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Kadomtsev–Petviashvili type equation for entirerange of relative water depths

Serdar Beji

To cite this article: Serdar Beji (2018): Kadomtsev–Petviashvili type equation for entire range ofrelative water depths, Coastal Engineering Journal, DOI: 10.1080/21664250.2018.1436241

To link to this article: https://doi.org/10.1080/21664250.2018.1436241

Published online: 12 Mar 2018.

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ORIGINAL RESEARCH PAPER

Kadomtsev–Petviashvili type equation for entire range of relative water depthsSerdar Beji

Faculty of Naval Architecture and Ocean Engineering, Istanbul Technical University, Istanbul, Turkey

ABSTRACTA Kadomtsev–Petviashvili type equation valid for the entire range of relative water depths;namely, shallow, intermediate, and deep, is derived. The new equation is capable of simulatingthe shallow water cnoidal waves and the deep water second-order Stokes waves equally wellbesides accounting for wave shoaling due to varying water depths. Linear shoaling properties ofthe equation for unidirectional sinusoidal waves are in complete agreement with the energy fluxconcept. A finite-difference scheme is adopted for numerical solution of the equation to demon-strate its performance against test cases and its wide range of possible applications.

ARTICLE HISTORYReceived 24 November 2017Accepted 19 December 2017

KEYWORDSKadomtsev–Petviashvili typeequation; cnoidal waves andStokes waves; arbitraryrelative depths; varyingbathymetry

1. Introduction

Since its first derivation by Kadomtsev and Petviashvili(1970) for the purpose of studying the stability ofsolitary waves the Kadomtsev and Petviashvili equationor shortly the KP equation has gained an appreciableinterest, particularly in the field of non-linear dispersivewater wave modeling. The KP equation is viewed asthe weakly directional version of the Korteweg anddeVries or KdV equation (Korteweg and deVries,1895), which in turn is a one-way propagation formof one-dimensional Boussinesq (1872) equation: aweakly non-linear weakly dispersive wave model.

Bryant (1982) derived a set of non-linear evolutionequations which, depending on the specification of coef-ficients, could describe fully dispersive waves as well asweakly dispersive waves of the KP type. Using theseequations, Bryant (1982) studied obliquely intersectingpermanent waves and indicated the rather good perfor-mance of the KP equation for the cases considered.

Tsuji and Oikawa (2007) considered the obliqueinteraction of two solitons of the same amplitude byemploying an extended KP equation with cubic non-linearity. Kodama (2010) gave an extensive survey ofsolitons and their modeling by KP equation. Yeh, Li,and Kodama (2010) investigated analytically andexperimentally Mach reflection of an obliquely inci-dent solitary wave while Li, Yeh, and Kodama (2011)made an in-depth review of the same subject.

In laboratory experiments, Hammack, Scheffner, andSegur (1989) generated finite-amplitude two-dimen-sional shallow water waves and mathematicallydescribed them by the exact solutions of the KP equa-tion. The capabilities of the KP equation are noted tobe not restricted to weakly directional waves; theextend of directionality encompassed by the KP

equation is observed to be beyond the usual validityrange of it but only slightly hindered by its linear non-dispersive character in transverse direction. Such satis-factory performance of the weakly non-linear modelshas quite recently been pointed out by Stiassnie (2017)with reference to the experimental confirmations ofBonnefoy et al. (2016). From this point of view, forpractical application purposes it becomes much moreimportant to place emphasis on the dispersion andshoaling characteristics of the equation rather thanhigher non-linearity. The work presented here aimsprecisely at developing such an equation that pos-sesses excellent linear dispersion and shoaling charac-teristics with just sufficient non-linearity.

Nadaoka, Beji, and Nakagawa (1994) introduced avertically integrated non-linear wave model withoutrestriction on relative depth. Likewise, making use ofthe variational principle of Luke (1967), Isobe (1994)gave various wave models for different depth-depen-dency functions. Continuing in the same line, Nadaoka,Beji, and Nakagawa (1997) presented a weakly non-linear wave model with full dispersion properties byemploying the Galerkin method which provided a setof coupled momentum equations for the componentsof horizontal velocity vector. Together with the conti-nuity equation the wave model results in 2Nþ 1 equa-tions, N being the number of wave numbers associatedwith velocity components contributing to the band-width of the wave field. Taking only three componentsN ¼ 3 is quite sufficient for simulating a broad-bandedwave spectrum while taking just a single componentN ¼ 1 is good enough for the simulation of waves witha narrow-banded spectrum. The special case of a singlecomponent was further elaborated by Beji andNadaoka (1997) to produce a non-linear mild-slope

CONTACT Serdar Beji sbeji@itu.edu.tr

COASTAL ENGINEERING JOURNAL, 2018https://doi.org/10.1080/21664250.2018.1436241

© 2018 Japan Society of Civil Engineers.

equation. Here, based on the work of Beji and Nadaoka(1997) a KP type wave equation is produced for mod-eling narrow-banded weakly non-linear water wavesover varying bathymetry for arbitrary relative depths.

The newly derived equation is discretized by aCrank–Nicolson type finite-difference formulationand various test cases are carried out. First, shoal-ing properties of the equation for sinusoidallychanging bathymetry are explored for incidentwaves of two different relative depths. Then, obli-quely intersecting two-dimensional cnoidal wavesand second-order Stokes waves are simulated.Finally, Whalin’s (1971) experimental measure-ments of non-linear waves over a convergingzone are compared with numerical results. Thesimulations clearly demonstrate that the KP typeequation derived here is capable of simulatingpropagation of linear and non-linear waves overthe entire range of relative water depths withacceptable accuracy.

2. One component wave model in combinedform

Before proceeding to the derivation of the KP typeequation, the combined form of the one componentwave model of Nadaoka, Beji, and Nakagawa (1997) isrecaptured. For the special case of a single compo-nent, Nadaoka, Beji, and Nakagawa (1997) give thefollowing continuity and momentum equations.

@ζ

@tþ � � C2

p

gþ ζ

!u0

" #¼ 0; (1)

CpCg@u0

@tþ C2

p� gζ þ ζ@w0

@tþ 12

u0 � u0 þ w20

� �� �

¼ @

@tCpðCp � CgÞ

k2�ð� � u0Þ þ �

CpðCp � CgÞk2

� �ð� � u0Þ

� �(2)

where ζ is the free surface displacement as measuredfrom the still water level, u0 is the horizontal velocityvector with components ðu0; v0Þ, and w0 the verticalvelocity component all at the still water level z ¼ 0. Cp,Cg, and k denote respectively the phase and groupvelocities and wave number, computed according tothe linear theory dispersion relationship ω2 ¼gk tanh kh for a prescribed dominant frequency ω anda local depth h ¼ hðx; yÞ. g is the gravitational accel-eration and � stands for two-dimensional horizontalgradient operator with components ð@=@x; @=@yÞ.

The above equations constitute a wave model fornarrow-banded weakly non-linear waves propagatingover varying depths. In the non-dispersive limit when

Cp ’ Cg ’ Cs ¼ ðghÞ1=2 the equations reduce to Airy’sshallow water equations. If Cp and Cg are approxi-

mated to the second-order as Csð1� k2h2=6Þ and

Csð1� k2h2=2Þ respectively and used in the dispersiveterms yields the Boussinesq equations for varyingdepth as given by Peregrine (1967).

By cross-differentiations and use of zeroth-orderrelations for non-linear terms Equations (1) and (2)may be combined into a single non-linear wave equa-tion for ζ as detailed in Beji and Nadaoka (1997).

ζtt � C2p

r �2ζ � C2

pð1�rÞrω2 �2ζtt � 3g

2r 3� 2r � ω2C2p

g2

� �2 ζ2� �

�� C2p

r

� � �ζ � 1

ω2 � ð1� rÞ C2p

r

� � C2

p

r �rh i

� �ζtt ¼ 0

(3)

where r ¼ Cg=Cp ¼ 1=2ð1þ 2kh= sinh 2khÞ and �stands for two-dimensional horizontal gradient operatoras before while subscript t denotes partial differentia-tion with respect to time. Equation (3) differs slightlyfrom the equation given in Beji andNadaoka (1997) withrespect to the last two terms. If the zeroth-order relationfor harmonic motion ζtt ¼ �ω2ζ is used in the last termand combined with the preceding linear shoaling termthe result becomes identical with that of Beji andNadaoka (1997). Further, removal of the harmonictime-dependency from the relevant terms and neglectof the non-linear term leads, after some manipulations,to Berkhoff’s (1972) mild-slope equation, which in turn,for constant depth gives the Helmholtz equation andLamb’s (1932) shallow water equation (p. 283) as specialcases.

3. New KP equation and its dispersionrelationship

A KP type equation based on the combined one-component model of Beji and Nadaoka (1997) asexpressed in Equation (3) is now derived. The classicalapproach proceeds by employing a non-dimensionalparameter for scaling the y-co-ordinate. Implicitly, theparameter is assumed proportional to the ratio oftransverse wave number to the main propagationwave number Ky=Kx . Then, except for the lowest-order term all the transverse terms are dropped. Inthis work, the discharge of these definite transverseterms is carried out in an informal manner by pointingout their physical functions. Namely, the dispersiveand non-linear terms containing the y � derivativeare all dropped as it is done by following purelymathematical arguments.

3.1. KP type equation for arbitrary relative waterdepths

For deriving a KP type equation from Equation (3)the x � direction is taken as the main propagationdirection with all the relevant terms included whilein the y � direction only the term representing non-dispersive linear propagation is retained. All the

2 S. BEJI

other y- dependent terms are dropped. Such a trun-cation renders the propagation in the y-directionlinear and non-dispersive; however, it does notimply a strictly weak-directionality as commonlytermed in the literature. The weakness in direction-ality may be associated with the asymmetric propa-gation properties of the final wave equation after theequation is cast into a one-way propagation modelvia co-ordinate transformation. The reduced form ofEquation (3) then reads

ζtt �C2p

rðζxx þ ζyyÞ �

C2pð1� rÞrω2 ζxxtt � 3g

2r3� 2r � ω2C2

p

g2

!ðζ2Þxx

� Cpr2

ð2rCpx � CprxÞζx � Cpr2ω2 ½2rð1� rÞCpx

� ð1þ rÞCprx�ζxtt ¼ 0

(4)

where Cpx and rx denote respectively the x � deri-vatives of Cp and r ¼ Cg=Cp. A co-ordinate systemmoving in the positive x � direction with thephase velocity Cp is introduced so that the evolu-tions of the wave form in this moving system areslow, permitting to write the following new co-ordinates:

σ ¼ x � Cpt; τ ¼ εt; (5)

where ε is a small parameter indicating the weakchanges of the wave form in time in the moving co-ordinate system. Expressing the terms in Equation (4)in the new co-ordinate system gives

ζtt ¼ C2pζσσ � 2εCpζστ þ εCpCpσζσ;

ζxtt ¼ C2pζσσσ � 2εCpζσστ þ εCpCpσζσσ; ζxx ¼ ζσσ;

ζxxtt ¼ C2pζσσσσ � 2εCpζσσστ þ εCpCpσζσσσ

(6)

where the terms containing the spatial derivative ofCp have also been labeled by ε to indicate they are an

order higher, and the terms proportional to ε2 are allneglected. Substituting the expressions in Equation(6) into Equation (4) and re-arranging results in

� 2εCpζστ �C2pð1� rÞ

rζσσ þ 2ε

C3pð1� rÞrω2

ζσσστ

� C4pð1� rÞrω2

ζσσσσ � αðζ2Þσσ þ εðCpCpσ � βÞζσ

� εC2p

Cpð1� rÞCpσrω2

þ γ

�ζσσσ þ 2εγCpζσστ ¼

C2p

rζyy;

(7)

where the coefficients α ¼ 3gð3� 2r � ω2C2p=g

2Þ=2r,β ¼ Cpð2rCpx � CprxÞ=r2, and γ ¼ Cp½2rð1� rÞCpx �ð1þ rÞCprx�=r2ω2 have been introduced for theease of notation.

The next step is to put Equation (7) into a formreadily integrable with respect to σ. Therefore, notingthat both Cp and r are spatially varying quantities, the

following equalities correct to the second spatial deri-vatives of Cp and r may be written

C2pð1� rÞ

rζσσ ¼ C2

pð1� rÞr

ζσ þ 2CpCpσ � Cpð2rCpσ � CprσÞr2

�ζ

" #σ

C3pð1� rÞrω2

ζσσστ ¼C3pð1� rÞrω2

ζσστ �3C2

pð1� rÞrCpσ � C3prσ

r2ω2

!ζστ

" #σ

C4pð1� rÞrω2

ζσσσσ ¼ C4pð1� rÞrω2

ζσσσ �4C3

pð1� rÞrCpσ � C4prσ

r2ω2

!ζσσ

" #σ

(8)

Making use of Equation (8) in Equation (7), re-arrangingand dividing the resulting equation by � 2Cp give

@

@σ½εζτ þ Cpð1� rÞ

2rζσ � ε

C2pð1� rÞrω2

ζσστ

þ C3pð1� rÞ2rω2

ζσσσ þ α

2Cpðζ2Þσ þ ε

Cpσ2

ζ

þ εCprω2

½ð1� rÞCpσ þ Cprσ�ζστ

� εC2p

2rω2½ð1� rÞCpσ þ Cprσ�ζσσ� þ Cp

2rζyy ¼ 0

(9)

The inverse transformation is carried out by using thefollowing expressions.

εζτ ¼ ζt þ Cpζx; εζστ ¼ ζxt þ Cpζxx; ζσ ¼ ζx; etc: (10)

Thus, the new KP type equation reads

@

@x½ζt þ Cpð1þ rÞ

2rζx �

C2pð1� rÞrω2

ζxxt �C3pð1� rÞ2rω2

ζxxx

þ α

2Cpðζ2Þx þ

Cpx2

ζ þ Cprω2

½ð1� rÞCpx þ Cprx�ζxt

þ C2p

2rω2½ð1� rÞCpx þ Cprx�ζxx� þ Cp

2rζyy ¼ 0

(11)

where the equation inside the square brackets corre-sponds to the KdV type equation given by Beji andNadaoka (1997). The only difference is in the linearshoaling terms, which may all be combined into asingle term by invoking the zeroth-order relationsζxx ¼ �k2ζ and ζxt ¼ kωζ, thus making it identicalwith the term in Beji and Nadaoka (1997). Expandedform of the new KP type equation is obtained byperforming the x � differentiation,

ζxt þ Cpð1þ rÞ2r

ζxx �C2pð1� rÞrω2

ζxxxt �C3pð1� rÞ2rω2

ζxxxx

þ α

2Cpðζ2Þxx þ

12r2

½rð1þ 2rÞCpx � Cprx�ζx

� Cpr2ω2

½rð1� rÞCpx � ð1þ rÞCprx�ζxxt

� C2p

2r2ω2½2rð1� rÞCpx � ð1þ rÞCprx�ζxxx

þ Cp2r

ζyy ¼ 0

(12)

where r ¼ Cg=Cp and α ¼ 3gð3� 2r � ω2C2p=g

2Þ=2r asdefined before. The above KP type equation

COASTAL ENGINEERING JOURNAL 3

represents all the known KP-like equations as specialcases: for constant depth when Cpx ¼ 0 and rx ¼ 0,

setting Cp ’ Cs ¼ ðghÞ1=2 and r ¼ 1 in the coefficients

of ζxx , ζyy , ðζ2Þxx , and r ¼ Cg=Cp ’ ½Csð1�k2h2=2Þ�=½Csð1� k2h2=6Þ� ’ 1� k2h2=3þ Oðk4h4Þ inthe coefficients of the dispersion terms ζxxxt and ζxxxxas approximated by weakly dispersive theoryresults in

ζxt þ Csζxx � h2

3ζxxxt � Cs

h2

6ζxxxx þ 3Cs

4hðζ2Þxx

þ Cs2ζyy ¼ 0;

(13)

where Cs ¼ffiffiffiffiffiffigh

pis the shallow water celerity. Equation

(13) has mixed dispersion terms but may be easily putinto the classical KP equation by using the zeroth-orderrelationship ζt ¼ �Csζx in the third term so that �ðh2=3Þζxxxt ¼ Csðh2=3Þζxxxx and the final dispersionterm becomes Csðh2=3Þζxxxx � Csðh2=6Þζxxxx ¼Csðh2=6Þζxxxx as expected. On the other hand, replacingthe fourth term ζxxxx by � ζxxxt=Cs gives� ðh2=3Þζxxxt þ ðh2=6Þζxxxt ¼ �ðh2=6Þζxxxt , which cor-responds to the dispersion term in the KP-like equationbased on the so-called BBM model (Benjamin, Bona,and Mahony, 1972). The terms proportional to Cpx andrx provide the wave equation with linear shoalingproperties which are exact for sinusoidal waves havingthe same frequency specified for the wave model. Thispoint is demonstrated in Section 5.1.

3.2. Dispersion relationship of KP equation

Dispersion relationship of the newly developed KP typeequation is now obtained. Let ζ ¼ ζ0 exp½iðKxx þ Kyy �ΩtÞ� represent a sinusoidal incident wave moving in anarbitrary direction on the xy plane, where the horizon-tal two-dimensional wave propagation takes place.Here, ζ0 is the constant wave amplitude, i the imagin-ary unit, Ω the circular incident wave frequency, Kx ¼K cos θ and Ky ¼ K sin θ the wave number componentsin the x � and y � directions, respectively. θ ¼arctanðKy=KxÞ is the angle wave propagation directionmakes with the x � axis. Substituting ζ into the linear-ized, constant depth form of Equation (12) gives

Ω

Kx¼ Cp

2

ð1þ rÞ þ C2pð1� rÞðKx=ωÞ2 þ ðKy=KxÞ2

r þ C2pð1� rÞðKx=ωÞ2

!;

(14)

where, as indicated before, Cp and r ¼ Cg=Cp arecomputed according to linear theory for a specifiedfrequency ω and local depth h ¼ hðx; yÞusing ω2 ¼ gk tanh kh.

Denoting CKP as the phase velocity vector of the KPtype equation and recalling the relationship betweenthe frequency and celerity Ω ¼ K � CKP with K ¼ Kxiþ

Ky2j for directional waves, the dispersion relationshipgiven by Equation (14) may be cast into the followingform for the phase velocity vector CKP of the newequation:

CKP ¼ Cp2

ð1þ rÞ þ C2pð1� rÞðKx=ωÞ2 þ ðKy=KxÞ2

r þ C2pð1� rÞðKx=ωÞ2

!KxK2

�K;

(15)

where K2 ¼ K2x þ K2

y denotes the magnitude square ofthe directional wave number vector K. The magnitudeof the phase celerity CKPj j divided by the shallowwater celerity Cs is

CKPCs

¼ 12

ð1þ rÞ þ ð1� rÞðKxh=khÞ2 þ ðKy=KxÞ2r þ ð1� rÞðKxh=khÞ2

!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tanh kh

khð1þ ðKy=KxÞ2Þ

s ;

(16)

where use has been made of ω ¼ kCp. The wave fre-quency specified for the wave model, ω, may or maynot be the same as the incident wave frequency Ω.When ω ¼ Ω then kh ¼ Kxh and for unidirectional caseKy ¼ 0 the linear dispersion and shoaling characteris-tics of the wave equation exactly matches with thelinear theory, CKP ¼ Cp. Linear theory dispersion rela-tionship for directional waves with wave number K is

CEXCs

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitanh Kh

Kh

r: (17)

For random waves with frequencies differentfrom the specified model frequency the matchwith linear theory is not exact anymore but quiteacceptable in a narrow band of frequencies and forrelatively small angles as demonstrated in Figure 1for directional waves by comparing CEX=Cs of lineartheory with CKP=Cs for four different directionangles. The specified model frequency ω is selectedsuch that kh ¼ π=2 (intermediate water waves) andKy=Kx is assigned to tan θ while Kh is varied overthe range 0� 2π. Note that for calculating CKP=Cs asa function of Kh, K2 ¼ K2

x þ K2y is employed so that

Kxh ¼ Kh=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðKy=KxÞ2

qwhich is the required vari-

able in Equation (16). As seen in Figure 1, the dis-persion relation of the new KP model CKP=Cs doesnot converge to unity when Kh tends to zero. Thereare two different reasons to this; the first reasonoriginates from the characteristics of the wavemodel of Nadaoka, Beji, and Nakagawa (1997). Thedispersion relationship of this particular wavemodel makes a tangential contact with the exactdispersion relation at the point Kh which corre-sponds to the specified kh value of the wavemodel. In the present example, ω for the wavemodel is selected such that kh ¼ π=2. Therefore,

4 S. BEJI

for θ ¼ 0o the dispersion relation of the model isexact at Kh ¼ kh ¼ π=2 but slowly diverges as Khgoes in either direction: Kh ! 0 or Kh ! 1. On theother hand, the dispersion relation of classical KPequation is different. It is an asymptotic expansionof the exact relation in the close vicinity of longwave limit Kh ! 0 but diverges relatively rapidly asKh gets larger.

The second reason is the effect of wave directionality.As the wave angle θ increases the dispersion relationship,Equation (16), diverges from the exact form due to the

presence of ðKy=KxÞ2 term in the numerator. This diver-gence is seen to be quite pronounced for θ ¼ 45o case ofFigure 1. For exactly the same reason exactly the samebehavior is observed for the dispersion relation of theclassical KP equation when the wave angle θ increases.Indeed, directional wave behavior of the dispersion rela-tionship for various wave angles demonstrated inFigure 1 serves as archetype for any kind of KP equation.

4. Numerical scheme

Feng and Mitsui (1998) proposed an implicit finite-difference scheme based on a Crank–Nicolson typeformulation for numerical solutions of the originalKdV and KP equations. In the same vein, Mekki andAli (2013) employed a Cranck–Nicolson discretizationfor the solution of a KP equation derived from theBBM equation of Benjamin, Bona, and Mahony (1972).Finite-difference schemes are preferable for practicalapplications therefore the new KP equation is discre-tized by a Crank–Nicolson type implicit formulationadapted from Feng and Mitsui (1998). While the pre-sent KP equation is quite different from the one con-sidered by Feng and Mitsui (1998) the finite-differenceformulation follows their scheme closely with the

exception of ζyy term, which is treated here inuncoupled manner with appreciable improvement incomputational efficiency.

4.1. Finite-difference representation of new KPequation

A Crank–Nicolson type finite-difference discretizationof Equation (12) results in

dxðζkþ1i;j � ðζkþ1

i;j ζki;jÞÞ2ΔtΔx

þ Ai;jδ2xðζkþ1

i;j þ ðζkþ1i;j ζki;jÞÞ

2Δx2

� Bi;jdxδ

2xðζkþ1

i;j � ðζkþ1i;j ζki;jÞÞ

2ΔtΔx3

� Ci;jδ4xðζkþ1

i;j þ ðζkþ1i;j ζki;jÞÞ

2Δx4

þ Di;jδ2xðf kþ1

i;j þ f ki;jÞ2Δx2

þ Ei;jdxðζkþ1

i;j ζki;jÞ2Δx

� Fi;jδ2xðζkþ1

i;j � ðζkþ1i;j ζki;jÞÞ

ΔtΔx2

� Gi;jdxδ

2xðζkþ1

i;j ζki;jÞ2Δx3

þ Hi;jδ2yðζkþ1

i;j þ ðζkþ1i;j ζki;jÞÞ

2Δy2¼ 0

(18)

where the coefficients Ai;j; . . . ;Hi;j, etc. correspondto those in Equation (12) in respective order. Notethat all the coefficients are spatially varying quan-tities and computed at each grid point. The cen-tered difference operators are defined as

dxζki;j ¼ ζkiþ1;j � ζki�1;j , δ2xζ

ki;j ¼ ζkiþ1;j � 2ζki;j þ ζki�1;j,

dxδ2xζ

ki;j ¼ ζkiþ2 � 2ζkiþ1;j þ 2ζki�1;j � ζki�2;j, and δ4xζ

ki;j ¼

KhC

/CS

0.5

1

0 π/2 π 3π/2 2π

CEX/CSCKP/CS

oθ=0

Kh

C/C

S

0.5

1

0 π/2 π 3π/2 2π

CEX/CSCKP/CS

oθ=15

Kh

C/C

S

0.5

1

0 π/2 π 3π/2 2π

CEX/CSCKP/CS

oθ=30

Kh

C/C

S

0.5

1

0 π/2 π 3π/2 2π

CEX/CSCKP/CS

oθ=45

Figure 1. Linear dispersion relationship of the new KP type equation for directional propagation compared with linear theory fora range of Kh values. Four different propagation angles θ ¼ 0o; 15o; 30o, and 45o shown for a model frequency ω specificationcorresponding to kh ¼ π=2 intermediate water waves.

COASTAL ENGINEERING JOURNAL 5

ζkiþ2 � 4ζkiþ1;j þ 6ζki;j � 4ζki�1;j þ ζki�2;j with i and j

denoting the indices multiplying the spacings Δxand Δy between the grid points in the x � and

y � directions, respectively. δ2yζki;j is the

y � direction analogy of δ2xζki;j . Superscript k stands

for the time level index and gives the actual timeof simulation when multiplied by the time incre-ment Δt. The non-linear function f ki;j is defined as

f ki;j ¼ ζki;jζki;j and the summation f kþ1

i;j þ f ki;j is

expressed in a semi-linear form f kþ1i;j þ f ki;j ¼

2ζkþ1i;j ζki;j as in Feng and Mitsui (1998) for carrying

out the implicit formulation without necessity ofiteration due to the non-linear term. The formalapproximation procedure may be found in Fengand Mitsui (1998); the informal way of expressing

ζkþ1i;j ζkþ1

i;j þ ζki;jζki;j as 2ζkþ1

i;j ζki;j may simply be accom-

plished by employing the approximation that

ðζkþ1i;j � ζki;jÞ2 ’ 0 hence ζkþ1

i;j ζkþ1i;j þ ζki;jζ

ki;j ’ 2ζkþ1

i;j ζki;j.Arranging Equation (18) by placing the unknown

new time level terms on the left and the known pre-vious time level values and the complete ζyy discretiza-tion on the right results in a penta-diagonal matrixequation. A penta-diagonal system may be solved byfour sweeps: the first two sweeps reduce the system toa tri-diagonal matrix and the next two sweeps solvethe tri-diagonal system. In the process employed herethe new time level values contained in ζyy on the rightare treated as known. However, in the first iterationonly the old time level values are used for computingζyy; the new time level values are introduced in sub-sequent iterations. For numerical stability, this treat-ment technique is found to be crucial. The uncoupledtreatment of ζyy necessarily requires iteration; onlythree iterations are observed to be sufficient for quitesatisfying accuracy for all the cases presented here.

4.2. Treatment of boundaries

Test cases and practical applications usually use anincoming boundary across which incident wave fieldis introduced and an outgoing boundary where wavesare radiated away outside the computational domain.Lateral boundaries are generally taken as imperme-able side walls with the so-called mirror condition,ζy ¼ 0, which is relatively simpler to implement.Specification of incoming waves at the first node ofthe domain is trivial; the prescribed incident waveform is assigned numerically to the new time level

surface elevation ζkþ11;j of the first node at each time

step. Normally, the second and following nodesshould be computed from the discretized wave equa-tion. Wave equations with spatially second-order deri-vatives result in tri-diagonal matrix systems and poseno problems in this aspect. However, in the present

case the presence of both third and fourth spatialderivatives gives rise to a penta-diagonal system, asindicated before. This problem may be overcomeeither by one-sided discretization of these higher-order derivatives or by simplifying the wave equationitself by appropriate means. One-sided discretizationshowed numerical instabilities and was abandoned.Instead, Equation (12) has been simplified by the useof the zeroth-order identities ζxx ¼ �k2ζ and ζxt ¼kωζ in the dispersion and shoaling terms so that thefollowing equation has been used for the second,third, and the ðn� 1Þ nodes.

ζxt þ Cpζxx þ αr2Cp

ðζ2Þxx þ12½ð1þ 2rÞCpx

þ Cprx�ζx þ Cp2ζyy ¼ 0

(19)

The last node in the x � direction, n, requires furthercare by backward differentiation of the second deriva-tive. Frequently a radiation boundary condition isimplemented on the outgoing boundary. Such a con-dition, being a further simplified and manipulated formof Equation (19), is used simply because the full waveequation cannot be discretized appropriately at or nearthe last nodal point. Ideally, it should be the waveequation itself propagating the waves out withoutany interruption. Pursuing such an idea forBoussinesq equations and comparing the results withthose obtained from the use of simple linearSommerfeld radiation condition Kiyokawa, Nadaoka,and Beji (1996) showed the remarkable advantage ofusing only the wave equations in preventing artificiallyreflecting waves. In the present case, the KdV versionof Equation (19) with only first-order derivatives andwithout 1

2 Cpζyy term is deemed suitable and used.

5. Simulations

Sample simulations are presented for exploring thecapabilities of the newly derived KP type equation.First, unidirectional wave propagation over a sinusoid-ally varying bottom topography is considered for inter-mediate and short waves. Simulations are depictedagainst theoretical wave envelopes drawn accordingto the energy flux concept. Second, similar to thecnoidal wave patterns experimentally produced byHammack, Scheffner, and Segur (1989), genuinelytwo-dimensional obliquely intersecting cnoidal andStokes waves are produced numerically. The remark-able point in these simulations is the ability of the newKP model to produce not only shallow water cnoidalwaves but also deep water Stokes waves. Finally, non-linear refraction-diffraction of waves over a convergingzone for three different periods are simulated andcorresponding harmonic amplitudes are comparedwith the measurements of Whalin (1971).

6 S. BEJI

6. 1-D linear waves over varying bathymetry

While accurate prediction of wave heights is an essentialengineering requirement that would be improved muchvia a reliable shoaling model, Stiassnie (2017), based onthe work of Bonnefoy et al. (2016), concludes that weaklynon-linear theories are satisfactory enough tomodel non-linearities. As the inclusion of higher-order non-linearity isnot crucial in practical applications the present workplaces particular emphasis on the accuracy of linear shoal-ing prediction and presents here two sample cases oflinear wave propagation over a sinusoidally varyingdepth. The water depth is initially h0 ¼ 10m, reduces tohm ¼ h0=3 ¼ 3:33m at mid-length of channel and thenincreases toh0 ¼ 10magain. Thewaveperiod is adjustedsuch that at the channel entrance and exit h0=L0 ¼ 1=10for the first case (long to intermediate incidentwaves) andh0=L0 ¼ 1=2 for the second case (short incident waves).The incident wave amplitude is set arbitrarily to a0 ¼ 1mas the simulation is performed with linearized equation.Figure 2 shows for both cases theperformanceof the newKP type equation against the wave envelope drawnaccording to the energy flux concept a2Cg ¼ Const:withCg taken from theexact linear theory. The agreementwith theory for both cases is nearly perfect; such anagreement confirms the reliability of the present modelin computing wave amplitude variations over varyingbathymetry for entire range of relative depths.

7. 2-D cnoidal and stokes waves

In order to demonstrate that the new KP equation iscapable of simulating shallow water cnoidal and deep

water Stokes waves equally well, two numerical simula-tions of obliquely intersecting wave trains are performed.For cnoidal waves, the period is T ¼ 2:89 s, the waveheight to water depth ratio H=h ¼ 0:05, and the ellipticparameter m ¼ 1� 0:215, as selected from the labora-tory experiments (CN2) of Goring and Reichlen (1980). ForStokes waves, the period is T ¼ 3:57 s and the non-line-arity parameter kSH ¼ 0:15 with kS denoting the wavenumber according to the third-order Stokes theory. Atthe incident boundary along the y � axis two separatewave trains directed at angles þ θ and � θ to thex � axis were generated and superposed. This superposi-tion doubled the values of non-linearity parameters ofthe incident wave fields for both cases. Generation of awave train with an angle to the x � axis was realized byintroducing a time phase lag of Δtl ¼ jΔy sin θ=c in theargument of the wave function at each y � nodej ¼ 0; . . . ;m. This method is exactly the same as theone used by Hammack, Scheffner, and Segur (1989) intheir laboratory experiments. The directed wave anglewas taken as θ ’ 19:5o so that the directed wavelengthin y � direction was three times the generating wave-length Ly ¼ L= sin θ ¼ 3L for both cases while L ¼ 4m,h=L ¼ 1=20 (shallow water) for cnoidal waves andL ¼ 20m, h=L ¼ 1=2 (deep water) for Stokes waves.Again for both cases the time resolution was Δt ¼T=60 s and the spatial resolution in the propagationdirection was adjusted to give a Courant number thatwas nearly unity. Due to numerical stability problems forΔy,Δx the resolution in the y � direction was severaltimes coarser in comparison with the x � direction. Thesimulation of deep water Stokes waves was more sensi-tive in this aspect and required even coarser transverse

Distance along channel centerline (m)

Nor

mal

ized

am

plitu

de

-1

0

1h0/L0=1/10

Distance along channel centerline (m)

Nor

mal

ized

am

plitu

de

0 250 500 0 250 500

-1

0

1h0/L0=1/2

Figure 2. Amplitude variation over a sinusoidally varying bathymetry. Envelopes (dashed lines) are computed from theconstancy of energy flux. Left: Long to intermediate incident waves, h0=L0 ¼ 1=10. Right: Short incident waves, h0=L0 ¼ 1=2.

Figure 3. Perspective views of obliquely intersecting cnoidal waves (left) and Stokes waves (right).

COASTAL ENGINEERING JOURNAL 7

grids. Figure 3 shows perspective views of fully devel-oped cnoidal and Stokes wave fields.

7.1. Non-linear waves over a topographical lens

The last case considers non-linear wave conver-gence over a bottom topography that acts as afocusing lens (Whalin, 1971). The physical wavetank used in the experiments was 84 ft ¼ 25:6mlong and 20 ft ¼ 6:096m wide. In the middle partof the wave tank, 11 semicircular steps were evenlyspaced to form a topographical lens. The equationsdefining the bottom are given in Whalin (1971).Experiments were carried out by generating regularsinusoidal waves with periods T ¼ 1; 2; and 3 s.Primary wave and harmonic amplitudes along thecenterline of the wave tank were obtained at var-ious stations.

Figure 4 shows the perspective view of the numer-ical wave tank after 38 periods of simulation. Theconverged and diverged regions of wave forms andthe relatively shortened length of the waves in theshallow region near the end of the domain are nota-ble features of the simulation. For all three cases, thesimulations were performed with a span-wise resolu-tion Δy of 1=12 of the wave tank width. For incidentwave period T ¼ 1 s, the incident wave amplitude isa0 ¼ 1:95 cm in water depth of h0 ¼ 0:4572m. Thetime-step and the x � direction resolution wererespectively Δt ¼ T=50 s and Δx ¼ Lm=50m with Lm ¼1:3m denoting the mean wavelength computed asthe average of the deep water h0 ¼ 1:5 ft ¼ 0:4572mand shallow water hs ¼ 0:5 ft ¼ 0:1524m wave-lengths, which are L0 ¼ 1:5m and Ls ¼ 1:1m. Notethat this case is very close to deep water conditionssince h0=L0 ¼ 1=3.

Figure 5 top graph depicts the measured dataand the computed results for the primary waveand the first harmonic amplitudes for T ¼ 1 swaves. In the middle the case for T ¼ 2 s and a0 ¼0:75 cm is shown for the primary wave and twoharmonics. The bottom graph makes the same

comparisons for T ¼ 3 s and a0 ¼ 0:68 cm. All thecomputations were carried out with Δt ¼ T=50and Δx ¼ Lm=50.

Figure 4. A perspective view of the fully developed numerical wave tank for Whalin’s experiment of T ¼ 1 s waves.

Distance along centerline (m)

Nor

mal

ized

ampl

itude

0

1

2

Distance along centerline (m)

Nor

mal

ized

ampl

itude

0

1

2

Distance along centerline (m)

Nor

mal

ized

ampl

itude

0 5 10 15 20 25

0 5 10 15 20 25

0 5 10 15 20 250

1

2

Figure 5. Top: Whalin’s experiments for incident waves ofT ¼ 1 s. Middle: Incident waves of T ¼ 2 s. Bottom: Incidentwaves of T ¼ 3 s. Measured and computed harmonic ampli-tudes along the centerline of the wave tank. Solid line:computation, scatter: experimental data.

8 S. BEJI

8. Concluding remarks

A Kadomtsev–Petviashvili type equation for the entirerange of relative water depths has been derived. Thenewly derived equation is capable of simulating shal-low and deep water waves equally well. Linear shoal-ing characteristics, which are essential in accuratewave height calculations, are tested for relativelylong and short waves propagating over varyingwater depths with very satisfactory results. The newwave model may be used for accurate estimation ofwave conditions for waves propagating from deep toshallow regions.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCIDSerdar Beji http://orcid.org/0000-0002-1927-9262

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