Resonance has recently been proposed as the fundamental underlying mechanism that shapes the amplification in coastal run-up for storm surges and surf beats, which are long-wavelength disturbances created by fluid velocity differences between the wave groups and the regions outside the wave groups. It is without doubt that the resonance plays a role in run-up phenomena of various kinds; however, we think that the extent to which it plays its role has not been completely understood. For incident waves, which we assume to be linear, the best approach to investigate the role played by the resonance would be to calculate the normal modes by taking radiation damping into account and then testing how those modes are excited by the incident waves. Such modes diverge offshore, but they can still be used to calculate the run-up. There are a small number of previous works that attempt to calculate the resonant frequencies, but they do not relate the amplitudes of the normal modes to those of the incident wave. This is because, by not including radiation damping, they automatically induce a resonance that leads to infinite amplitudes, thus preventing them from predicting the exact contribution of the resonance to coastal run-up. In this study we consider two different coastal geometries: an infinitely wide beach with a constant slope connecting to a flat-bottomed deep ocean and a bay with sloping bottom, again, connected to a deep ocean. For the fully 1-D problem we find significant resonance if the bathymetric discontinuity is large.The linearisation of the seaward boundary condition leads to slightly smaller run-ups. For the 2-D ocean case the analysis shows that the wave confinement is very effective when the bay is narrow. The bay aspect ratio is the determining factor for the radiation damping. One reason why we include a bathymetric discontinuity is to mimic some natural settings where bays and gulfs may lead to abrupt depth gradients such as the Tokyo Bay. The other reason is, as mentioned above, to test the role played by the depth discontinuity for resonance.

During the last decades, several analytical and numerical studies of coastal
run-up were published (see

In the past, several researchers looked at resonance aspect of the coastal
run-up. Among those, the ones that are the most relevant to the discussion in
the present study are

In our modelling, we will be considering monochromatic incident waves, simply because this makes it easier to shed light on the resonance. However, the mathematical algorithm we will develop is not limited to monochromatic waves but is capable of calculating run-up for any kind of offshore source including the earthquakes, submarine landslides or atmospheric pressure perturbations.

Panel

This article deals with the transient run-up response of a sloping
channel (or bay) to an incident wave. Here, the term

Our purpose in the present work is to determine the way in which the free
modes near the coast are excited by the incident waves by taking the
radiation damping into account. We will examine resonance in two different
geometric settings: first in a 1-D slope which connects to a 1-D channel with a
flat bottom and then, again, in a 1-D slope that connects to a semi-infinite 2-D
ocean with a flat bathymetry (see Fig.

Model-1 is actually solvable, through fast Fourier transforms (see

Model-1 consists of a channel of constant slope

The governing equations we shall use over the sloping part of the geometry
are nonlinear shallow water equations:

A hodograph transformation introduced by

The nonlinear shallow water equations accordingly become

For the flat part of the domain (

The corresponding free surface elevation near

The linearised free surface and flux continuity conditions at the toe (

Both

The complex natural frequencies multiplied by 2 using both the Müller
method and the asymptotic approach (see Eq.

Recently

Now consider an incident wave of the following form:

Any incident wave can be expressed in terms of a linear superposition of
Dirac functions; the response,

It is important to note that as

To perform the free mode expansion approach, let us rewrite the convolution
(Eq.

In an effort to calculate solitary wave run-up,

To do our calculations for wave evolution, we will need the non-vanishing
poles of

As seen in this table, as

The geometry considered by

Now let us return to the integral (Eq.

In this section we consider a monochromatic incident wave of type

The limiting amplitude of run-up,

In the limit of large

One last remark relates to the power laws for run-up, provided by

Continuous and dashed blue curves display the run-up
normalised to the amplitude of incident wave for Model-1 with

The run-up normalised to the amplitude of the incident wave for
Model-1. In panel

The resonant phenomena we discussed above do not set in immediately upon the entrance of the incident wave into the slope region. It is important to know how fast the limiting amplitude of oscillation of the run-up will be reached, because in a real situation the incident wave will have a finite duration, a fact that was not taken into account in the large time limit analysis.

Panel

For that purpose the residue series in Eq. (

Run-ups normalized to the amplitude of the incident wave for
Model-1. The incident wave is in the form

We assume the incident wave to be linear. Considering that we are dealing
with a monochromatic incident wave, this makes sense, because any
nonlinearity over the flat part of the ocean would have generated higher
harmonics during the propagation. As long as the waves do not break, the
nonlinearity arising from the shoaling over the slope is accounted for by the
CG approach. This particular nonlinearity, as indicated by

For

At the near-resonant frequencies nonlinear effects will become important even
in the deep part of the slope, rendering the

In this section we investigate the resonant frequencies of the waves produced
by a wave maker placed on an infinite, constant slope. The reason for this
practice is that the work that claims significant resonance

When the water is sufficiently deep, in the vicinity of the wave maker, linear
shallow water equations will apply. The effect of the wave maker, which starts
its action at a given time, will be equivalent to hypothetical volume
injections and suctions at a rate

Analytical studies that consider run-up in 2-D are rare. Among the most
relevant, we can mention

The governing equations, linearised in the deeper part of the channel and the
open sea, for the potential,

The bulk of the incident wave will be reflected back by the solid boundary at

We want the solution in the open sea to satisfy the no-flux condition at

The complex natural frequencies multiplied by 2 are tabulated for
Model-2. These are essentially those

Note that in Eqs. (

A quick look at Table 2 reveals that for any mode, with the decreasing
channel width (

The rays that the waves follow in the sloping channel are straight lines at
the shallower parts, and they bend towards the corners as they get near to the
mouth of the channel, due to geometrical spreading (see the streamlines in
Fig.

Now let us turn our attention to the transient response for Model-2. As
before, we shall model the transient response to an incident wave of the form

Remember that in Model-1 we had an analytical expression for

The blue continuous curve is the run-up (calculated using
Eq.

The maximum run-up normalized to the amplitude of the incident wave
for the standing wave case for two different channel half-width values. The
continuous blue curves are computed using the integral equation, and the red
broken curves are obtained from the conformal mapping (see
Eq.

In Fig.

To summarise, for both Model-1 and Model-2, when the incident
wavelength is larger than the channel length, the run-up amplitude tends to

In this work we studied the resonance aspect of the coastal run-up as a
response to incident waves. The analysis follows a normal mode approach and
examines the sensitivity of those normal modes to a given linear incident
wave to produce coastal run-up. These modes diverge offshore, but, since we are
only interested in the coastal run-up, we can still use them. In Model-1, significant run-up sensitivity, in other words resonance, occurs
only when

The residue method developed here can actually be generalized for more
complicated channel geometries (such as piecewise constant slopes with
varying width) by performing a “fusion” of this method with the boundary
elements technique. This is because the boundary elements technique has
recently proven to be very efficient for solving the Helmholtz equation in
multiple dimensions

No data sets were used in this article.

In this appendix, the natural frequencies of Model-1 will be evaluated
for large

The determinant of the matrix becomes zero if

The solution we obtained in Eq. (

Our aim is to calculate an additional term,

For the velocities, a similar approach based on the continuity of the flux
(this time taking into account the real depth, given by

Let us now propose a general solution for

The problem of an incident wave into a rectangular bay of uniform depth was
solved in

The blue curves are

The flow displays a complex pattern in the vicinity of the mouth of the
channel (see Fig.

The no-flux condition is also invariant under the conformal mapping and it becomes

For small

Matching Eq. (

Solving Eqs. (

The authors declare that they have no conflict of interest. Edited by: Ira Didenkulova Reviewed by: Takenori Shimozono and three anonymous referees