Examination of Extraordinary Transmission of Waves Propagation through Gaps of Vertical Thin Barriers in Channels by A Hypersingular Boundary Element Method

The extraordinary transmission (ET) phenomenon is examined for waves propagating through gaps of vertical thin barriers in channels with a hypersingular boundary element method model on the linear potential theory, and an estimate formula based on small gap approximation for predicting the number of ET frequencies is proposed. Numerical computations are carried out to examine the influences of barrier number, barrier interval, gap size, gap position and barrier arrangement on extraordinary transmission and wave height in the channel. It shows that all of those factors evidently affect the extraordinary transmission frequencies. The contours of wave amplitude show that very high waves can be excited in the basins between barriers at the extraordinary transmission frequencies. Proper arrangement of barriers in a channel can avoid the occurrence of ET phenomenon and reduce wave height in the channel.


Introduction
The extraordinary transmission (ET) of water waves is a phenomenon of greatly enhanced transmission of waves for wave interaction with bodies. The extraordinary transmission may occur for wave interaction with a group of partially arranged vertical barriers or a group of horizontally arranged obstacles with gaps in a channel. At certain frequencies, it may give rise to total transmission of wave energy and larger wave height in the basin between two barriers.
Considering the problem of water waves past over a submerged long obstacle, Newman (1965) derived an approximation analysis by considering the effect of diffraction at each end of the long submerged obstruction, and showed that complete transmission can occur with a certain obstacle length. For the problem of scattering of surface waves by two submerged vertical plane barriers, the extraordinary transmission phenomenon can also happen. As for two submerged vertical parallel plane barriers with an interval b and a submerged depth a, Levine and Rodemich (1958) firstly obtained accurate but complex expressions for the reflection and transmission coefficients in terms of six definite integrals, the integrands of which were complicated functions of elliptic integrals. Under the assumption that the b/a ratio of separation to depth is not very small, Evans and Morris (1972) derived complementary approximations which may be computed easily. Newman (1974) derived an asymptotic solution for a pair of vertical flat plates intersecting the free surface, assuming that the separation between the two plates is small. Evans (1975) considered a similar structure, two vertical parallel flat plates each perforated with a small hole. Under certain conditions, energy can be totally transmitted or totally reflected.
As for the vertical bottom-mounted structures with constant cross section intersecting the free surface of an incompressible fluid of constant depth, the problems of wave scattering have been extensively studied as well. The two-dimensional problems of interaction between obliquely incident plane waves and periodic arrays of circular cylinders, rectangular cylinders and breakwaters were investigated by Linton and Evans (1993), Fernyhough and Evans (1995) and Porter and Evans (1996), respectively. By using a matrix formulation, those scattering problems were solved by representing the solution with unknown coefficients, and total transmission is also found under specific conditions. For wave scattering from bodies in channels, Evans and Porter (2016) investigated the problem of the transmission of plane waves through a periodically-spaced array of thin screens with gaps in a narrow channel by using the small gap approximation method. They demonstrated that total transmission of waves is possible through multiple thin barriers with small gaps.
That work led Evans et al. (2018) speculate zero reflection should also be possible for cylinders in channels as the cylinder approaches the channel walls. Therefore, Evans et al. (2018) investigated wave transmission for two geometries: cylinders of circular and rectangular cross-sections. They repeated the numerical calculations of Linton and Evans (1992a) for cylinders of circular and found that the total transmission exists when the ratio of diameter to channel width is about 0.9. Evans et al. (2018) also performed experiments to measure the reflection and transmission from double vertical half-cylinders mounted against the side of a wave tank. They confirmed that a high proportion of water energy can propagate through a tiny aperture between the two half-cylinders with energy loss because of nonlinear surface and viscous wall effects. For a rectangular cylinder at the center of a channel, they provided a closed form expression for the transmitted wave energy and found that total transmission could happen when a rectangular cylinder occupies almost all width of the narrow channel.
In this study, a hypersingular boundary element method (BEM) model is established for wave scattering from thin vertical barriers with gaps in channels, which is applicable for broad channels, wide gaps, arbitrary gap positions and non-uniformly distributed barriers without those limitations of small gap approximation method, as the auxiliary propagation waves are included in the BEM model and in the computation of reflection and transmission wave energy fluxes. With application of this model, the influences of barrier number, barrier interval, gap size, gap position, and barrier arrangement on wave transmission are examined, and the patterns of wave amplitude in basins between two barriers at ET frequencies are also plotted. An estimation formula is derived based on small gap approximation for predicting the number of ET frequencies in a channel with a series of uniformly distributed barriers. From the numerical examinations, it is found that at ET frequencies, the wave amplitude inside basins may be several times the incident wave amplitude due to the water resonance inside basins, and ET phenomenon can be avoided by suitable arrangement of barriers. Those results have practical significance in breakwater planning for ports along estuaries.

Mathematical formulations
The problem of wave normal transmission through a channel distributed with barriers is studied, as shown in Fig. 1. A Cartesian coordinate system Oxyz is defined that the x-axis is at the centerline of the channel, the y-axis is in the transverse direction, and the z-axis is upright with its origin on the undisturbed water plane. The water domain is surrounded by the channel wall and the wetted barrier surface . A series of barriers perpendicular to the channel walls are located at . Each barrier contains a gap of the same width 2a and the coordinates of gap center are such that . Particularly when barriers are arranged uniformly, the barrier intervals are the same as .
The potential theory is applied to study this problem. On the linear approximation, the time factor and the vertical eigenfunction can be separated out for the problem of monochromatic wave scattering from erect uniform bodies. Therefore, the velocity potential and wave elevation can be written as: where h is the water depth. ϕ(x, y) The horizontal complex potential satisfies the Helmholtz equation in the fluid, where wave number k is the positive real root of the dispersion relationship with wave frequency as: The wave elevation can be calculated by wave potential as The wave potential can be decomposed into the incident potential and the diffraction potential as: The incident potential for waves with an amplitude of A is in the form The diffraction potential satisfies the non-penetration conditions S W on the channel walls , and S B on the barrier surface . When satisfying the channel wall condition Eq. (8), the diffraction potential can be expanded with its eigen functions as: at the wave side of the barriers, and as at the lee side of the barriers, where and are the amplitudes of each wave modes to be determined. The eigenvalues for the transverse oscillations are , j = 0, 1, 2, · · · (12) Correspondingly, the eigenvalues for the longitudinal oscillation are When the channel width 2d is larger than the wave length and k is in the range of , Eq. (13) will have J imaginary roots. For the sake of convenience, those longitudinal eigenvalues are defined as: . Thus, at a large distance from the barriers in the channel, the diffraction potential can be approximated as:

Hypersingular BEM for vertical thin barriers in a channel
S B Application of the second Green's theorem to the diffraction potential and channel Green's function leads to the following Fredholm integral equation over the body surface x 0 (x, y) x (x, y) α n where and are the coordinates of the source and filed points, is the solid angle of fluid domain, and is the unit normal vector pointing out of the fluid domain.
If the barrier under consideration has negligible thickness, the normal directions at its two sides are opposite By application of this condition, the integral equation (16) is reduced to (Linton and McIver, 2001) where is the difference of the diffraction potentials on the positive and negative sides of a barrier. Allocating the source point on the body surface and taking normal derivative of the integral equation (18) at the source point yield with the body surface condition Eq. (9), where HD denotes the Hadamard integration. In Eq. (19), the kernel of integrand includes a hypersingularity with an order of when . To evaluate the hypersingular integrals, we use the method proposed by Guiggiani (1998) with a semi-analytical technique of general applicability for the direct evaluation of hypersingular integrals in the boundary element implementation.
In the numerical evaluation of Eq. (19), the positive body surfaces are divided into a number of straight line segments, and on each of them is set as a constant. Then, a hypersingular BEM model can be set up. After solving Eq. (19), the difference of the diffraction potentials on barrier's two sides can be determined.
x 0 Diffraction potential in the fluid domain can be computed directly by using the integral equation (18) and the corresponding wave elevation can be obtained by Eq. (5). When is located on the barrier surface, the integral equation will give the average of the diffraction potentials on the positive and negative sides of barriers as follows (see Appendix A for detail derivation) ds.

(20)
The diffraction potentials at two sides of a barrier can be obtained by combining the summation and difference of the diffraction potentials on the two sides of the barrier. Then, the wave elevation at the barrier surface can be determined.
Moreover, the channel Green's function, applied in the above integral equations, satisfying the channel wall boundary conditions (8) and the Helmholtz equation (3) except at the source point can be written in terms of a function of dimensionless variables as: , and . The image series are taken as: where , and is the Hankel function of the first kind. On account of the slow convergence of the series, it is usually ineffective to apply it directly from the perspective of numerical computation. Linton (1998) has summarized various analytic techniques to speed up the numerical evaluation of the two-dimensional Green's function for Helmholtz equation in a periodic domain and highlighted Ewald's method as being superior to other representations.

Reflection and transmission coefficients in channels
Reflection and transmission are far field properties of wave propagation in a channel. At far field from the source points in the channel, the channel Green's function can be approximated as (Linton and Evans, 1992b): With substitution of the diffraction potential Eq. (15) and the Green's function (23) into the integral equation (18), the amplitude of each propagation wave mode of reflection and transmission waves can be computed by The mean wave energy flux across a channel section can be calculated by the integration of mean wave power over a channel section as: where is the wave group velocity, and the superscript * represents the conjugate.
At the wave side of the channel, the wave energy flux across a section can be divided into the fluxes of incident and reflection waves as: (28) With the substitute of the incident and reflection potentials into Eq. (27), the fluxes of the incident and reflection waves are derived as Analogously, the wave energy flux across the lee side of the channel is All the propagating mode coefficients are correlated to energy propagation in Eqs. (30) and (31).
Following Srokosz (1980), the wave energy fluxes across the wave and lee sides of the channel should satisfy the energy conservation law. Thus, the propagating modes of the reflection and transmission waves satisfy the following equation This is a very helpful relationship for checking the accuracy of numerical solutions. For convenience in comparison, we define the non-dimensional energy of the reflection and transmission waves as: The reflection and transmission coefficients are defined by the ratios of the energy fluxes of reflection and transmission waves with that of incident waves as: For validation, the hypersingular BEM model developed in this research was firstly checked by the energy conservation relationship, and compared with Evans and Porter's (2016) small gap approximation solution and Porter and Evans's (1996) eigen-function expansion results for two pairs of thin barriers with gaps. Then, numerical computations are carried out to examine the influences of barrier number, barrier interval, gap size, gap position, and barrier arrangement on wave transmission and wave elevation in channels. When the gaps are arranged at the center of the channel, , only half of the channel is computed for velocity potential due to the geometry symmetry. In all of the figures, only the transmission coefficient is plotted for brevity, as the computed results of the present model have been checked by the energy conversion relationship (Eq. (32)).

Validation
To demonstrate the accuracy of the presented method, wave scattering problems for two pairs of identical barriers in a channel with the same gap size but different barrier intervals are examined. In Fig. 2, the non-dimensional energy factors are plotted and it can be seen that the energy conser-vation relationship is satisfied very well.
Then the present method is compared with Evans and Porter's (2016) small gap approximation (SGA). It can be seen that the present results agree very well with the results of SGA method for the cases of channels with relatively large interval b/d=1.0 and 0.5 (Figs. 2a and 2b). However, there are some divergences between the present results and SGA results, and the divergence increases with the decrease of the interval b/d between two barriers for the cases of channels with relatively small interval b/d=0.25 and 0.125 (Figs. 2c and 2d).
To further validate the present model, Porter and Evans's (1996) results from eigen-function expansion with Galerkin approximation method are included in the figures. It can be seen that the agreement between the present method and Porter & Evans's is much better. The reason for the divergences has been indicated by Evans and Porter (2016) as: 'the loss of agreement is likely to be because the small gap assumption ( ) is forced to compete with a comparable dimensionless length scale (i.e. ) which has not been subjected to a similar or consistent approximation'.
The fact is that the present method satisfies the energy conversion relationship well and has good agreement with SGA method for channels with larger barrier interval b/d, and good agreement with Porter and Evans's (1996) for channels with small interval b/d. These prove that the present hypersingular BEM model is accurate and available for practical application. Fig. 3 shows curves of the transmission coefficients in

Effect of barrier number
channels with N pairs of barriers versus non-dimensional wave number . The barriers are distributed equally with an interval of and the gaps are located at the center of the channel. From the results, it can be clearly seen that perfect transmission may occur when there is only one propagating mode and the number of ET frequencies is except for the zero frequency. It means that the total transmission can only happen when there are two or more barriers, which is consistent with Evans and Porter (2016). The ET frequencies with different number of barriers are summarized in Table 1. The fundamental total transmission frequency decreases with the increase of barrier number, and it seems to be related with the total basin length .
kd > π In addition, although total transmission ceases to exist at large wave numbers ( ), enhanced transmission could appear due to the generation of auxiliary propagation waves, and the number of high transmission peaks also depends on the number of barriers.
In the cases of four pairs of barriers distributed in a channel with , and , the contours of wave elevation at three ET frequencies are illustrated in Fig. 4. It can be seen that the wave amplitude inside basins is much larger than the incident wave amplitude at ET frequencies, and the wave amplitude at higher ET frequency is larger than that at lower ET frequency. The maximum wave amplitude in the basin is 4.5 times the incident wave amplitude at the highest ET frequency. While outside the basins, for and , the wave amplitude is nearly equal to the incident wave amplitude at the ET frequencies. More specifically, the wave amplitude in the first and third basins increases with the increase of ET frequencies.
However, the wave amplitude in the middle basin is larger than that in the first and third basins at the first and third ET frequencies (kd=0.42 and 1.06), but lower than that at the second ET frequency (kd=0.80). Those phenomena are due to the reason that wave oscillations have three different modes in the channel with four periodically arranged thin barriers. Animation examinations show that the wave oscil-lations in the first and third basins are out of phase at the second ET frequency, but are in phase at the third ET frequency. Fig. 5 shows the variation of the wave transmission coefficients for the case of N pairs of barriers uniformly arranged in channels with different barrier intervals. The barriers have the same size gaps at the center of the channel, . To study the phenomenon of extraordinary transmission, the wave transmission coefficient corresponding to one propagation mode is only plotted in the figures. It can be seen that for there are only ET frequencies in the frequency range of , while for there are groups of ET frequencies in the frequency range and each group has ET frequencies. It means that the number of ET frequencies depends not only on the number of barriers but also on the interval of barriers. N ET Actually, an estimation formula based on small gap approximation to estimate the number of ET frequencies can be derived from Evans and Porter (2016). After some algebra, the number of ET frequencies can be expressed as (see Appendix B for detail derivation)  Fig. 6 shows the contours of wave elevation at the fundamental ET frequencies for channels with different barrier intervals, , 0.25 and 0.5, but with the same barrier number and gap size . It can be seen that the wave amplitude at the ET frequency is much larger than    Fig. 7 plots the contours of wave elevation at the 4 th , 5 th and 6 th ET frequencies in channels with , and

Effect of barrier interval
. It can be seen that wave oscillations have other modes, which are similar to the oscillations in a closed rectangular basin at their eigenfrequencies. In a closed rectangular basin, there exists a wave node at the transverse centerline of the basin. However, in the basin between barriers with gaps in a channel, the wave amplitude of 'wave node' is not zero and the ET frequencies are a little bit higher than the eigenfrequencies of the closed rectangular basin. TENG Bin, HUANG Jin China Ocean Eng., 2019, Vol. 33, No. 5, P. 509-521 515

Effect of gap size
approximation for symmetrically arranged barriers, , in channels with the same barrier interval but different gap sizes . From the comparison it can be seen that for the cases with smaller gap size the two methods agree very well, but for larger gap size the two methods diverge. For the present computation, the small gap assumption is reliable for .

(d − a)
From the figures, it can also be seen that the gap width has evident influence on ET frequencies. With the increase of gap width, the ET frequencies increase and the high transmission range also broadens. Table 2 lists the fundamental ET frequencies for channels with different gap widths. It shows that the ET frequency increases with the increasing gap size, as the transverse oscillation of water wave in basins is related to the barrier length .

Effect of gap position
We now consider a problem in which gaps are not ar- ranged at the centerline of a channel. As it is not a geometry symmetric problem anymore, both barriers beside a gap are discretized in BEM computation. Four asymmetric cases with different gap positions are computed, and compared with the symmetric case. In all computations, four pairs of barriers with gap width of are distributed uniformly in the channel and barrier interval is . Table 3 lists the positions of gaps on barriers for the symmetric and asymmetric arrangements. In Case 0 the gaps are positioned at the centerline of the channel, in Case 1 and Case 2 gaps are arranged in a transverse staggered manner, and in Case 3 and Case 4 gaps are distributed symmetrically about the middle basin.
(N − 1) kd ∈ (0, π) kd ∈ (π/2, π) Fig. 9 shows the comparison of the transmission coefficients for different arrangements of gaps. It can be seen that ET phenomenon still exists and there are ET frequencies in the frequency range of although the gap positions are not in alignment. However, by comparing the symmetric arrangement of Case 0, there is a significant change of ET frequencies to lower frequency when gaps are located near a channel wall. It can also be seen that other enhanced transmission could appear in the frequency range of as one of the barriers will be longer than the corresponding one in the symmetrical arrangement.
The contours of wave elevation at ET frequencies for the computation cases are plotted in Fig. 10. By comparing Fig. 4, it can be seen that the maximum wave amplitude of each basin has a slight increase as gap position moves toward the channel wall. The location of gap affects the wave amplitude distribution in each basin, and in general, the maximum wave amplitude appears at the longer barrier side. As mentioned previously, one aim of the present research is to investigate if the ET phenomenon could be avoided, or ET frequency could be moved to other frequency, so as to provide design plans for arrangement of breakwaters along a channel or an estuary. Fig. 11 shows the results for channels with different arrangements of barriers with center-located gap of width . In all computations, four pairs of barriers are distributed in channels and the total barrier span is . Eight arrangements are computed, and barrier locations for the eight cases are listed in Table 4. Fig. 10. Contours of wave elevation at ET frequencies for different gap positions in channels with , and .   Table 3 Gap positions for the computation cases Figs. 11a and 11b show the trends of transmission coefficients against kd for the first four cases, where the barriers are located symmetrically about and the middle basin length increases gradually. From the computation results, it can be seen that the distribution of the barriers has little effect on the first ET frequency, as are the same, while the second and third ET frequencies are affected obviously. There are ET frequencies in Cases 2 and 3, where all three barrier intervals have comparable scale to channel width. But for Case 1 and Case 4, it has more than ET frequencies which occur at higher frequencies because of some long barrier intervals being.
Figs. 11c and 11d show the variations of transmission coefficients versus kd for the last four cases, where the barriers are arranged non-uniformly. It can be seen that for some non-uniform arrangements, the ET phenomenon can be avoided, and replaced by some high peaks in transmission coefficients. The peak values decrease with the increase of wave number kd. In addition, it can also be seen that the arrangements of barriers have little influence on the first peak frequency, but evident influence on the second and third peak frequencies.
Furthermore, the maximum wave amplitude at peak frequencies in a channel with non-uniformly distributed barriers is investigated. Fig. 12 shows the contours of wave elevation at peak frequencies for Case 5, which can be compared with the uniformly distributed case, Case 2, as plotted in Fig. 4. The wave amplitude in the basins for Case 5 is slightly larger than that for Case 2 at the first peak/ET frequency, while the wave amplitude for Case 5 is notably smaller than that for Case 2 at the second and third peak/ET frequencies, especially for relatively wide basin in Case 5.
Through the examination, it can be concluded that with careful barrier arrangement plan, the occurrence of ET phenomenon can be avoided, the low transmission range can be widened, and the wave amplitude in a channel can be reduced, which is significant in practical application to re- Fig. 11. Wave transmission coefficients for different arrangements of barriers in channels with , , and . Fig. 12. Contours of wave elevation at peak frequencies for Case 5. duce the intensity of wave action in ports along estuaries.

Conclusions
Water waves propagating through gaps of vertical thin barriers in channels are studied in the present research with a hypersingular BEM model. By distributing dipoles on the positive side of barrier surface with channel Green's functions, a Hadamard integral equation is established and solved numerically to achieve velocity potential. With the application of approximate diffraction potential and Green's function at large distance from bodies, the reflection and transmission coefficients are derived. The model is greatly flexible for different barrier distribution in a channel. Numerical calculations are carried out to examine the influences of the barrier's number, barrier interval, channel width, gap size, gap position and barrier arrangement on wave transmission and wave amplitude in channels. A formula is also proposed to predict the number of ET frequencies in a channel.
It is found that all those factors have influence on wave ET frequencies, and total transmission occurs only when there is one propagating mode. The number of ET frequencies depends on barrier's number and barrier interval. The fundamental ET frequency is related to the total span of barrier intervals, but a little affected by the arrangement of barriers, and higher ET frequencies are related to barrier interval. At higher frequency auxiliary propagation waves may occur in a wider channel, and enhanced transmission will appear at those frequencies.
Inside a basin, the wave elevation at the highest ET frequency is higher than that at the fundamental ET frequency as the near resonance oscillation occurs at the ET frequency in the basins. In a channel with smaller barrier interval or larger channel width, larger wave height may appear. The spatial position of the maximum wave amplitude depends on the wave oscillation mode in the basins. At ET frequencies, the wave amplitude is larger in a wide channel than that in a narrow channel. But the position of the gap does not have much influence on the wave amplitude.
Non-uniform arrangement of barriers can avoid wave total transmission, or change the positions of higher ET frequencies. Especially the enhanced transmission frequency band can be reduced by proper barrier arrangement. It provides a possibility to avoid large wave transmission and wave elevation in ports along estuaries by suitable arrange-ment of breakwaters.
al thin barrier placed arbitrarily in a channel. The water domain is divided into two parts: and . is the barrier surface and is the interface of the two domains. Their two sides are defined as and .
is the channel wall. In the domain , we apply the diffraction potential and the channel Green's function to the second Green theorem, and then obtain an integration equation for the diffraction potential at the positive side of the barrier as follows: In the same way, we can establish an integral equation for the diffraction potential at the negative side of the barrier as: Adding Eqs. (A1) and (A2) together, we can obtain an integral equation for the average of diffraction potentials on the two sides of the barrier as: with applying the relation of normal directions on barrier surfaces , and the relations To combine the summation Eq. (A3) of the diffraction potentials on the two sides of the barrier with the difference Eq. (19) of them, the potential on each side of the barrier can be determined.

Appendix B: Number of ET frequencies
Considering several vertically thin barriers placed uniformly in a channel, Evans and Porter (2016) derived the reflection and transmission coefficients analytically by small gap approximation as: where p = 2 tan δ − cot(kb) + E 2 ; (B3) q = − 1 sin(kb) When barrier interval b is large enough relative to the channel width d, and tend to zero so that the reflection and transmission coefficients can be written as  . For a prescribed , the periodic function (B11) has a period of . Fig. B2 shows the variation of versus with different based on Eq. (B10). It can be seen that for different , has different periods, and different number of zeros in the range of . For example, has zeros at , and zeros at .
When is not an integer, there are zeros, where represents the largest integer not exceeding . Thus, for N periodically arranged thin barriers in a channel, the number of ET frequencies can be estimated by When barrier interval is smaller than channel width , it is reasonable to conclude that is equal to as shown by the numerical results in Fig. 5.