface, as shown in figure 17(a,b). The two phase constraints in (4.46) lead to two groups of $L$–$\omega$ lines. For the $x$-symmetric system in figure 9, $x$-symmetric and $x$-antisymmetric TMs happen at the crossing points of the two groups of lines with $m_1$ and $m_2$ both being even and odd. The squares in figure 17(c) indicate that (4.46) is accurate for all accidental TMs of the $x$-symmetric system. For the point-symmetric system in figure 10, point-symmetric and point-antisymmetric TMs, two of which are denoted by the diamonds in figure 17(c), happen at the crossing points of the two groups of $L$–$\omega$ lines from (4.46) with $m_1$ and $m_2$ having opposite parities so that $k_1^{\pm }$ and $k_2^{\pm }$ loops have the same point symmetry.

Figure 17. (a,b) Modulus and argument of $C_1^-/C_1^+$ and $C_2^-/C_2^+$ in the interface scattering sketched in figure 12(e) with the incidence combination (4.36). (c) Solid and dashed lines: $L$–$\omega$ relation from (4.46) with $m_{1,2}$ being even and odd. Thick and line lines: $k_1^{\pm }$ and $k_2^{\pm }$ loops. Squares: accidental TMs of the system in figure 9; diamonds: accidental TMs shown in figure 10(a,e).

Figure 12(f) corresponds to the accidental TMs in figure 11(f,k), which happen under necessary conditions that $k_n^{\pm }$ ($n=1,2,3$) DMs are cut-on in $d3$ and $k_n^{\pm }$ ($n=1,2$) DMs are cut-on in $d1$ and $d2$. In figure 12(f), $C_{1,2}^{\pm }$ denote the amplitudes of $k_{1,2}^{\pm }$ DMs in $d1$, $C_{3,4}^{\pm }$ denote the amplitudes of $k_{1,2}^{\pm }$ DMs in $d2$ and $B_{1,2,3}^+$ denote the amplitudes of $k_{1,2,3}^+$ DMs in $d3$. We may understand the TMs as the result of co-protection: the $y$-symmetry protection as in § 4.1 and the most simple accidental TM mechanism above. If the incident waves satisfy

(4.47a,b)\begin{equation} C_1^+=-C_3^+ \quad \mbox{and}\quad C_2^+=C_4^+, \end{equation}

then

(4.48)\begin{equation} B_1^+=B_3^+=0, \end{equation}

owing to the $y$-symmetry of the system. If the incident waves also satisfy

(4.49)\begin{equation} \frac{C_1^+}{C_2^+} =- \frac{T_{22}}{T_{21}}, \end{equation}

where $T_{21}$ and $T_{22}$ respectively denote the transmission from $k_1^+$ and $k_2^+$ DMs in $d1$ to $k_2^+$ DM in $d3$, then

(4.50)\begin{equation} B_2^+=0. \end{equation}

In a similar way, we can obtain total reflection as (4.45) and two phase constraints as (4.46) for this case. The incidence combination $C_1^+ /C_2^+$ and the reflection coefficients $C_1^- /C_1^+$ and $C_2^- /C_2^+$ are shown in figure 18(a,b), and the graphical solution of TM locations in the $L$–$\omega$ two-parameter space is verified by the two squares in figure 18(c).

Figure 18. (a,b) Modulus and argument of $C_1^-/C_1^+$ and $C_2^-/C_2^+$ in the interface scattering sketched in figure 12(f) with the incidence combination (4.47a,b) and (4.49). (c) Solid and dashed lines: $L$–$\omega$ relation from (4.46) with $m_{1,2}$ being even and odd. Thick and line lines: $k_1^{\pm }$ and $k_2^{\pm }$ loops. Squares: accidental TMs shown in figure 11(f,k).