3.1. An underlying curved coast

Figure 1.

The curvilinear coordinate system ( ${\it\sigma},{\it\eta}$ ). The solid line denotes the coast (Dirichlet boundary condition) and the dashed line represents the shelf–ocean boundary.

Consider a smoothly curving shelf and follow Johnson et al. (Reference Johnson, Levitin and Parnovski2006) by introducing curvilinear coordinates $({\it\sigma},{\it\eta})$ , as in figure 1, with ${\it\sigma}$ the arclength along the coast, ${\it\eta}$ the coordinate offshore and ${\it\gamma}({\it\sigma})$ the signed curvature of the coast. Let $L$ be a typical cross-shelf length scale associated with each position ${\it\sigma}=\text{constant}$ along the coast. Let $\ell ^{\ast }$ be the alongshore length scale of any localised mode. The small parameter in the expansion below is then taken to be

(3.1) $$\begin{eqnarray}{\it\epsilon}=\frac{\ell ^{\ast }}{{\it\gamma}}\frac{\partial {\it\gamma}}{\partial {\it\sigma}}.\end{eqnarray}$$

This does not require the shelf to be narrow compared to $\ell ^{\ast }$ : small ${\it\epsilon}$ means that the curvature changes by only a small fraction of its value along the section of shelf supporting the $\ell$ CSW. In the limit ${\it\epsilon}\rightarrow 0$ the geometry reduces to an island of fixed radius, $1/{\it\gamma}$ . In this limit, provided the shelf profile does not vary too strongly, modes propagate freely around the entire island (Rhines Reference Rhines1969b ). Trapped modes in the asymptotic limit $0<{\it\epsilon}\ll 1$ require a section of increased curvature (or increased slope or coast–shelf-break displacement (Johnson & Kaoullas Reference Johnson and Kaoullas2011)) where the local frequency ${\it\omega}$ is only of order ${\it\epsilon}$ above the cutoff frequency ${\it\omega}_{c}$ . Then at some distance of order $\ell ^{\ast }$ from the propagating region the wave becomes cut off and evanescent, permitting a trapped $\ell$ CSW. The geometry outside the region of support of the $\ell$ CSW is immaterial to the trapping (except for the possibility of tunnelling (Stocker & Johnson Reference Stocker and Johnson1991), discussed below): once the wave becomes evanescent at cutoff its energy flux falls to zero and energy is reflected, giving a trapped mode. There are two distinct cases. Firstly, ${\it\epsilon}$ could fall to zero outside the trapping region. The two arms of the shelf outside the trapping region would then rejoin, giving an $\ell$ CSW trapped on a section of the coast of an island. Alternatively, the shelf arms could straighten, corresponding to ${\it\gamma}\rightarrow 0$ and ${\it\epsilon}\rightarrow \infty$ . Since the $\ell$ CSW is already evanescent, with zero energy flux, in this region the effect on the trapped mode is negligible. Other variations in shelf geometry lie between these two and thus give trapping with significant tunnelling only in the unlikely possibility that the shelf after cutoff rapidly returns to a geometry that allows propagating waves at the $\ell$ CSW frequency.

Introduce ${\it\xi}={\it\epsilon}{\it\sigma}$ and let ${\it\xi}^{\prime }={\it\xi}/L$ and ${\it\eta}^{\prime }={\it\eta}/L$ , where $L$ is the shelf width, then, allowing also for depth profiles varying alongshelf over the length scale of ${\it\epsilon}^{-1}$ , the non-dimensional governing equations (dropping the primes) are

(3.2) $$\begin{eqnarray}{\it\omega}[{\it\epsilon}^{2}{\it\kappa}^{2}{\it\Phi}_{{\it\xi}{\it\xi}}+{\it\Phi}_{{\it\eta}{\it\eta}}-{\it\epsilon}^{2}({\it\kappa}^{3}{\it\eta}{\it\gamma}_{{\it\xi}}+{\it\kappa}^{2}{\it\beta}_{{\it\xi}}){\it\Phi}_{{\it\xi}}-({\it\beta}_{{\it\eta}}-{\it\kappa}{\it\gamma}){\it\Phi}_{{\it\eta}}]+\text{i}{\it\epsilon}{\it\kappa}({\it\beta}_{{\it\xi}}{\it\Phi}_{{\it\eta}}-{\it\beta}_{{\it\eta}}{\it\Phi}_{{\it\xi}})=0,\end{eqnarray}$$

subject to (2.7). Here ${\it\beta}({\it\xi},{\it\eta})=\ln H({\it\xi},{\it\eta})$ and ${\it\kappa}=(1+{\it\eta}{\it\gamma})^{-1}$ .

3.1.1. The long WKBJ approximation

For typical rectilinear depth profiles on a straight coast, such as the exponential profile of Buchwald & Adams (Reference Buchwald and Adams1968), there is a maximum frequency of propagation ${\it\omega}_{max}$ so that for ${\it\omega}<{\it\omega}_{max}$ there are two roots for the wavenumber corresponding to two waves with unidirectional phase propagation and bi-directional energy propagation. The energy propagation associated with the long waves propagates with the phase whereas the energy associated with the short waves propagates in the opposite direction. Once the frequency exceeds ${\it\omega}_{max}$ the two solutions for the wavenumber form a complex conjugate pair and the modes are evanescent. Alongshore variations in shelf geometry or coastline curvature mean that a CSW mode can propagate, as a superposition of two waves carrying energy in opposite directions, within some finite region of the coast but be evanescent outside this region. This prompts the ansatz

(3.3) $$\begin{eqnarray}{\it\Phi}({\it\xi},{\it\eta})={\it\phi}^{+}({\it\xi},{\it\eta})\exp [\text{i}S^{+}({\it\xi})/{\it\epsilon}]+{\it\phi}^{-}({\it\xi},{\it\eta})\exp [\text{i}S^{-}({\it\xi})/{\it\epsilon}],\end{eqnarray}$$

where each term in (3.3) is an independent solution of (3.2). Expanding the amplitudes in powers of ${\it\epsilon}$ , i.e.

(3.4) $$\begin{eqnarray}{\it\phi}^{\pm }=\mathop{\sum }_{j=0}^{j=\infty }{\it\epsilon}^{j}{{\it\phi}_{j}}^{\pm }({\it\xi},{\it\eta}),\end{eqnarray}$$

and substituting (3.3) and (3.4) into (3.2) leads to a hierarchy of equations. The leading order, of order ${\it\epsilon}^{0}$ , gives

(3.5) $$\begin{eqnarray}{\it\omega}[{\it\phi}_{0{\it\eta}{\it\eta}}^{\pm }-({\it\beta}_{{\it\eta}}-{\it\kappa}{\it\gamma}){\it\phi}_{0{\it\eta}}^{\pm }]+[{\it\beta}_{{\it\eta}}{\it\kappa}S_{{\it\xi}}^{\pm }-{\it\omega}{S_{{\it\xi}}^{\pm }}^{2}{\it\kappa}^{2}]{\it\phi}_{0}^{\pm }=0,\end{eqnarray}$$

and the next order, of order ${\it\epsilon}$ , gives

(3.6) $$\begin{eqnarray}\displaystyle & & \displaystyle {\it\omega}[{\it\phi}_{1{\it\eta}{\it\eta}}^{\pm }-({\it\beta}_{{\it\eta}}-{\it\kappa}{\it\gamma}){\it\phi}_{1{\it\eta}}^{\pm }]+[{\it\beta}_{{\it\eta}}{\it\kappa}S_{{\it\xi}}^{\pm }-{\it\omega}{S_{{\it\xi}}^{\pm }}^{2}{\it\kappa}^{2}]{\it\phi}_{1}^{\pm }=-{\it\omega}\text{i}2{\it\kappa}^{2}S_{{\it\xi}}^{\pm }{\it\phi}_{0{\it\xi}}^{\pm }-\text{i}{\it\omega}{\it\kappa}^{2}S_{{\it\xi}{\it\xi}}^{\pm }{\it\phi}_{0}^{\pm }\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\text{i}{\it\omega}{\it\kappa}^{3}{\it\eta}{\it\gamma}_{{\it\xi}}{\it\phi}_{0}^{\pm }S_{{\it\xi}}^{\pm }+\text{i}{\it\omega}{\it\kappa}^{2}{\it\beta}_{{\it\xi}}S_{{\it\xi}}^{\pm }{\it\phi}_{0}^{\pm }+\text{i}{\it\kappa}{\it\beta}_{{\it\eta}}{\it\phi}_{0{\it\xi}}^{\pm }-\text{i}{\it\kappa}{\it\beta}_{{\it\xi}}{\it\phi}_{0{\it\eta}}^{\pm }.\end{eqnarray}$$

System (3.5), (2.7) is precisely the system that determines the local dispersion relation at each station ${\it\gamma}=\text{constant}$ along the coast.

Let ${\it\psi}({\it\xi},{\it\eta})^{\pm }$ be the two local eigenmodes, with corresponding eigenvalues $k^{\pm }$ ( $k^{+}>k^{-}$ ), of the problem in the cross section $\mathscr{D}({\it\xi})=\{({\it\xi},{\it\eta}):0\leqslant {\it\eta}\leqslant \partial \mathscr{D}\}$ of the waveguide:

(3.7) $$\begin{eqnarray}{\it\omega}[{\it\psi}_{{\it\eta}{\it\eta}}^{\pm }-({\it\beta}_{{\it\eta}}-{\it\kappa}{\it\gamma}){\it\psi}_{{\it\eta}}^{\pm }]+[{\it\beta}_{{\it\eta}}{\it\kappa}k^{\pm }-{\it\omega}{k^{\pm }}^{2}{\it\kappa}^{2}]{\it\psi}^{\pm }=0,\end{eqnarray}$$

subject to (2.7), where the ${\it\psi}^{\pm }$ are normalised so that

(3.8) $$\begin{eqnarray}\int _{0}^{\partial \mathscr{D}}H^{-1}{\it\kappa}{{\it\psi}^{\pm }}^{2}\,\text{d}{\it\eta}=1.\end{eqnarray}$$

Then ${\it\phi}_{0}^{\pm }$ can be expressed as an undetermined multiple of the local eigenmodes ${\it\psi}^{\pm }$ , i.e.

(3.9) $$\begin{eqnarray}{\it\phi}_{0}^{\pm }({\it\xi},{\it\eta})=f_{0}^{\pm }({\it\xi}){\it\psi}^{\pm }({\it\xi},{\it\eta}),\end{eqnarray}$$

with the functions $S_{{\it\xi}}^{\pm }$ given by the two roots $k^{\pm }({\it\xi})$ for the wavenumber. It is convenient to introduce $P({\it\xi})$ , $Q({\it\xi})$ defined as

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle P({\it\xi})=\frac{1}{2}[S^{+}({\it\xi})+S^{-}({\it\xi})]=\frac{1}{2}\int ^{{\it\xi}}[k^{+}({\it\tau})+k^{-}({\it\tau})]\,\text{d}{\it\tau}, & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle Q({\it\xi},{\it\xi}_{0})=\frac{1}{2}[S^{+}({\it\xi})-S^{-}({\it\xi})]=\frac{1}{2}\int _{{\it\xi}_{0}}^{{\it\xi}}[k^{+}({\it\tau})-k^{-}({\it\tau})]\,\text{d}{\it\tau}. & \displaystyle\end{eqnarray}$$

Then $P$ gives the unidirectional (fast) phase and $Q$ is proportional to the (slow) group velocity (which vanishes at the turning points defined by $k^{+}=k^{-}$ ). The lower limit of integration in (3.11), the phase reference level (Heading Reference Heading1962; Berry & Mount Reference Berry and Mount1972), is determined by the location of the transition points where the group velocity $c_{g}=0$ . Since the phase $P$ is unidirectional and continuous over the whole domain the matching of $P$ across the singular regions requires only multiplication by an arbitrary complex constant of modulus one. Therefore the lower limit of integration has been omitted from (3.10) and the localised CSW can be regarded as a mode with a unidirectional phase propagating through a slowly varying envelope defined by the direction of energy propagation. An expression for the group velocities $c_{g}^{\pm }$ of the propagating waves follows from multiplying (3.7) by $({\it\kappa}H)^{-1}{\it\psi}^{\pm }$ and integrating over the region $0\leqslant {\it\eta}\leqslant \partial \mathscr{D}$ , to give

(3.12) $$\begin{eqnarray}A^{\pm }-2{\it\omega}k^{\pm }=c_{g}^{\pm }I^{\pm },\end{eqnarray}$$

where

(3.13a ) $$\begin{eqnarray}\displaystyle & \displaystyle A^{\pm }=\int _{0}^{\partial \mathscr{D}}{\it\beta}_{{\it\eta}}H^{-1}{{\it\psi}^{\pm }}^{2}\,\text{d}{\it\eta}, & \displaystyle\end{eqnarray}$$

(3.13b ) $$\begin{eqnarray}\displaystyle & \displaystyle I^{\pm }={k^{\pm }}^{2}+\int _{0}^{\partial \mathscr{D}}({\it\kappa}H)^{-1}({\it\psi}_{{\it\eta}}^{\pm })^{2}\,\text{d}{\it\eta}. & \displaystyle\end{eqnarray}$$

The inhomogeneous eigenvalue problem (3.6), (2.7) is solvable if the right-hand side of (3.6) is orthogonal to the eigenfunctions of the corresponding homogeneous adjoint operator (Nayfeh Reference Nayfeh1993). Instead of constructing the adjoint problem, the operator on the left-hand side can be transformed into self-adjoint form by multiplying by $({\it\kappa}H)^{-1}$ . The eigenfunctions of the transformed self-adjoint operator are then ${\it\psi}^{\pm }$ . Multiplying (3.6) by $({\it\kappa}H)^{-1}{\it\psi}^{\pm }$ and integrating across the shelf gives the solvability condition

(3.14) $$\begin{eqnarray}2(A^{\pm }-2{\it\omega}S_{{\it\xi}}^{\pm })f_{0{\it\xi}}+(A^{\pm }-2{\it\omega}S_{{\it\xi}}^{\pm })_{{\it\xi}}f_{0}=0,\end{eqnarray}$$

which for the propagating modes gives the conservation of the alongshore kinetic energy flux,

(3.15) $$\begin{eqnarray}2c_{g}^{\pm }I^{\pm }{f_{0}}_{{\it\xi}}+(c_{g}^{\pm }I^{\pm })_{{\it\xi}}f_{0}=0,\end{eqnarray}$$

with solution, to within an arbitrary multiplicative constant,

(3.16) $$\begin{eqnarray}f_{0}^{\pm }=|c_{g}^{\pm }I^{\pm }|^{-1/2}.\end{eqnarray}$$

For the evanescent modes $f_{0}$ is given by

(3.17) $$\begin{eqnarray}f_{0}^{\pm }=[A^{\pm }-2{\it\omega}S_{{\it\xi}}^{\pm }]^{-1/2},\end{eqnarray}$$

so that $f_{0}$ remains on the same branch of its complex square root (since $A$ and $S$ are complex in the evanescent regions).

Equation (3.15) shows that the WKBJ solutions break down in the neighbourhood of the transition points $c_{g}^{\pm }=0$ , denoted here by $\pm {\it\xi}_{c}$ . Therefore in the interval $(-{\it\xi}_{c},{\it\xi}_{c})$ , excluding the width of order ${\it\epsilon}^{2/3}$ near the endpoints (Bender & Orszag Reference Bender and Orszag1978), the first-order WKBJ solution is a superposition of the forward and propagating waves given by

(3.18) $$\begin{eqnarray}\displaystyle {\it\Phi}({\it\xi},{\it\eta}) & = & \displaystyle \left\{{\it\alpha}_{1}f_{0}^{-}{\it\psi}^{-}({\it\xi},{\it\eta})\text{exp}\left[-\frac{\text{i}}{{\it\epsilon}}Q({\it\xi},-{\it\xi}_{c})\right]\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,{\it\alpha}_{2}f_{0}^{+}{\it\psi}^{+}({\it\xi},{\it\eta})\text{exp}\left[\frac{\text{i}}{{\it\epsilon}}Q({\it\xi},-{\it\xi}_{c})\right]\right\}\text{exp}\left[\frac{\text{i}}{{\it\epsilon}}P({\it\xi})\right],\end{eqnarray}$$

where $P$ and $Q$ are given by (3.10) and (3.11) respectively. The solution decaying in ${\it\xi}<-{\it\xi}_{c}$ is

(3.19) $$\begin{eqnarray}{\it\Phi}({\it\xi},{\it\eta})=C_{1}f_{0}^{-}{\it\psi}^{-}({\it\xi},{\it\eta})\text{exp}\left[\frac{\text{i}}{{\it\epsilon}}P({\it\xi})-{\displaystyle \frac{1}{{\it\epsilon}}}|Q(-{\it\xi}_{c},{\it\xi})|\right].\end{eqnarray}$$

For ${\it\xi}>-{\it\xi}_{c}$ , the WKBJ connection formula (e.g. (3.24) of Berry & Mount (Reference Berry and Mount1972)) gives

(3.20) $$\begin{eqnarray}{\it\Phi}({\it\xi},{\it\eta})=2C_{1}f_{0}^{-}{\it\psi}^{-}({\it\xi},{\it\eta})\text{exp}\left[\frac{\text{i}}{{\it\epsilon}}P({\it\xi})\right]\cos \left[\frac{1}{{\it\epsilon}}Q({\it\xi},-{\it\xi}_{c})-\frac{{\rm\pi}}{4}\right].\end{eqnarray}$$

Note that (3.20) is not of the form (3.18) for the entire interval $(-{\it\xi}_{c},{\it\xi}_{c})$ . To satisfy both the connection formula and governing equations consider the overlap region defined by $-{\it\xi}_{c}+{\it\epsilon}^{2/3}<{\it\xi}<-{\it\xi}_{c}+{\it\epsilon}^{{\it\delta}}$ where ${\it\delta}<2/3$ . In this region $|{\it\xi}-(-{\it\xi}_{c})|$ is small so $[k^{+}({\it\xi})-k^{-}({\it\xi})]^{2}\sim a({\it\xi}-(-{\it\xi}_{c}))$ , where $a$ is a positive constant. Therefore $S_{{\it\xi}}^{\pm }\sim k({\it\xi}_{c})\pm a^{1/2}({\it\xi}-(-{\it\xi}_{c}))^{1/2}$ , and

(3.21) $$\begin{eqnarray}c_{g}^{\pm }I^{\pm }\sim \mp {\it\omega}a^{1/2}({\it\xi}-(-{\it\xi}_{c}))^{1/2}.\end{eqnarray}$$

Thus

(3.22) $$\begin{eqnarray}f_{0}^{\pm }\sim |{\it\omega}a^{1/2}({\it\xi}-(-{\it\xi}_{c}))^{1/2}|^{-1/2}.\end{eqnarray}$$

Then matching (3.18) and (3.20) in the overlap region determines the constants ${\it\alpha}_{1}$ and ${\it\alpha}_{2}$ as (Rodney & Johnson Reference Rodney and Johnson2012)

(3.23a,b ) $$\begin{eqnarray}{\it\alpha}_{1}=C_{1}\text{exp}(\text{i}{\rm\pi}/4),\quad {\it\alpha}_{2}=C_{1}\text{exp}(-\text{i}{\rm\pi}/4).\end{eqnarray}$$

The solution decaying in ${\it\xi}>{\it\xi}_{c}$ is given by

(3.24) $$\begin{eqnarray}{\it\Phi}({\it\xi},{\it\eta})=C_{2}f_{0}^{+}{\it\psi}^{+}({\it\xi},{\it\eta})\text{exp}\left[\frac{\text{i}}{{\it\epsilon}}P({\it\xi})-\frac{1}{{\it\epsilon}}|Q({\it\xi},{\it\xi}_{c})|\right].\end{eqnarray}$$

For ${\it\xi}<{\it\xi}_{c}$ , the WKBJ connection formula gives

(3.25) $$\begin{eqnarray}{\it\Phi}({\it\xi},{\it\eta})=2C_{2}f_{0}^{+}{\it\psi}^{+}({\it\xi},{\it\eta})\text{exp}\left[\frac{\text{i}}{{\it\epsilon}}P({\it\xi})\right]\cos \left[\frac{1}{{\it\epsilon}}Q({\it\xi}_{c},{\it\xi})-\frac{{\rm\pi}}{4}\right].\end{eqnarray}$$

Matching (3.18) and (3.25) in the overlap region ${\it\xi}_{c}-{\it\epsilon}^{{\it\delta}}<{\it\xi}<{\it\xi}_{c}-{\it\epsilon}^{2/3}$ gives the constraint

(3.26) $$\begin{eqnarray}\frac{1}{{\it\epsilon}}Q({\it\xi}_{c},-{\it\xi}_{c})\sim \left(n+\frac{1}{2}\right){\rm\pi}+O({\it\epsilon}),\quad n=0,1,2,\ldots ,\end{eqnarray}$$

and $C_{2}=(-1)^{n}C_{1}$ , for $n$ from (3.26). Since $Q$ and ${\it\xi}_{c}$ depend on the frequency ${\it\omega}$ , (3.26) determines the frequency of the localised continental shelf wave of alongshore mode number $n$ , as required. The integral in (3.26) increases as ${\it\xi}_{c}\rightarrow \infty$ , giving an upper bound on the total number of trapped modes

(3.27) $$\begin{eqnarray}n\leqslant \frac{1}{{\rm\pi}{\it\epsilon}}\lim _{|{\it\xi}_{c}|\rightarrow \infty }Q({\it\xi}_{c},-{\it\xi}_{c})-\frac{1}{2}.\end{eqnarray}$$

3.1.2. The short WKBJ approximation

Within regions of thickness ${\it\epsilon}^{2/3}$ about ${\it\xi}=\pm {\it\xi}_{c}$ the evanescent waves (3.19), (3.24) are matched to the propagating modes (3.18) smoothly by Airy functions (Bender & Orszag Reference Bender and Orszag1978; Rodney & Johnson Reference Rodney and Johnson2012), giving solutions with enhanced amplitudes near ${\it\xi}=\pm {\it\xi}_{c}$ (as in figure 5 b below). If the alongshore region where modes propagate is sufficiently short (of order ${\it\epsilon}^{1/2}$ , still long compared to the shelf width of order ${\it\epsilon}$ ) then both turning points lie within the matching region and waves propagate only within this region. For smoothly varying shelves with propagating modes only near ${\it\xi}=0$ , the integrand in (3.11) takes its maximum value at the origin. Thus define

(3.28) $$\begin{eqnarray}{\it\Delta}={\textstyle \frac{1}{2}}[k^{+}(0)-k^{-}(0)],\end{eqnarray}$$

so $2{\it\Delta}$ gives the maximum difference between the left and right propagating waves (and is of order ${\it\epsilon}^{1/2}$ ). Since the integrand vanishes at ${\it\xi}=\pm {\it\xi}_{c}$ ,

(3.29) $$\begin{eqnarray}Q({\it\xi}_{c},-{\it\xi}_{c})=\int _{{\it\xi}_{c}}^{-{\it\xi}_{c}}{\it\Delta}[1-({\it\xi}/{\it\xi}_{c})^{2}]^{1/2}\,\text{d}{\it\xi}={\rm\Delta}{\it\xi}_{c}{\rm\pi}/2{\it\epsilon},\end{eqnarray}$$

and so the eigenrelation (3.26) for the $n$ th alongshore mode becomes, to leading order,

(3.30) $$\begin{eqnarray}{\rm\Delta}{\it\xi}_{c}=(2n+1){\it\epsilon},\end{eqnarray}$$

determining the frequencies of the trapped modes as both ${\it\xi}_{c}$ and ${\it\Delta}$ are functions of ${\it\omega}$ . For this parameter regime the leading-order trapped wave has the explicit expression

(3.31) $$\begin{eqnarray}{\it\Phi}({\it\xi},{\it\eta})=C_{3}{\it\psi}^{\pm }(0,{\it\eta})\exp \left[-\frac{\text{i}}{{\it\epsilon}}P({\it\xi})\right]X({\it\xi}),\end{eqnarray}$$

where the envelope $X({\it\xi})$ satisfies

(3.32) $$\begin{eqnarray}{\it\epsilon}^{2}X_{{\it\xi}{\it\xi}}+{\it\Delta}^{2}[1-({\it\xi}/{\it\xi}_{c})^{2}]X=0,\end{eqnarray}$$

with solutions bounded as ${\it\xi}\rightarrow \infty$ only for frequencies satisfying (3.30) and given explicitly by the parabolic cylinder functions

(3.33) $$\begin{eqnarray}X({\it\xi})=H_{n}[(2n+1)^{1/2}({\it\xi}/{\it\xi}_{c})]\exp [-(n+{\textstyle \frac{1}{2}})({\it\xi}/{\it\xi}_{c})^{2}],\end{eqnarray}$$

where the $H_{n}$ are the Hermite polynomials of order $n$ .

3.2. An underlying straight shelf

The analysis in § 3.1 applies to small variations in the curvature of a coast whose underlying local curvature is non-zero. If the underlying shelf is straight then ${\it\gamma}=0$ in (3.1) and so local curvature changes are no longer small compared to the underlying curvature. Trapped modes may still be obtained by expanding about the structure and frequency of the maximum-frequency propagating mode on the shelf but the analysis differs. The small parameter ${\it\epsilon}$ , which for the underlying curved shelf gives the non-dimensional alongshelf length scale ${\it\epsilon}^{-1}$ for both the curvature changes and the cross-shelf profile changes, must be taken to determine only the magnitude of the alongshore variations in bend angle, with ${\it\epsilon}=0$ giving a straight coast. Follow Postnova & Craster (Reference Postnova and Craster2008) and Johnson et al. (Reference Johnson, Rodney and Kaoullas2012) by initially taking the scale for ${\it\gamma}$ to be ${\it\epsilon}$ and introducing the expansions

(3.34) $$\begin{eqnarray}\displaystyle & {\it\Phi}({\it\xi},{\it\eta})\sim \exp (\text{i}{\it\xi}{\it\mu}/2{\it\omega}{\it\epsilon})(f_{0}({\it\xi}){\it\psi}^{c}({\it\eta})+{\it\epsilon}{\it\psi}_{1}({\it\xi},{\it\eta})+{\it\epsilon}^{2}{\it\psi}_{2}({\it\xi},{\it\eta})+\cdots \,), & \displaystyle\end{eqnarray}$$

(3.35) $$\begin{eqnarray}\displaystyle & {\it\omega}^{-2}={\it\omega}_{c}^{-2}+{\it\epsilon}{\it\lambda}_{1}+{\it\epsilon}^{2}{\it\lambda}_{2}+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.36) $$\begin{eqnarray}{\it\mu}=\int _{0}^{\partial \mathscr{D}}{\it\beta}_{{\it\eta}}H^{-1}{{\it\psi}^{c}}^{2}\,\text{d}{\it\eta}=2{\it\omega}_{c}k_{c},\end{eqnarray}$$

and ${\it\psi}^{c}$ is the cutoff streamfunction, satisfying

(3.37) $$\begin{eqnarray}{\it\psi}_{{\it\eta}{\it\eta}}^{c}-{\it\beta}_{{\it\eta}}{\it\psi}_{{\it\eta}}^{c}+({\it\beta}_{{\it\eta}}k_{c}/{\it\omega}_{c}-{k_{c}}^{2}){\it\psi}^{c}=0,\end{eqnarray}$$

the boundary conditions (2.7), and normalised so

(3.38) $$\begin{eqnarray}\int _{0}^{\partial \mathscr{D}}H^{-1}{{\it\psi}^{c}}^{2}\,\text{d}{\it\eta}=1,\end{eqnarray}$$

with ${\it\omega}_{c}$ and $k_{c}$ the cutoff frequency and cutoff wavenumber for a straight coast. Substituting (3.34) and (3.35) into (3.2) leads to a hierarchy of equations. The leading-order system is satisfied automatically. The next order, of order ${\it\epsilon}$ , gives

(3.39) $$\begin{eqnarray}{\it\psi}_{1{\it\eta}{\it\eta}}-{\it\beta}_{{\it\eta}}{\it\psi}_{1{\it\eta}}+({\it\beta}_{{\it\eta}}k_{c}/{\it\omega}_{c}-{k_{c}}^{2}){\it\psi}_{1}=-{\it\gamma}{\it\psi}_{{\it\eta}}^{c}-{\it\beta}_{{\it\eta}}{\it\lambda}_{1}{\it\psi}^{c}/2+\text{i}\mathscr{F}f_{0{\it\xi}}{\it\psi}^{c},\end{eqnarray}$$

subject to (2.7), with $\mathscr{F}({\it\eta})={\it\omega}_{c}^{-1}({\it\beta}_{{\it\eta}}-{\it\mu})$ .

Multiplying (3.39) by $H^{-1}{\it\psi}^{c}$ and integrating over the domain $0\leqslant {\it\eta}\leqslant \partial \mathscr{D}$ gives the solvability condition

(3.40) $$\begin{eqnarray}2{\it\gamma}\int _{0}^{\partial \mathscr{D}}H^{-1}{\it\psi}^{c}{\it\psi}_{{\it\eta}}^{c}\,\text{d}{\it\eta}+{\it\lambda}_{1}{\it\mu}=0.\end{eqnarray}$$

Since the integrand in (3.40) is non-zero and ${\it\gamma}$ varies with ${\it\xi}$ , no choice of the number ${\it\lambda}_{1}$ can satisfy (3.40) for all ${\it\xi}$ and the curvature must be weakened by introducing ${\it\gamma}={\it\epsilon}\hat{{\it\gamma}}$ . This determines the alongshelf length scale for changes in the curvature as one order higher in ${\it\epsilon}$ when the underlying shelf is straight compared with the curvature when the underlying shelf is curved. The leading-order equation is unchanged. However, equating terms of order ${\it\epsilon}$ now gives

(3.41) $$\begin{eqnarray}{\it\psi}_{1{\it\eta}{\it\eta}}-{\it\beta}_{{\it\eta}}{\it\psi}_{1{\it\eta}}+({\it\beta}_{{\it\eta}}k_{c}/{\it\omega}_{c}-{k_{c}}^{2}){\it\psi}_{1}=-{\it\beta}_{{\it\eta}}{\it\lambda}_{1}{\it\psi}^{c}/2+\text{i}\mathscr{F}f_{0{\it\xi}}{\it\psi}^{c},\end{eqnarray}$$

and the equivalent solvability condition to (3.40) gives ${\it\lambda}_{1}=0$ with the first-order system becoming

(3.42) $$\begin{eqnarray}{\it\psi}_{1{\it\eta}{\it\eta}}-{\it\beta}_{{\it\eta}}{\it\psi}_{1{\it\eta}}+({\it\beta}_{{\it\eta}}k_{c}/{\it\omega}_{c}-{k_{c}}^{2}){\it\psi}_{1}=\text{i}\mathscr{F}f_{0{\it\xi}}{\it\psi}^{c},\end{eqnarray}$$

subject to (2.7). This system can be solved using variation of parameters by introducing a solution $\tilde{{\it\psi}}_{0}$ of (3.37) independent of ${\it\psi}^{c}$ so that

(3.43) $$\begin{eqnarray}{\it\psi}_{{\it\eta}}^{c}\tilde{{\it\psi}_{0}}-\tilde{{\it\psi}_{0{\it\eta}}}{\it\psi}^{c}=H.\end{eqnarray}$$

The solution to (3.42) is then

(3.44) $$\begin{eqnarray}{\it\psi}_{1}=f_{1}({\it\xi}){\it\psi}^{c}+\text{i}f_{0{\it\xi}}{\it\psi}^{c}\int _{0}^{{\it\eta}}H^{-1}\mathscr{F}{\it\psi}^{c}\tilde{{\it\psi}}_{0}\,\text{d}{\it\eta}-\text{i}f_{0{\it\xi}}\tilde{{\it\psi}_{0}}\int _{0}^{{\it\eta}}H^{-1}\mathscr{F}{{\it\psi}^{c}}^{2}\,\text{d}{\it\eta},\end{eqnarray}$$

where $f_{1}$ gives an $O({\it\epsilon})$ correction to the leading-order solution. The ray paths of low-frequency topographic Rossby waves are determined by the geostrophic vector $\boldsymbol{G}=h\boldsymbol{{\rm\nabla}}(f/h)$ (Smith Reference Smith1970). Thus for depth profiles satisfying $\mathscr{F}\equiv 0$ ray paths are parallel to the coast and the imaginary terms in (3.44) vanish. For non-zero $\mathscr{F}$ the imaginary terms on the right-hand side of (3.44) can be regarded as an $O({\it\epsilon})$ phase shift to the wave (since the ray paths are no longer parallel to the coast). The next order, of order ${\it\epsilon}^{2}$ , gives

(3.45) $$\begin{eqnarray}{\it\psi}_{2{\it\eta}{\it\eta}}-{\it\beta}_{{\it\eta}}{\it\psi}_{2{\it\eta}}+({\it\beta}_{{\it\eta}}k_{c}/{\it\omega}_{c}-{k_{c}}^{2}){\it\psi}_{2}=-{\it\psi}^{c}f_{0{\it\xi}{\it\xi}}-\hat{{\it\gamma}}{\it\psi}_{{\it\eta}}^{c}f_{0}-{\it\beta}_{{\it\eta}}{\it\lambda}_{2}{\it\psi}^{c}f_{0}/2+\text{i}\mathscr{F}{\it\psi}_{1{\it\xi}}.\end{eqnarray}$$

In deriving (3.45), $f_{1}$ in (3.44) has been absorbed into $f_{0}$ with the extra terms on the right side of (3.45) generated by this modification appearing only at higher order. Multiplying (3.45) by $H^{-1}{\it\psi}^{c}$ and integrating over the domain $0\leqslant {\it\eta}\leqslant \partial \mathscr{D}$ gives

(3.46) $$\begin{eqnarray}(1+\mathscr{I}_{1})f_{0{\it\xi}{\it\xi}}+(\hat{{\it\gamma}}\mathscr{I}_{2}+{\it\mu}{\it\lambda}_{2}/2)f_{0}=0,\end{eqnarray}$$

where

(3.47) $$\begin{eqnarray}\displaystyle \mathscr{I}_{1} & = & \displaystyle \int _{0}^{\partial \mathscr{D}}H^{-1}\mathscr{F}{\it\psi}^{c}\tilde{{\it\psi}}_{0}\left(\int _{0}^{{\it\eta}}H^{-1}\mathscr{F}{{\it\psi}^{c}}^{2}\,\text{d}{\it\eta}\right)\,\text{d}{\it\eta}\nonumber\\ \displaystyle & & \displaystyle -~\int _{0}^{\partial \mathscr{D}}H^{-1}\mathscr{F}{{\it\psi}^{c}}^{2}\left(\int _{0}^{{\it\eta}}H^{-1}\mathscr{F}{\it\psi}^{c}\tilde{{\it\psi}}_{0}\,\text{d}{\it\eta}\right)\,\text{d}{\it\eta}.\end{eqnarray}$$

and

(3.48) $$\begin{eqnarray}\mathscr{I}_{2}=\int _{0}^{\partial \mathscr{D}}H^{-1}{\it\psi}_{{\it\eta}}^{c}{\it\psi}^{c}\,\text{d}{\it\eta}.\end{eqnarray}$$

Solutions of (3.46) decaying exponentially at infinity determine the leading-order wave envelope $f_{0}({\it\xi})$ and eigenvalue correction ${\it\lambda}_{2}$ .

3.2.1. An explicit example

For the exponential topography

(3.49) $$\begin{eqnarray}H({\it\eta})=\left\{\begin{array}{@{}ll@{}}\text{e}^{2b({\it\eta}-1)}\quad & 0\leqslant {\it\eta}\leqslant 1,\\ 1\quad & {\it\eta}>1,\end{array}\right.\end{eqnarray}$$

the function $\mathscr{F}\equiv 0$ and the leading-order eigenfunctions, ${\it\psi}^{c}$ , can be found analytically. For cases I, II and III

(3.50) $$\begin{eqnarray}{\it\psi}^{c}=2[{\it\alpha}/(2{\it\alpha}-\sin 2{\it\alpha})]^{1/2}\text{e}^{b({\it\eta}-1)}\sin {\it\alpha}{\it\eta},\end{eqnarray}$$

where

(3.51) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{case I:}\quad {\it\alpha}=n{\rm\pi},\quad n=1,2,3,\ldots ,\end{eqnarray}$$

(3.52) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{case II:}\quad b\tan {\it\alpha}=-{\it\alpha},\end{eqnarray}$$

(3.53) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{case III:}\quad (b+k_{c})\tan {\it\alpha}={\it\alpha}.\end{eqnarray}$$

For the full open-ocean boundary condition (2.6)

(3.54) $$\begin{eqnarray}\text{case IV:}\quad {\it\psi}^{c}=\left\{\begin{array}{@{}ll@{}}A\,\text{e}^{b({\it\eta}-1)}\sin {\it\alpha}{\it\eta}\quad & \text{if }0\leqslant {\it\eta}\leqslant 1,\\ A\sin ^{2}{\it\alpha}\,\text{e}^{-k_{c}({\it\eta}-1)}\quad & \text{if }{\it\eta}>1,\\ \quad \end{array}\right.\end{eqnarray}$$

where $A=2{\it\alpha}^{1/2}\sin {\it\alpha}/(2-2\sin 2{\it\alpha}+2{\it\alpha}\sin ^{2}{\it\alpha}/k_{c})^{1/2}$ and ${\it\alpha}=\sqrt{2bk_{c}/{\it\omega}-(k_{c}^{2}+b^{2})}$ .

For curvature

(3.55) $$\begin{eqnarray}\hat{{\it\gamma}}=({\rm\pi}/4)\text{sech}^{2}{\it\xi},\end{eqnarray}$$

giving a total bend angle of ${\it\epsilon}{\rm\pi}/2$ , the solution of (3.46) is given by

(3.56a ) $$\begin{eqnarray}\displaystyle & f_{0}=\cosh ^{m-s}F(-m,2s+1-m,s+1-m,(1-\tanh ({\it\xi})/2)), & \displaystyle\end{eqnarray}$$

(3.56b ) $$\begin{eqnarray}\displaystyle & {\it\lambda}_{2}=-2(s-m)^{2}/{\it\mu}, & \displaystyle\end{eqnarray}$$

where $s=(\sqrt{1+{\rm\pi}{\it\beta}}-1)/2$ , $m$ is a non-negative integer, $F$ is the confluent hypergeometric function of degree $m$ and ${\it\beta}$ is given by

(3.57) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{case I:}\quad {\it\beta}=b+n{\rm\pi}(1-\cos 2n{\rm\pi})/(2n{\rm\pi}-\sin 2n{\rm\pi})\end{eqnarray}$$

(3.58) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{case II:}\quad {\it\beta}=b+{\it\alpha}(1-\cos 2{\it\alpha})/(2{\it\alpha}-\sin 2{\it\alpha}),\quad {\it\alpha}\tan {\it\alpha}=-b,\end{eqnarray}$$

(3.59) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{case III:}\quad {\it\beta}=b+{\it\alpha}(1-\cos 2{\it\alpha})/(2{\it\alpha}-\sin 2{\it\alpha}),\quad ({\it\alpha}+k_{c})\tan {\it\alpha}=-b,\end{eqnarray}$$

(3.60) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \text{case IV:}\quad {\it\beta}=\frac{b(2{\it\alpha}-\sin 2{\it\alpha})}{2{\it\alpha}(1+\sin ^{2}{\it\alpha}/k_{c})-\sin {\it\alpha}},\quad {\it\alpha}=\sqrt{2bk_{c}/{\it\omega}-(k_{c}^{2}+b^{2})}.\end{eqnarray}$$

Here (3.58) is the result in Johnson et al. (Reference Johnson, Rodney and Kaoullas2012).

Figure 2.

Dispersion curves for the first propagating mode over the depth profile (5.1) with $g\equiv 1$ and the full open-ocean boundary condition (2.6) applied at infinity for (a) different values of ${\it\gamma}$ with $b=2$ , (b) different values of $b$ with ${\it\gamma}=0$ .

5.1. Alongshore variations in coastline curvature

As a CSW mode of fixed frequency propagates along a coastline with slowly varying curvature its local characteristics are governed by the leading-order system (3.7), (2.7) which, at each position ${\it\xi}=$ constant, reduces to an annular region with constant radius of curvature ${\it\gamma}^{-1}$ . In general the local wavenumbers $k^{\pm }$ and corresponding eigenfunction ${\it\psi}^{\pm }$ of the eigensystem (3.7), (2.7) can be obtained only numerically but this is a straightforward one-dimensional problem (see the appendix A). Consider the depth profile

(5.1) $$\begin{eqnarray}H({\it\xi},{\it\eta})=\left\{\begin{array}{@{}ll@{}}\text{e}^{2bg({\it\xi})({\it\eta}-1)}\quad & 0\leqslant {\it\eta}\leqslant 1,\\ 1\quad & {\it\eta}>1,\end{array}\right.\end{eqnarray}$$

where $g({\it\xi})$ controls the alongshore variation in shelf slope, which reduces to (3.49) when $g\equiv 1$ . Figure 2 shows the local dispersion curves, computed using the spectral method in the appendix A, for the fundamental cross-shelf mode over the depth profile (5.1) with $g\equiv 1$ , for different values of the curvature ${\it\gamma}$ and slope $b$ . Larger positive curvature has a similar effect to larger bottom slope, raising the cutoff frequency and so permitting $\ell$ CSWs encompassing the section of maximum curvature. This agrees with the result in Johnson et al. (Reference Johnson, Rodney and Kaoullas2012) that, in the limit of small curvature, there is always an $\ell$ CSW associated with a region of positive curvature on an otherwise straight shelf (a ‘cape’) but no $\ell$ CSW exists associated with a region of negative curvature (a ‘bay’).

Table 2.

Numerical eigenfrequencies, ${\it\omega}_{0,1}^{N}$ , and asymptotic eigenfrequencies, ${\it\omega}_{0,1}^{A}$ , calculated using (3.56b ), for the weak curvature function (3.55), depth profile (3.49) with $b=2$ and with ${\it\epsilon}=0.1$ , and so a total bend angle of ${\rm\pi}/20$ .

$\ell$ CSWs can be characterised by their alongshore mode number $n$ (with $n=0$ corresponding to the fundamental mode) and cross-shelf mode number $m$ . Denote the frequency of the $(n,m)$ mode by ${\it\omega}_{n,m}$ , with corresponding eigenfunction ${\it\Phi}_{n,m}$ . Table 1 compares the $\ell$ WKBJ eigenvalues, ${\it\omega}_{n,m}^{\ell \mathit{WKBJ}}$ , calculated using (3.26); the sWKBJ eigenvalues, ${\it\omega}_{n,m}^{\text{s}\mathit{WKBJ}}$ , calculated using (3.30); and the numerical eigenvalues, ${\it\omega}_{n,m}^{N}$ , for the curvature function

(5.2) $$\begin{eqnarray}{\it\gamma}=(\bar{{\it\gamma}}/2)\text{sech}^{2}{\it\xi},\end{eqnarray}$$

so that the total bend angle is $\bar{{\it\gamma}}/{\it\epsilon}$ , and depth profile (5.1), with $b=2$ and $g\equiv 1$ , for different values of ${\it\epsilon}$ , with full open-ocean boundary condition (2.6) applied at infinity. The WKBJ eigenvalues are indeed extremely accurate, to well within 1 % of the numerical solutions. As expected, the accuracy of the $\ell$ WKBJ frequencies increases with alongshore wavenumber, since the ratio of the variation in the wavelength to the variation in the curvature increases as the number of alongshore modal oscillations increases (Bender & Orszag Reference Bender and Orszag1978). The agreement with the full numerical solution even for ${\it\epsilon}=1$ is remarkable. The accuracy of the sWKBJ frequencies decreases with alongshore wavenumber since the length over which the modes propagate increases with alongshore wavenumber. Figure 3 shows the modulus of the corresponding numerical eigenfunctions $|{\it\Phi}_{0,1}|$ and $|{\it\Phi}_{1,1}|$ for ${\it\epsilon}=0.1$ (figure 3 a,b) and ${\it\epsilon}=1$ (figure 3 c,d). Most of the wave disturbance is concentrated in the region of maximum curvature with modes decaying exponentially along the straight section of coast. From figure 3(c,d), it is clear that the alongshore decay scale of the modes increases as the mode number increases. The frequencies of $\ell$ CSWs decrease with alongshore mode number, with modes coupling to modes on the straight section of coast having smaller value of $\text{Im}\,k$ , and so decaying on a slower scale.

Figure 4.

Isobaths (a) and depth profiles (b) of the shelf with local perturbation on shelf slope given by (5.1) and (5.3) with $b=2$ , $a={\rm\pi}/4$ and ${\it\epsilon}=0.1$ . In the right panel the solid line is the unperturbed far-field profile and the dashed line gives the maximally perturbed profile at $x=0$ .

Table 2 compares the numerical eigenfrequencies, ${\it\omega}_{0,1}^{N}$ , and the asymptotic eigenfrequencies, ${\it\omega}_{0,1}^{A}$ , calculated using expansion (3.34) with ${\it\lambda}_{2}$ given by (3.56b ), for the weak curvature (3.55) and depth profile (3.49) with $b=2$ and ${\it\epsilon}=0.1$ , for boundary conditions (2.3)–(2.6). The asymptotic eigenfrequencies are again extremely accurate, with the absolute error well below 1 % for all four boundary conditions. Eigenfrequencies calculated using the full open-ocean condition (2.6) and the near-cutoff approximation (2.5) differ by less than 0.2 %, showing the high accuracy and thus usefulness of (2.5) when computing $\ell$ CSWs. The cutoff and full open-ocean eigenfrequencies lie above the Dirichlet but below the Neumann eigenfrequencies as expected (Johnson Reference Johnson1989). Table 3 compares the $\ell$ WKBJ eigenvalues and the weak curvature eigenvalues (with error derived from the full numerical eigenvalues) for the $(0,1)$ mode, for the weak curvature (3.55) and depth profile (3.49), with $b=2$ , ${\it\epsilon}=0.1$ and the full open-ocean boundary condition (2.6) applied at infinity. The comparison shows that even for extreme values of (weak) curvature the $\ell$ WKBJ eigenvalues remain in close agreement with the full numerical solutions, and in fact are more accurate than the eigenvalues derived from the method of § 3.2, specifically designed for small curvature.

5.2. Alongshore variations in offshore depth profile

The frequencies of CSWs are greatly affected by changes in the gradient of the shelf slope. Figure 2(b) shows dispersion curves for the depth profile (5.1), with $g\equiv 1$ and ${\it\gamma}\equiv 0$ , for different values of $b$ . The steepening of the shelf slope (i.e. increasing  $b$ ) raises the maximum frequency of propagation of the waves. Consider the depth function (5.1) with local variation in shelf slope governed by

(5.3) $$\begin{eqnarray}g({\it\xi})=1+\bar{a}\text{sech}^{2}({\it\xi}),\end{eqnarray}$$

so that the maximum perturbation in shelf slope occurs at ${\it\xi}=0$ and then disappears as ${\it\xi}\rightarrow \infty$ , with $\bar{a}>0$ corresponding to a submerged continental shelf ridge (figure 4).

Table 4 compares the numerical eigenfrequencies, ${\it\omega}_{n,m}^{N}$ , the $\ell$ WKBJ asymptotic eigenfrequencies, ${\it\omega}_{n.m}^{\ell \mathit{WKBJ}}$ , and the sWKBJ asymptotic eigenfrequencies, ${\it\omega}_{n.m}^{\text{s}\mathit{WKBJ}}$ , for the Dirichlet boundary condition (2.3), the curvature function ${\it\gamma}\equiv 0$ and depth profile (5.1) with $g$ given by (5.3), $\bar{a}={\rm\pi}/4$ , $b=2$ and ${\it\epsilon}=0.1$ . Again the asymptotic eigenvalues are extremely accurate, with the relative error well below 1 % with the accuracy of the $\ell$ WKBJ eigenvalues increasing, and that of the sWKBJ eigenvalues decreasing, with alongshore wavenumber, $n$ . The accuracy of the both the $\ell$ WKBJ eigenvalues and sWKBJ eigenvalues also increases with increasing offshore number, $m$ . Higher offshore modes propagate, locally, over a shorter alongshore distance. They are confined over a shorter strip of shelf and the wavelength varies on a faster scale compared to the variation in the curvature, and both the $\ell$ WKBJ and sWKBJ approximations improve. For boundary conditions (2.4)–(2.6) the accuracy of the $\ell$ WKBJ eigenvalues mirrors the accuracy shown in table 4, and so these results are not displayed.

Table 5 compares the corresponding $\ell$ WKBJ eigenvalues with boundary conditions (2.3)–(2.6). Comparison between the eigenfrequencies calculated using the cutoff ocean boundary condition (2.5) and the full open-ocean boundary condition (2.6) shows agreement to well within 1 % with accuracy increasing with alongshore mode number, as the offshore decay scale approaches cutoff. Again the cutoff and full open-ocean eigenfrequencies lie above the Dirichlet but below the Neumann eigenfrequencies. Figure 5 compares $\ell$ WKBJ (dot-dashed line), sWKBJ (dashed line) and numerical eigenfunctions (solid line) $|{\it\Phi}_{0,1}|$ and $|{\it\Phi}_{n,m}|$ (from table 4), as a function of ${\it\xi}$ in the cross section $y=0.5$ . The caustics, located at ${\it\xi}=\pm {\it\xi}_{c}$ are represented by the dotted lines. Most of the wave disturbance is concentrated above the maximum perturbation in shelf slope and decays exponentially alongshore, with the $\ell$ WKBJ eigenfunctions becoming singular at the caustics.

Table 3.

Comparison between the WKBJ eigenvalues, calculated using (3.26), and the asymptotic eigenvalues calculated using (3.56b ) with ${\it\lambda}_{2}$ given by (3.56b ), for the weak curvature function (3.55), depth profile (3.49) with $b=2$ and ${\it\epsilon}=0.1$ and so a total bend angle of ${\rm\pi}/20$ .

Figure 5.

(a) Comparison of the modulus of the numerical eigenfunction with eigenvalue ${\it\omega}_{0,1}$ (solid line) as a function of the coordinate ${\it\xi}$ at $y=0.5$ , with the corresponding $\ell$ WKBJ (dot-dashed line) and sWKBJ eigenfunctions (dashed line, but indistinguishable from the numerical solution), for the depth profile (5.1) with local slope perturbation (5.3) and the Dirichlet boundary condition applied at the coast and shelf–ocean boundary. (b) As in (a) but for the eigenvalue ${\it\omega}_{2,1}$ . In both panels $b=2$ , ${\it\epsilon}=0.1$ and $\bar{a}={\rm\pi}/4$ .