# The Solution of the Long-Wave Equation for Various Nonlinear Depth and
Breadth Profiles in the Power-Law Form

Cihan Bayındır<EMAIL_ADDRESS>Associate Professor, Engineering Faculty,
İstanbul Technical University, 34469 Maslak, İstanbul, Turkey.
Adjunct Professor, Engineering Faculty, Boğaziçi University, 34342 Bebek,
İstanbul, Turkey.
International Collaboration Board Member, CERN, CH-1211 Geneva 23,
Switzerland.  Sofi Farazande  Graduate Student, Engineering Faculty, Boğaziçi
University, 34342 Bebek, İstanbul, Turkey.

###### Abstract

In this paper, we derive the exact solutions of the long-wave equation over
nonlinear depth and breadth profiles having power-law forms given by
$h(x)=c_{1}x^{a}$ and $b(x)=c_{2}x^{c}$, where the parameters
$c_{1},c_{2},a,c$ are some constants. We show that for these types of power-
law forms of depth and breadth profiles, the long-wave equation admits
solutions in terms of Bessel functions and Cauchy-Euler series. We also derive
the seiching periods and resonance conditions for these forms of depth and
breadth variations. Our results can be used to investigate the long-wave
dynamics and their envelope characteristics over equilibrium beach profiles,
the effects of nonlinear harbor entrances and angled nonlinear seawall breadth
variations in the power-law forms on these dynamics, and the effects of
reconstruction, geomorphological changes, sedimentation, and dredging to
harbor resonance, to the shift in resonance periods and to the seiching
characteristics in lakes and barrages.

###### pacs:

92.10.Hm

##  I Introduction

In ocean and coastal modeling and engineering fields, one of the most
challenging problems for the designs is to suppress the effects of long-waves
which lead to very large period oscillations that are very hard to dissipate,
large amounts of wave runup, overwash, and inundation. The literature on this
well-studied subject is vast Lamb ; Chrystal ; Wilson ; Goldsbrough ; DeanDal
; Mei . Linear long-wave equation is first published by Lamb in 1879 Lamb ,
and in an approximate form by Chrystal in 1905 Chrystal . Since then these
equations are analyzed extensively Lamb ; Chrystal ; Wilson ; Goldsbrough ;
DeanDal ; Mei . While the majority of these studies are focused on the
analysis of long-wave equations for different estuary, bay, and harbor
geometries, there are also many studies on their interaction with sediments,
beaches, barred beaches, and islands such as those discussed in Munk ; Lettau
; Bowen ; HigginsStewart ; Suhayda ; SmithSprinks ; Kirby1981 ; Eckart ;
Ball67 .

Many investigations of the standing long-waves with oblique incidence to the
beach have been conducted for long by the scientific community. Multiple bar
formation due to the response of loose sands under such standing waves is one
of the observed and studied phenomena. The solution for the long-wave equation
for the reflection of normally incident, non-breaking waves coming from a
horizontal beach slope has been given by Lamb Lamb , and the motion of
suspended sediments towards the antinodes of such waves is demonstrated by
Lettau Lettau  and Noda Noda . Bowen Bowen  has also predicted the
accumulation of sediments under standing wave nodes and antinodes which is
shown to impel the linear offshore bar formation.

Munk Munk  has named the ”surf beat” the occurrence of long-period progressive
waves on the beach, a phenomenon which has been observed for the first time
back then. Longuet-Higgins and Stewart HigginsStewart  have analyzed the
mechanism of wave groups of incident wind waves driving a second-order wave of
long-period that is traveling at the wind wave group velocity. Gallagher
Gallagher  has studied the surf beat generation by non-linear interaction of
37 wind waves. Long-wave resonance and their scattering around circular
islands have also been studied by Smith and Sprinks SmithSprinks , and by
Summerfield Summerfield  with a numerical approach. Eckart Eckart  studied the
surface waves where the water depth is variable, and Ball Ball67  made an
analysis on the behavior of waves of different orders moving in various
directions.

One of the interesting and crucial phenomena studied within the frame of the
long-wave equations are the edge waves Kirby1981 ; Ursell . Due to the
vanishing depth related singularity at the shoreline, edge wave solutions are
studied using the Frobenius series and solutions for some cases turned out to
be in the form of the Laguerre polynomials Kirby1981 . It is also demonstrated
that coastal edge waves can be excited in the infragravity spectral range with
a limited analysis including only low edge-wave mode numbers Gallagher . These
results were extended to edge waves with high mode numbers, by Bowen and Guza
BowenGuza . Infragravity waves with offshore surface profiles similar to the
Bessel function solution were shown to be existent by the fieldwork conducted
by Suhayda Suhayda  in which it is also shown that energy spectrum and phase
difference in standing waves are shown to be coherent with the linear long-
wave theory developed by Lamb Lamb . Some nonlinear theories have also been
applied to edge waves to investigate the effects of wave interactions Ursell ;
Stokes ; Huntley ; HolmanBowen ; GuzaInman .

Long-waves are also commonly used in the field of sound and optical
engineering Auld ; Lakinacousticresonator ; LigeometryResonator ;
Chevalieropticalresonator . The sound and optical resonators are tuned to
resonate at specific frequencies of the excitations. To model such phenomena
with very different and complex geometries, long-wave equations and some of
their extensions are commonly employed.

Long-wave equations have well-known solutions for various simple geometries
such as uniform depth Lamb ; Wilson ; Goldsbrough ; DeanDal ; Mei , linearly
varying depth or linearly varying breadth (pie-shaped beach) Lamb ; Wilson ;
Goldsbrough ; DeanDal ; Mei , some combinations of trenches with these types
of simple geometries Wilson , parabolic and elliptical depth variations Lamb ;
Wilson ; Goldsbrough . However, to our best knowledge, long-wave equations are
not solved for the geometries where the depth and breadth profiles have
nonlinear power-law variations. A special case of such variations in the ocean
environment is the equilibrium beach profiles. With this motivation, we derive
the analytical solutions of the long-wave equation for such profiles in this
paper. The solutions are obtained in terms of Bessel-Z Functions and Cauchy-
Euler Series, the harbor resonance and seiching hydrodynamics are analyzed for
such depth and breadth variations. Long-wave solutions, harbor resonance
conditions, and seiching periods derived for such profiles are compared with
their well-known counterparts existing in the literature. The uses,
applicability, and limitations of the proposed approach, our findings, and
possible future research directions are discussed.

##  II Solution of the Long-Wave Equation

###  II.1  Review of the Long-Wave Equation

Linear long-wave equation is first derived by Lamb Lamb  and in a simplified
form by Chrystal Chrystal , and extensively studied by many researchers since
then Lamb ; Chrystal ; Wilson ; Goldsbrough ; DeanDal ; Mei . The long-wave
equation is given by

$\frac{g}{b(x)}\frac{d}{dx}\left[b(x)h(x)\frac{d\eta(x)}{dx}\right]+\sigma^{2}\eta=0,$
(1)

where $h(x)$ denotes the depth and $b(x)$ denotes the breadth (width) of the
estuary. After the solution for $\eta(x)$ are obtained, transient solutions
can be constructed by using the time harmonics, i.e.
$\widehat{\eta}(x,t)=\eta(x)\cos(\sigma t)$. It is well-known that if linearly
varying profiles are used for either the depth function, $h(x)$, or the
breadth function, $b(x)$, Eq. 1 admits solution in terms of the Bessel
functions of order zero Lamb .

To our best knowledge, the nonlinear variations of the estuary depth and
breadth in the power-law forms are not studied in the existing literature.
With this motivation, in this paper, we generalize the solutions of the long-
wave equation to account for these types of variations of $h(x)$ and/or
$b(x)$.

###  II.2 Solutions in terms of Bessel Functions

In order to model the long-waves in an estuary having a power-law form of
nonlinear variation of depth and breadth, such variations can be formulated as
$h(x)=c_{1}x^{a}$ and $b(x)=c_{2}x^{c}$, where $c_{1},c_{2},a,c$ are some
constants. For these profiles, the long-wave equation given in Eq. (1) becomes

$\frac{d}{dx}\left[x^{a+c}\frac{d\eta(x)}{dx}\right]+\frac{\sigma^{2}}{c_{1}g}x^{c}\eta=0.$
(2)

This equation can be reduced to the Bessel equation using power-law change of
variables and transformations Greenberg . The solution of this equation can be
written as

$\eta(x)=x^{\nu/\alpha}Z_{|\nu|}\left(\alpha\sqrt{\left|\frac{\sigma^{2}}{c_{1}g}\right|}x^{1/\alpha}\right),$
(3)

where $\alpha=\frac{2}{2-a}$ and $\nu=\frac{1-a-c}{2-a}$. Here, $Z_{|\nu|}$
denotes Bessel function of first kind and order $|\nu|$, $J_{|\nu|}$, and
Bessel function of second kind and order $|\nu|$, $Y_{|\nu|}$, if
$\frac{\sigma^{2}}{c_{1}g}>0$, and the modified Bessel function of the first
kind order $|\nu|$, $I_{|\nu|}$, and the modified Bessel function of the
second kind and order $|\nu|$, $K_{|\nu|}$, if $\frac{\sigma^{2}}{c_{1}g}<0$.

Example 1: Equilibrium Beach Profile, $h_{eq}(x)=c_{1}x^{a}=Ax^{2/3}$, and
Uniform Breadth, $b_{uni}(x)=1$: In order to illustrate the possible usage of
our findings, we present the solution of the long-wave equation over an
equilibrium beach profile as our first example. Using an energy balance
approach, Dean DeanEq  derived the geometric equations that describe the
bathymetry of the equilibrium beach profiles which can be summarized as
$h_{eq}(x)=Ax^{2/3}$. Here, $A$ is the profile scale factor that depends on
the sediment size at the sea bottom DeanEq ; DallyDeanDal . Letting
$h_{eq}(x)=c_{1}x^{a}=Ax^{2/3}$ and $b(x)=1$, we obtain the long-wave profile
over an equilibrium beach profile

$\eta_{eq}(x)=x^{\nu_{eq}/\alpha_{eq}}Z_{|\nu_{eq}|}\left(\alpha_{eq}\sqrt{\left|\frac{\sigma^{2}}{Ag}\right|}x^{1/\alpha_{eq}}\right),$
(4)

where the parameters are $\alpha_{eq}=3/2$ and $\nu_{eq}=1/4$. We depict the
Bessel function solutions in Fig. 1 for three different periods of
$T=5s,10s,20s$. Since the condition $\frac{\sigma^{2}}{c_{1}g}>0$ is satisfied
the Bessel function solutions, not the modified Bessel function solutions, are
used. For the solutions plotted in Fig. 1, a profile scale factor of
$A=0.1m^{1/3}$ is used.

Figure 1: Long-wave solution over the equilibrium beach profile with a shape
factor of $A=0.1m^{1/3}$ for three different wave periods of $T=5s,10s,20s$ at
time $t=0$. a) Water surface fluctuation b) Depth variation.

As one can see Fig. 1, the typical Bessel solution behavior, that is the
oscillatory behavior and decaying amplitudes is apparent in the offshore
direction as expected. However, more strikingly, if the behavior in the
nearshore region is investigated, it is clear that the peak water surface
elevation occurs at shifted position compared to the shoreline position at
$x=0$. The location of peaks can be determined by checking the zeroes of the
Bessel function, as explained in more detail in Sec. II.5. This shift becomes
more prominent for larger wave periods. This shift and the changes in the peak
surface elevation can lead to many interesting engineering challenges. First
of all, it is clear that the resonance conditions are significantly affected
by equilibrium beach profiles and similar topographic features. The harbor
lengths, resonant frequencies, and positioning of quay walls and floating
units inside harbors and marinas should be carefully examined to take the
effect of such depth variations. Additionally, the maximum runup and breaking
locations can be affected by such depth and breadth profiles. This would lead
to many important lessons in tsunami modeling and swash zone hydrodynamics.

Example 2: Linearly Varying Bathymetry, $h_{lin}(x)=c_{1}x^{a}=0.05x$, and
Various Breadth Profiles-From Uniform to Parabolic,
$b_{uni}(x)=c_{2}x^{c}=0.05x^{0}=0.05$ to $b_{par}(x)=c_{2}x^{c}=0.05x^{2}$
for various values of $c$: In our second example, we consider the case of
linearly varying bathymetry which can be formulated as
$h_{lin}(x)=c_{1}x^{a}=0.05x$ if the parameters are selected as
$c_{1}=0.05,a=1$. Together with this linear depth variation, we consider 6
different cases of breadth variation, from uniform to parabolic profiles. The
parameters used for the breadth profiles are given in Fig. 2.

Figure 2: Long-wave solution for uniform depth and 6 different breadth
profiles for a wave period of $T=10s$ at time $t=0$. a) Water surface
fluctuation b) Breadth variation c) Depth variation.

We depict the water surface fluctuation for the various depth-breadth
combinations considered in Fig. 2. As it can be seen in the figure, as the
breadth of the estuary becomes larger, the oscillation amplitudes decrease and
the wavelength increases. However, in the nearshore region, no shift in the
location of the peak of the water surface fluctuation is observed.

###  II.3 Solutions in term of Cauchy-Euler Series

It is well-known that the solutions of the long-wave equation given by Eq. 1
are not described by Bessel or modified Bessel functions for a uniform estuary
breadth and a parabolic bathymetry. The solutions for this case are in terms
of the Cauchy-Euler series as described in many texts such as Greenberg ;
Arfken ; Whittaker . This is also true for the combinations of parabolic
bathymetry in the form $h_{par}(x)=c_{1}x^{2}$ obtained for $a=2$ with the
arbitrary breadth profiles in the power-law nonlinear form of
$b(x)=c_{2}x^{c}$. For this case governing equation reduces to

$x^{2}\eta_{xx}+(c+2)x\eta_{x}+\sigma^{2}/(c_{1}g)\eta=0.$ (5)

This equation is known as the Cauchy-Euler equation Greenberg . It is possible
to solve this equation by seeking a solution in the form of $\eta=Mx^{r}$
where $M$ is a constant Greenberg . Seeking a solution in this form and
solving characteristic equation gives two roots

$r_{1,2}=\frac{-c-1\pm\sqrt{(c+1)^{2}-4\sigma^{2}/(c_{1}g)}}{2}.$ (6)

The sign of the term inside the square-root in Eq. 6 determines the behavior
of the solutions Greenberg . It is possible to identify three families of
solutions:
i) If $(c+1)>4\sigma^{2}/(c_{1}g)$ the roots of the characteristic equation,
$r_{1,2}$ become real and solution becomes $\eta=Mx^{r_{1}}+Nx^{r_{2}}$ where
$M$ and $N$ are some constants Greenberg .

ii) If $(c+1)=4\sigma^{2}/(c_{1}g)$ the roots of the characteristic equation,
$r_{1,2}=(-c-1)/2$ are repeated and the solution becomes
$\eta=Mx^{r_{1}}+Nx^{r_{2}}\ln(x)$ Greenberg .

iii) If $(c+1)<4\sigma^{2}/(c_{1}g)$ the roots of the characteristic equation,
$r_{1,2}$ become complex and the solution becomes
$\eta=Mx^{r_{1}}+Nx^{r_{2}}$. If we re-express the complex roots by
$r_{1,2}=\lambda\pm\mu i$, then the solution can be expressed as
$\eta=Mx^{\lambda}\cos(\mu\ln(x))+iNx^{\lambda}\sin(\mu\ln(x))$ Greenberg .
Due to the linearity of the governing equation, the superposition principle
holds true, thus linear combinations of these expressions are also valid
solutions to the problem investigated. For the physical significance of the
problem, we also require the water surface fluctuation to be real.

Example 3: Parabolic Bathymetry, $h_{par}(x)=c_{1}x^{a}=c_{1}x^{2}$, and
Linearly Varying Breadth, $b_{lin}(x)=c_{2}x^{c}=0.05x^{1}$ for various values
of $c_{1}$:

In Fig. 3, we depict the solution of the long-wave problem for 4 different
parabolic depth profiles and linearly varying breadth. The results are plotted
for a wave period of $T=10s$.

Figure 3: Long-wave solution for 4 different parabolic depth profiles and
uniformly varying breadth profile for a wave period of $T=10s$ at time $t=0$.
a) Water surface fluctuation b) Breadth variation c) Depth variation.

As indicated in Fig. 3, we observe more dips and local crests with larger
oscillation amplitudes for shallower parabolic bathymetries. There is also a
slight shift in the dips observed in the water surface fluctuation in the
nearshore region. For deeper parabolic bathymetries, the amplitudes of
oscillation on $x$ are very insignificant, whole wave field exhibits a
Helmholtz mode characteristic.

Example 4: Parabolic Bathymetry, $h_{par}(x)=c_{1}x^{a}=0.005x^{2}$, and
Various Linear and Nonlinear Varying Breadth Profiles in the form of
$b(x)=c_{2}x^{c}=0.01x,0.1x,1x,0.01x^{2},0.01x^{3}$:

Lastly we present the solution of the long-wave equation over a parabolic
depth profile of $h_{par}(x)=c_{1}x^{a}=0.005x^{2}$ with 5 different linear
and nonlinear breadth profiles described by the functions
$b(x)=c_{2}x^{c}=0.01x,0.1x,1x,0.01x^{2},0.01x^{3}$ which are plotted in Fig.
4.

Figure 4: Long-wave solution for a parabolic depth profile with 5 different
parabolic breadth profiles for a wave period of $T=10s$ at time $t=0$. a)
Water surface fluctuation b) Breadth variation c) Depth variation.

Fig. 4 confirms that as the breadth of the estuary becomes larger, the
oscillation amplitudes decrease and the wavelength increases similar to the
case of Bessel function solutions given and discussed in Fig. 2. However, no
shifts in the locations of the peaks or dips in the water surface fluctuation
are observed.

###  II.4 Resonance of the co-oscillating tides in a channel with nonlinear
depth and breadth variations in the power-law form

Considering that the water surface elevation found in Eq. 3 should give a
finite solution at $x=0$ for physical significance and requiring that
$\frac{\sigma^{2}}{c_{1}g}>0$, Bessel-Z equation can be reduced to Bessel
function of the first kind of order $|\nu|$, $J_{|\nu|}$. For an unit
amplitude, the time-dependent form of the equation can be written as

$\widehat{\eta}(x,t)=x^{\nu/\alpha}J_{|\nu|}\left(\alpha\sqrt{\left|\frac{\sigma^{2}}{c_{1}g}\right|}x^{1/\alpha}\right)\cos(\sigma
t),$ (7)

where $\alpha=\frac{2}{2-a}$ and $\nu=\frac{1-a-c}{2-a}$. Matching the the
propagating wave solution at the entrance of the channel, $x=l$, which is
given by

$\widehat{\eta}(l,x)=\frac{H}{2}\cos(\sigma t),$ (8)

yields

$\widehat{\eta}(x,t)=\frac{H}{2}\frac{x^{\nu/\alpha}J_{|\nu|}\left(\alpha\sqrt{\left|\frac{\sigma^{2}}{c_{1}g}\right|}x^{1/\alpha}\right)}{l^{\nu/\alpha}J_{|\nu|}\left(\alpha\sqrt{\left|\frac{\sigma^{2}}{c_{1}g}\right|}l^{1/\alpha}\right)}\cos(\sigma
t),$ (9)

where $H$ is the amplitude of the wave coming from offshore to the entrance of
the channel. Accordingly, resonance conditions and characteristics of the
problem can be analyzed by evaluating the ratio
$R(x)=\left|\frac{\eta(x)}{\eta(l)}\right|$ which eventually leads to

$R(x)=\left|\frac{x^{\nu/\alpha}J_{|\nu|}\left(\alpha\sqrt{\left|\frac{\sigma^{2}}{c_{1}g}\right|}x^{1/\alpha}\right)}{l^{\nu/\alpha}J_{|\nu|}\left(\alpha\sqrt{\left|\frac{\sigma^{2}}{c_{1}g}\right|}l^{1/\alpha}\right)}\right|.$
(10)

Resonant modes occur where the denominator of this function is equal to zero.
Namely, the zeros of
$J_{|\nu|}\left(\alpha\sqrt{\left|\frac{\sigma^{2}}{c_{1}g}\right|}l^{1/\alpha}\right)$
should be found to determine resonant frequencies of different modes.

Example 5: Comparison between Uniform Depth, $h_{uni}(x)=h_{0}$, and Linearly
Varying Breadth, $b_{lin}(x)=c_{2}x$, Case with Equilibrium Beach Profile,
$h_{eq}(x)=c_{1}x^{a}=Ax^{2/3}$, and Linearly Varying Breadth,
$b_{lin}(x)=c_{2}x$, Case:
For $a=0$ and $c=1$, the problem reduces to the well-known long-wave solution
for the constant depth and pie shaped beach Lamb ; DeanDal  and parameters of
the Bessel-Z function become $\alpha=1$ and $\nu=0$. For this scenario, the
ratio $R(x)$ turns out to be in term of the Bessel functions of the first kind
of order zero, which can be formulated as Lamb ; DeanDal

$R(x)=\left|\frac{J_{0}\left(\sqrt{\left|\frac{\sigma^{2}}{c_{1}g}\right|}x\right)}{J_{0}\left(\sqrt{\left|\frac{\sigma^{2}}{c_{1}g}\right|}l\right)}\right|.$
(11)

For the second case of equilibrium beach profile with linearly varying
br8eadth, parameters become $a=\frac{2}{3}$, $c=1$, and accordingly,
$\alpha=\frac{3}{2}$ and $\nu=-\frac{1}{2}$. The resonance condition for this
can be formulated by setting
$\sqrt{\left|\frac{\sigma^{2}}{c_{1}g}\right|}l=z_{n}$, where $z_{n}$ are the
zeroes of the Bessel function, $J_{|\nu|}$. The values that make $J_{0}$ and
$J_{1/2}$ zero are obtained from Abramowitz and Stegun Abramowitz . In order
to illustrate the typical resonance conditions in an example, $c_{1}$ values
are selected as $3.4200$, $0.1000$ and $0.0171$ for the cases of constant
depth, equilibrium beach profile (note that $c_{1}=A$ for equilibrium beach
profile case) and uniformly varying depth, respectively, so that maximum depth
for each of these profiles is $h_{1}=3.42m$. The domain of the problem for
this example is selected to be $x\in[0m,200m]$, thus the value of $l=200m$ is
used as a representative channel length. Accordingly, angular frequencies of
first 5 resonant modes for this three cases are numerically calculated and
tabulated in Tab. 1.

Table 1: Resonant modes and angular frequencies ($rad.s^{-1}$). Resonant modes | 1 |  2 |  3 |  4 |  5
---|---|---|---|---|---
| Resonance frequency ($rad.s^{-1}$)
---
for $h_{uni}(x)=h_{0}$ and $b_{lin}(x)=c_{2}x$
0.0696 | 0.1599 | 0.2506 | 0.3415 | 0.4324
| Resonance frequency ($rad.s^{-1}$)
---
for $h_{eq}(x)=Ax^{2/3}$ and $b_{lin}(x)=c_{2}x$
0.0607 | 0.1213 | 0.1820 | 0.2426 | 0.3033
| Resonance frequency ($rad.s^{-1}$)
---
for $h_{eq}(x)=c_{1}x$ and $b_{lin}(x)=c_{2}x$
0.0555 | 0.1016 | 0.1473 | 0.1929 | 0.2385

As can be seen in the Tab. 1, resonant frequencies for equilibrium beach
profile are smaller than the ones for the constant depth, for the same breadth
shape of $b(x)=c_{2}x$. This is an expected result since in larger average
depths, the shallow water celerity $V=\sqrt{gh}$ is larger, thus the angular
frequency, $\sigma=\frac{2\pi}{T}=\frac{2\pi V}{L}$ is larger. It is also
possible to compute the resonant wavelengths by solving the dispersion
relation.

###  II.5  Seiching period for nonlinear depth variation in the power-law form

Transient solution of the long-wave equations given by Eq. 3 can be obtained
using simple time harmonics which leads to

$\widehat{\eta}(x,t)=x^{\nu/\alpha}J_{|\nu|}\left(\alpha\sqrt{\frac{\sigma^{2}}{c_{1}g}}x^{1/\alpha}\right)\cos(\sigma
t),$ (12)

where the form is derived for constant breadth, $b$, for the case of
$\frac{\sigma^{2}}{c_{1}g}>0$. For constant breadth, the power parameter of
the breadth variation is $c=0$, thus the parameters read as
$\alpha=\frac{2}{2-a}$ and $\nu=\frac{1-a}{2-a}$. The seiching period for this
wave can be obtained by checking the minimum of the $\eta$ function. For this
reason, $\frac{\partial\eta}{\partial x}$ should be set equal to zero.
Inserting $\alpha=\frac{2}{2-a}$ and $\nu=\frac{1-a}{2-a}$, and evaluating the
derivative one can obtain

$\frac{\partial\eta}{\partial
x}=\sqrt{\frac{\sigma^{2}}{c_{1}g}}x^{\frac{1-2a}{2}}J_{\left|\frac{1-a}{2-a}\right|-1}\left(\frac{2}{2-a}\sqrt{\frac{\sigma^{2}}{c_{1}g}}x^{\frac{2-a}{2}}\right).$
(13)

The fundamental seiching periods for different $a$ values are calculated by
evaluating the zeroes of the function at $x=l$ given by Eq. 13 numerically in
terms of the wavelength $L=2l$, $g$ and $c_{1}$. The corresponding seiching
period becomes $T=\mu\frac{2l}{\sqrt{gh_{1}}}$. Different $\mu$ values
corresponding to different $a$ values are tabulated in Tab. 2 for the case
where the parameters illustrated in Fig. 5 is taken as $h_{1}=20m$ and
$l=200m$.

Table 2: Coefficients of fundamental seiching periods for different values of $a$. a | $c_{1}$ | $\mu$
---|---|---
0 | 20.0000 | 1.0000
0.10 | 11.7741 | 1.0403
0.20 | 6.9314 | 1.0849
0.30 | 4.0806 | 1.1316
0.40 | 2.4022 | 1.1837
0.50 | 1.4142 | 1.2409
0.60 | 0.8326 | 1.3040
2/3 | 0.5848 | 1.3499
0.70 | 0.4901 | 1.3740
0.80 | 0.2885 | 1.4522
0.90 | 0.1699 | 1.5402
1.00 | 0.1000 | 1.6398

As it can be seen in Tab. 2, the fundamental seiching period for the
equilibrium beach profile depicted in Fig. 5 defined by the function
$h_{eq}=Ax^{2/3}$ is found out to be $T\approx 1.3499\frac{2l}{\sqrt{gh_{1}}}$
for the case of $A=0.1m^{1/3}$.

Figure 5: Equilibrium beach profile with $h_{eq}=Ax^{2/3}$.

The coefficient $\mu$ is independent of $c_{1}$, however, $T$ depends on
$c_{1}$ since it control the depth $h_{1}$. As expected, when the value of
$a=1$ are used, depth variation becomes linear thus the well-known value of
$T\approx 1.6398\frac{2l}{\sqrt{gh_{1}}}$ is obtained. Besides, at $a=0$,
again the expected value of $T=\frac{2l}{\sqrt{gh_{1}}}$ is found.
Additionally, as the parameter $a$ increases, the average depth decreases
which leads to higher fundamental seiching periods. This result is also
commensurate with previous findings of the literature. It is also possible to
find the seiching periods for higher modes by evaluating the other zeros of
the expression given by Eq. 13.

##  III Conclusion and Future Work

In this paper, we have derived the analytical solutions of the long-wave
equation for coastal regions having a nonlinear depth and breadth variation in
the power-law form. We have shown that the solutions of the long-wave equation
in such regions can be described by Bessel-Z functions. We depicted and
discussed the applicability and possible usage of our findings, including the
analysis of wavefields over equilibrium beach profiles. Additionally, we have
developed the seiching periods and possible resonance conditions over
bathymetries where such power-law variation can be observed. In addition to
the analysis of the wavefields over the nonlinear depth and breadth described
above, our results can also be used to investigate harbor resonance and shift
in resonant periods due to reconstruction, sedimentation, and dredging.
Hydrodynamics in the vicinity of sufficiently long seawalls with linear or
power-law nonlinear layout variations together with bathymetric variations and
around energy focusing structures and islands, wave reflection at inward
corners, runup on revetments such as those discussed in Goda , can also be
investigated in the frame of our analytical findings. Kirby et al. KirbyHaller
showed that low frequency motions which are indicative of wave seiching are
also observed in laboratory flume experiments, thus our results can be used to
investigate long-wave and seiching phenomena in laboratory environment for
various beach profiles including equilibrium beach profiles.

In a more general setting, our results can be used to estimate the resonance
frequencies of the acoustic and optical resonators, when different geometries
such as prismoidal shapes are used to resonate sound or light waves Auld ;
Lakinacousticresonator ; LigeometryResonator ; Chevalieropticalresonator . The
effects of doping, accretion and erosion of materials such as dust (see for
example FerreiraOliveira ; KimChoi ; Matsko ) can also be investigated within
the analytical frame we presented.

In near future, we aim to expand our analysis to various problems of great
importance for the ocean modeling community. We plan to extend our findings to
the waves with frictional damping and under the geostrophic effects. We also
aim to investigate the solutions of the mild-slope equation proposed by
Berkhoff Berkhoff  which can be formulated as

$\nabla\cdot(p\nabla\eta)+\sigma^{2}q\eta=0,$ (14)

where $\sigma$ is the angular wave frequency, $h(x,y)$ is the depth,
$q=\frac{1}{2}\left(1+\frac{2kh}{\sinh 2kh}\right)=\frac{C_{g}}{C}$ and
$p=gh\frac{\tanh kh}{kh}q=CC_{g}$ where $C$ denotes the celerity, $C_{g}$
denotes the group velocity, with the dispersion relationship given by
$\sigma^{2}=gk\tanh kh$ to the cases where depth and breadth variations obey
nonlinear power-law forms. Additionally, the 2D version of the long-wave
equation given by Eq. (1) is treated in Kirby1981 ; Ursell  for analysis of
the edge waves. It remains an open question that how the edge waves behave
over nonlinear depth and breadth variations in the power-law form. Possible
Bessel-Z analogs of the Laguerre polynomials and their reducibility will be
investigated for this purpose. It is also possible to extend our analysis for
the long-waves presented for various nonlinear depth and breadth profiles in
the power-law form to obtain solutions in terms of Laguerre, Chebyshev,
Hermite, or Legendre polynomials Arfken ; Whittaker ; HilbertCourant  and
their possible reducible extensions to their self-adjoint forms. Such
extensions would be useful for analyzing the effects of various harbor inlet
shapes on harbor circulations and hydrodynamics, as well as for analysis of
sound and light resonators having various geometries. Another possible
research direction to follow is to extend the tsunami runup calculations such
as those presented in Synolakis  to take the effects of equilibrium beach
profiles into consideration with solutions given in terms of Bessel-Z
functions and analyze the effects of realistic profiles on tsunami runup and
inundation. Investigation of fully nonlinear wave characteristics and their
evolution over equilibrium beach profiles using numerical approaches such as
those discussed in Baysci  is also planned.

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