modeling of wave characteristics parameters in the

Advances in Theoretical and Applied Mechanics, Vol. 12, 2020, no. 1, 1 - 12HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/atam.2020.423

Modeling of Wave Characteristics Parameters in

the Shoaling Zone with Saint-Venant Equations

M. A. Houekpoheha1,2, B. Kounouhewa3,2

and C. N. Awanou3

1 Institut de Mathématique et de Sciences Physiques/Universitéd’Abomey Calavi(IMSP/UAC) 01BP 613 Porto-Novo Rép. du Bénin

2 Centre Béninois de la Recherche Scientifique et Technique(CBRST) 03BP 1665 Cotonou Rép. du Bénin

3 Laboratoire de Physique du Rayonnement/Université d’AbomeyCalavi(LPR/UAC) 01BP 526 Cotonou Rép. du Bénin

This article is distributed under the Creative Commons by-nc-nd Attribution License.Copyright c© 2020 Hikari Ltd.

Abstract

In coastal areas, even in the absence of any disturbance, ocean wavesfeel the effect of the seabed. This perturbation, function of the localwater depth d, causes the variation of almost all their characteristicparameters. Shallow water, is divided into two areas: the intermediatezone (zone lift or area shoaling : area before the bathymetric breaking)and Swash and Surf areas (area after breaking). In the shoaling area,the wave amplitude will gradually increase and their profile is herebyamended while their wavelength decreases.

To take into account nonlinear effects generated by the action of theseabed on the deformation of the free surface of the ocean, we used theSaint-Venant equations with topography term (Green Naghdi equations)to model the variations of these parameters according to the local waterdepth from the intermediate zone to the point of breaking depth.

The results show that the amplification of the height is proportionalto d−

14 while the remaining wavelength decreases proportional to the

square root of the local water depth d. This seabed action also inducesa phase delay in the wave propagation.

Keywords: swell; Saint-Venant equations; wave height; wavelength; waveshoaling; breaking point

2 M. A. Houekpoheha et al.

1 Introduction

The swells are surface waves (oscillations of ocean-atmosphere interface)that are maintained by the continuous oscillation between kinetic and poten-tial gravity energies: they are gravity waves. They are generally created bythe wind action on the acean surface. As for swells, they are adults waves andpropagate following an direction almost straight. The coastal zone, in whichthe swell feels the perturbation effect of the seabed, is divided into two ar-eas: the intermediate zone (which carries the lifting swell before bathymetricdéferlemnt) called shoaling area and the area after bathymetric breaking zonesconsisting of Surf and Swash. In fact in this area, the waves lose their natureand take a sinusoidal shape almost sawtooth.

In previous work, Phillipes Bonneton, Vijay G. Panchang and other au-thors have shown that the basic parameters that characterize waves are theirwavelength (L), their period (T ), the vertical elevation of sea level (η) or peakto valley height (H), their phase velocity (Cϕ) and group velocity Cg. Theywere described by presenting their forms of variation and analytical expressionsof some from the linear theory which the dispersion relation can be writtenω2 = gk tanh(kd). This dispersion relation, established for the Airy wave(sinusoidal waves which have a small amplitude) can not better describe theevolution of certain parameters such as the peak-to-trough which amplifies andeven the length of height wave which contracts.

To better describe analytically the variations of these parameters duringpropagation in the area shoaling (including height and wavelength), we com-bined the Saint-Venant equations and those of Green Naghdi [11,19,24]. To thisend, a system of equations that will provide analytical expressions that justifythe numerical and experimental results [16,23,28] obtained by Phillipes Bonneton,Vijay G. Panchang and others.

Table of figures:Figure 1a: Distortion of the free surface of the ocean when the increases waveFigure 1b: Variations for the height and wavelength of the waveFigure 1c: Evolution of phase late induced by the seabed according to slopeFigure 2a: Variations of the phase velocity and group velocity as µ functionFigure 2b: Evolutions of the local water depth db and wave height Hb at thebreaking pointFigure 3a: Variation of the breaking criterion according to the wave height Ho

Figure 3b: Variations of the local depth at the breaking point according to Ho

Figure 3c: Variations of the wave height at breaking point Hb according to Ho.

Modeling of wave characteristics parameters 3

2 Materials and Methods

2.1 Variations of parameters according to linear theory

The only parameter which remains practically constant along the wavepropagation is the time period T ≈ Cste [20]

The dispersion relation of regular waves is ω2 = gk tanh(kd). From thisrelationship, it follows the expression of the wavelength L, depending on thelocal depth of water is L = gT 2

2πtanh(kd). In deep waters: d ≥ Lo

2, [18,20]

tanh(kd) ≈ 1, then L = Lo =gT 2

2π∀ µ =

d

Lo≥ 1

2(1)

In the area shoaling, using the dispersion relation of the waves Airy (lineartheory), we have:

L = Lo tanh(kd) if µ ≤ 1

2(2)

Let δ′ = LLo

the coefficient of variation of the wavelength according to thelinear theory, posing

kd = 2π dLo

LoL

= 2πµδ, we have: δ′ =

L

Lo=

{1 if µ ≥ 1

2

tanh(

2πµδ

)if µ ≤ 1

2

(3)

According to this theory, the expressions of the phase velocity and groupvelocity are respectively

Cϕ = Cϕo

√δ′ tanh

(2πµ

δ′

)with Cϕo =

√gLo2π

Cg = Cgo

(1 +

2kd

sinh(2kd)

)√L

Lotanh(kd) with Cgo =

1

2

√gLo2π

(4)

If δ′1 = CϕCϕo

and δ′2 = CgCgo

are coefficients of variations of the phase andgroup velocities, then:

δ′1 =

1 if µ ≥ 12√

δ′ tanh(

2πµδ′

)if µ ≤ 1

2

and δ′2 =

1 if µ ≥ 1

2(1 +

( 4πµδ′ )

sinh( 4πµδ′ )

)√δ′ tanh

(2πµδ′

)if µ ≤ 1

2

(5)

Taking δ′3 as the ratio between group velocity Cg and phase velocity Cϕ,we have:

δ′3 =CgCϕ

=

12

if µ ≥ 12

12

(1 +

( 4πµδ′ )

sinh( 4πµδ′ )

)if µ ≤ 1

2


Assuming that the total energy carried by wave is preserved in the shoalingarea, according Peronno G. (2003), we have:[13]

H = Ho

[tanh(kd) +

kd

cosh2(kd)

]−1/2

(6)

If κ′ = HHo

is the coefficient of variation of height along the linear theory(coefficient shoaling):

κ′ =

1 if µ ≥ 1

2[tanh

(2πµδ′

)+

( 2πµδ′ )

cosh2( 2πµδ′ )

]−1/2

if µ ≤ 12

(7)

2.2 Breaking criterion

Kaminsky and Kraus [1993] conducted a comparative study based on ananalysis of previous studies (409 cases) covering a wide range of curvaturesof wave and beach slopes. They show a positive relationship between thebreaking index and slope (with a good representation of γ = 0.78 for theslopes tan β ≤ 0, 1). They advise to use the following formula where Hb is thewave height and db to local water depth at breaking point:[2,9]

γ =Hb

db= 1, 2

(√LoHo

tan β

)0,27

(8)

2.3 Boundary Conditions

The vertical elevation of the free surface of the sea, in its complex form ata deep water, obtained by solving the Laplace equation in the Airy’s theoryis ηo = Ho

2ei(kox−ωt). In the area shoaling where ϕ is the phase shift induced

by the seabed, let η = H2ei(kx−ωt+ϕ) [6]. From these expressions, we deduce the

conditions at the limit of deep water and the intermediate zone:

L(Lo2

)= Lo =

gT 2

2π; H

(Lo2

)= Ho and ϕ

(Lo2

)= 0 (9)

2.4 Characterization of the swell in the shoaling area

To describe changes in the wavelength and the peak to valley height incoastal areas during propagation, we combined the Saint-Venant equations andthose of Green Naghdi. The system of equations below to establish analyticalexpressions that justify the numerical and experimental results obtained by P.Bonneton, Vijay G. Panchang and other authors.[16,23,28]


{ ∂h∂t

+ ∂∂x(hu) = 0

∂∂t(hu) + ∂

∂x(hu2) + ∂

∂x

(gh2

2

)= gh ∂d

∂x

with

η = −1

g

(∂Φ∂t

)h = η + du = ∂Φ

∂x

[19,22,23]

(10)

As the amplitude of the wave is negligible, all non-linear terms are negli-gible, and neglecting the convective effect to that induced by the acceleration(∂(ηu)

∂x≈ 0 and u∂u

∂x� ∂u

∂t), the Saint-Venant equations leads to:

1

g

∂2η

∂t2− ∂d

∂x

∂η

∂x− d∂

2η

∂x2= 0 (11)

In the coastal zone, Xiao-Bo Chen, (2006), implies that the effect of seabedon the deformation of the free surface induces a phase shift ϕ on wavepropagation[6] and includes η = H(x)

2ei(kx−ωt+ϕ) = ηa(x)e

−iωt withηa(x) =

H(x)2ei(kx+ϕ). Thus, the above equation becomes:

d∂2ηa∂x2

+∂d

∂x

∂ηa∂x

+ω2

gηa = 0 (12)

I. I. Didenkulova and al. (2008), have chosen the origin of the space at thefree surface on the beach. In this condition, the equation of the tangent atalmost flat seabed, inclined at an angle β relative to the horizontal, is: [15,16]

z = ax = −d where a = tan β (13)

By the change of variable ηo(x) = ηo(d) and using the equation of thetangent to the seabed local point of water depth d, we obtain:

d∂2ηa∂d2

+∂ηa∂d

+ω2

ga2ηa = 0 (14)

2.5 Solution of the differential equation

The general solution of this equation is any linear combination of Besselfunctions of ν order , Jν and Yν the first kind and the second kind or Hankelfunctions ν order H(1)

ν and H(2)ν the first kind and the second kind.[5]

ηa(α) = A.Jν(α) +B.Yν(α) or


ηa(α) = A.H(1)ν (α) +B.H(2)

ν (α) with α = 2aω

√d

g(15)

And since ηa is a complex function whose real part must be a decreasingfunction of d before the breaking, the general solution can be written:

η(x) = η(d) = A.

√2

παe[i(α−

π(2ν+1)4−ωt)] with α = 2aω

√d

g(16)

3 Results and Discussion

3.1 Height and wavelength in the shoaling zone

To general solution ηa(d) = A.√

2παe[i(α−

π(2ν+1)4 )], to the previous boundary

conditions, and identification with ηa(d) = 12H(d)ei(kx+ϕ), we deduced for

µb ≤ µ ≤ 12that:

H(d) = Ho

(Lo2d

) 14 = Ho

(1

2µ

) 14 ; L = Lo

√2dLo

= Lo√2µ

ϕ =(2a√π + π

a

) (√2dLo− 1

)=(2a√π + π

a

)(√2µ− 1)

(17)

If δ = LLo

and κ = HHo

are respectively the coefficients of variation of thewavelength and the peak to valley height of the swell before the breaking point,their expressions according to the parameter of the wave train µ = d

Loare:

κ =H

Ho

=

1 if µ ≥ 12(

12µ

)1/4if µb ≤ µ ≤ 1

2

δ =L

Lo=

{1 if µ ≥ 1

2

(2µ)12 if µb ≤ µ ≤ 1

2

and ϕ(d) =

{0 if µ ≥ 1

2(2√π tan β + π

tanβ

)(√2µ− 1) if µb ≤ µ ≤ 1

2

(18)

3.2 Wave height and water depth at the breaking point

Using criterion breaking expression proposed by Kaminsky and Kraus since1993, for low slopes tan β < 0.1 and the height previously found, we have:

For tan β < 0, 1 we have:

db =

[LoH4

o

4,1472

(√LoHo

tan β)−1,08

]1/5

Hb = Ho

[0, 6

(LoHo

)1,135(tan β)0,27

]1/5 (19)


3.3 Variations of phase and group velocities

Let δ1 = CϕCϕo

, δ2 = CgCgo

and δ3 = CgCϕ

the coefficients of variation of thephase and group velocities and the ratio of these two speeds.

δ1 =

1 if µ ≥ 12√

δ tanh(

2πµδ

)if µb ≤ µ ≤ 1

2

δ2 =

1 if µ ≥ 1

2(1 +

( 4πµδ )

sinh( 4πµδ )

)√δ tanh

(2πµδ

)if µb ≤ µ ≤ 1

2

and δ3 =

12

if µ ≥ 12

12

(1 +

( 4πµδ )

sinh( 4πµδ )

)if µb ≤ µ ≤ 1

2

(20)

Figure 1:a Figure 1:b





• Figure 1: a) shows the evolution of the deformation of the free surfacein the absence of seabed ηo and in the presence of seabed η. The evolution ofthese two parameters shows that the perturbation effect of the seabed inducesa phase delay of η from ηo. The amplitude remains constant in the deep waterand grows for effect of the seabed in the area shoaling.• Figure 1: b) shows the evolution of coefficients of variations of height and

wavelength according with the linear theory one hand, and according to theresults obtained from the Saint-Venant equations other.

- According to the results of Saint-Venant equations, the height is amplifiedto the breaking point, but the linear theory shows that the height decreases to0.2 ≤ µ ≤ 0, 5 before start grow up at the breaking point.

- For the wavelength, it slower further according to our results, but itcontracts slower for linear theory.C. Potel, M. Bruneau• Figure 1: c) represents the variation of late phase ϕ induced by a pertuba-

tive action of seabed on the incident waves from deep water. This phase variesdepending on the local water depth d and depending on the seabed slope.• These results show that the height increases when the local water depth

d decreases to the breaking point and it is proportionally to d−1/4. The wave-length decreases and is proportionate to the square root of the local waterdepth d. The perturbative effect of this background also induces a phase delayϕ < 0 ∀ tan β during propagation in the coastal zone compared to that of theincident wave comes from deep water. This phase shift ϕ < 0 depends on theangle β sea breaking relative to the horizontal and the local water depth d.• Variations of δ1 and δ′1 show that the phase velocity is constant in deep

water but decreases in area shoaling when the local water depth d decreases.But this decrease is more marked in the case of the results obtained with theSaint-Venant equation than the linear theory.• Variations of δ2 and δ′2 show that the group velocity is constant in the

deep waters. But in the area shoaling, this speed decreases according to theresults obtained while she believes using the linear theory and to the point


where µ ≈ 0, 16 prior to start decrease. In reality, the speed should not growwhen dispersive system is taking up a slope (probable source of energy loss).• The coefficient δ3 and δ′3 are 1

2in deep water, they reveal that the phase

velocity is twice the group in deep water. These reports increase in shoalingarea and tend to 1 when d tends to 0. But growth is more pronounced in thecase of the linear than the results obtained by the equations of Saint-Venanttheory.• The curves of figure 2 b) show firstly that the local water depth at the

breaking point decreases when the bottom slope increases. This decrease isgrowing more when the height Ho decreases, but very little when the wave-length decreases. On the other hand, the wave height at breaking point Hb

increases when the bottom slope increases. This growth is more pronouncedwhen the height Ho increases, but very little when the wavelength increases.• The curves in figure: 3 b) and figure: 3 c) show that the bathymetric

breaking point of wave, characterized by (db, Hb), depends not only on thebottom slope but also the value of Ho.• As for the curves in Figure 3 a), they show that the criterion of a breaking

wave is not constant (0.78 ≤ γ ≤ 1, 2 for tan β < 0, 1): it increases whenHo decreases but increases when the bottom slope increases.

4 Conclusion

From previous results, we deduce that all the parameters studied are almostcontants in deep waters. The period of propagation of a wave is almost constantalong its propagation. The wavelength, the phase velocity and the groupvelocity decreases when the local depth of water decreases in the shoaling zone.This wavelength is proportional to the square root of the local water depth d.Unlike the Airy theory gives the impression that the group velocity believes inthis area, the group velocity decreases because the system is dispersive.

For the peak to valley height, it is increased in the intermediate zone whenthe remaining depth decreases and is proportional to d−1/4 but reaches itsmaximum value at the breaking point. The effect of seabed pertubatif inducesa phase delay ϕ < 0 in the propagation of the wave associated with the swell.This phase shift depends not only on the local depth of water d but also theinclination of the bottom with respect to the horizontal.

Overall, the use of Saint-Venant equations is better to model the variationsof the characteristics parameters of the swell in the area shoaling unlike thelinear theory which does not provide information, not only on the decreaseof the group velocity, but also on the evolution of the phase shift induced bythe perturbative action of seabed on the deformation of the free surface of theocean in this area.


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Received: February 15, 2014; Published: December 29, 2020

modeling of wave characteristics parameters in the

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