RUN-UP OF CNOIDAL WAVES

•

RUN-UP OF CNOIDAL \1AVES by

Geir Pedersen

PREPRINT SERIES - Maternatisk intitutt, Universitetet i Oslo

•

..

..

ABSTRACT

Run-up of cnoidal waves .

Run-up heights for cnoidal waves on sloping beaches are calclulated by time integration starting with an incoming wave- train in the off shore region. To get a simple treatment of the freely moving shoreline the governing Boussinesq equations are expressed in Lagrangian coordinates. The equations are solved. by a finite difference techinque. A nearly periodic motion at the shoreline is usually reached after a few periods. The results are discussed by spectral analysis and related to run-up heights for

solitary waves and sinusoidal waves described by linear hydro-·

static theory .

..

•

of papers motivated by different applications in connection with

~. ··i ·;;·~··" , __ -·--~~-· ... -.. :, . . . '. <r: ·rJ·:

tsunanrles~'waves gen~rated

bi

slides or avalanches, utilization of

~-!' · -:···~\:::; i 1 '~.~~.-:! .. 'I. • - .. J; . .:..·_ -~. -·t : \. , . ; '.. ~ _; ;.- · i ,r· ·· -;; .. : ~ ...

wave energy, sedimentation and excitation of edge wav.es accompa-

'· ·' ^'.>·.

nied by cusp formation .. ·

'.,\ · Q-(!"~fft' ·-\ ~e~n.Ylih¹ ;f''1"9;5'8 >

iitta S\fti!"\

.Jb~l.,.,

:( 11i7tf

~o·niH.cfe'red' ' run~uef Qfi\ :an iri~i\tt:':C:e TncY£nije ·pi'arie · u't;1h9 non· linear

hydrostatfc

:th9ory··'. :The' t•p1'-tccr:bt1itiy .E)if' the&e dal.o\.i1a;t:lon$ are' 94!~rely

reatr,iet:.e.d . byt ;d,i;f;ficulties ·· in .d:p;f ining a.pP1rnpriate incoming wave•s.- earrier ( 196'6) patched solutions obtained by this theory tuc ·an-'

qtf~•hore solution based on linear theory and qonsta:nt:FJ1epth. A

si.milar patching is .used in chapter 4 of this paper. Calculation

dl

icf1i.taJi

~~d;J~;,

ruriJJp have beet\ s'tudi.ed

;by,

'Pedersen & Gjevik

;<.\~'~3).,..~~J Kfff ~~~ ~nd .~i~~t.~ .o~~~) .. ,.lflH~h~,;.~;,•,~~~

^.+ep9^rt

;~,,

are.

;~ft~~~' ~l-.:!'t~u~r ·r:~~;-,u~ 3 ,:?f: ~h~, pbef,io~~~!'· cn?i~~l, 'f.~v;8.~,.,us~Jl9

.. , . dispersive and nonlinear equations·. in bpth offshore al'ld near shore

~~i.,~·' .. __ . ~r~ ,:·~:_,_.,\,'.,~ ),··~·::'),;j l ~---:! ·.-~;6;. :.~·-'~ r···i< \'I'" f''• ~ :~:·.~-:<,~~' 1 -#:· .,-,'. .:

regions. Our theory does only de.scribe nonbreaking cases. There-.. ^{~ .} .,

• ..

fore calculations cannot be performed for very high amplitudes or

" ,:) , ::_. -~ .r.J~ , ,_:::·· ~ -~~-,r_-, :.:_ . f .·. C _;·, /

too gently sloping beaches.

·-r

Run-up of periodic waves have been proved to be unstable with respect to edge wave excitation in the't 'c'ase of a cbntst'antly slop- ing bottom (Gusa & Davis (1974)). With a topography consistin9 of a .sloping beach joined by an offshore basin df d9nstant,depth these instabilities may be expected to occur fot short incoming w,aves- and g:ep.:;tt,~''ll'.lf~tr1rr-~~!m SJlcrr-.~ae-~~-: the pl~Jl•'W'ave solutions

-~-~---· -·--· ---··---~----·--- - - - · ---·--·---.·----·-·-·----~---· ···---- -------··----

reported in this paper will be realizable only in narrow channels.

Ap~imate

:llmH:'.s'·

~dz- the parameter ranges with p6saible 'edge··

•

- 3 -

After developing the ~qu~tions we have no further use o~ ..

-, ' ;: ,_ ~-, _, ,-.. '- :: ;·' ' i ;' .. ~ ·; ' . t.'. ,;,;

the the forJDCil parameter _{• i'.;•} E, and for convenience we rescale .. the'.

' " " . ' < ' _ , • : • - . • - ' " - } ~ • ,,,';::l ,·:,)•'.,-:1.:··< c , .. ·,:,,_',~ ' , • • •

equations.: To obtain a scalin~ dependent only_ on th~ .:i.nher~nt .,

.> •-. ^{. i , '}

·,

ge~~~t~ies w,e _us~ .. h_{0 .} ~lso as a hor~~oi:it_al +~ngth scale~· The rescaled form of the equations are fm1n,d by substituting 1 . for

E in (2.4).

P'or uniform depth;,e may write H=l+TJ where ^T) is. the surface displacement from the equilibrium position. In this case there e.~i,st,,._~q:ent wa,>v.e torn.: J'l!=Y·( +), u=u ( <P) . where · +sx+Gt •' The wa,v~ ;propaga'\:~$ ,~i,1;h constant, speed c· which d~rtds on " · amplitude and wavelength. One group of sua.h solution:•· :ta the solitary waves described by the formulas:

Y =A sech² pC++cori.st) (2.5)

U = -cY/(l+Y)

'. {12 .f~)

'• .• _, : :c:t.:.: (-1

+Ar~

', .

p~('~A;;.)\ I

:ic.

c2.

^{1 >}

···{; '-···.'·' .,~;···.J-i.~ •.·. ~:-·-:.C!.~' ,,\ , .... ~, '<:;~~'' ."' •.-.. /! .·:'~_.): :~;-·,.·.

where

JC

^·.is·· the amplitude.· The run-up behaviour of the solitary wave is'' ihvestlgat;;d: :by

P~d~r-~~n

^&

Gje~.i~

. ( 198,3 ) .•

The''>1,llffh:·

group' :of

stati~nary sh~ped

waves is the periodic cnoi.da'i·

ii~v€~:.

·In '6rder

to

^obtain

u~i~~e

soluti.bns for given am-

' . ,. ., ... ~ ^{; .• -} ~' ^{- ·. j ' . .} i3 . . ·_ . ~-7·." " j \ . " : , ' • -~- ~ ^•

plitUde1 A, and' wavelength, A, we have chosen the mean horizontal

. ' .. .... ^~ .

•C;..• .~ ':)o--°"r 1 f1'. 1 -:

volume flux and the mean surface elevation to be zero. The surface

•' ·-·. -, ^...

displacement can be expressed in terms of the Jacobi elliptic function

where· A=(Y· _.y : ·: )/2 ^and' 6 max min .

m=2A(l-(1+6 2-A2)-1+o+A)-l. Eq.

< 2.

et ..

^~

].

...

=

.:....(y . +Y ) and 2 max miµ

(2.6) applies beth for cnoidal and solitary waves. The relations between c, 6 and A, and the wave

•

introQilced ··.,for ·e,Qnvenience. The overba.cs· corresponding to de.pith

seoQnd.Grder:,a:ccuzraoy: di:fference equations:

where

= -

cfi~>

² H . J

OJ

6 u. n a J

ff~

=

H~+~

+

j6tq~-l

and J '

.l.. ·:

^.J

(3.2)

(3.3)

where all lower indexes equal j +

~

and all superscripts equal n -

2

1 and therefore are omitted.·Values·of h

x are obtained by

computing the ratio: '·: ·. ·1

' - ,1 .. '

-a n-~ -a n-~

= 6ah(x >j+~/oa(x >j+~ (3.4)

'·' ...

Thf·s·

'.k-epresentation is chosen to match the repre•ntalidn of the

H bH

Ho ba t~:C:~\,·M~~r s~ly~~91"'the e~H.ti-~Aon~ "'(3'.·}), ^{(3 .4_)} for- the .. n+~ · n+~

and . q .. valp.-es at. t~t1t , we cotapute' Hj ····; · x)j+~ by:,.

. ^{. ;•} , U.+~ n· I

J ^{j .}

q. n J '

wll'idh; ~6niptet~s'"t.h~ computation~ 'cycle' o"f 'on~;, time step.

( 3. 5)

. t .

·'.At a

~reely

·m6t,in:g shoreline," locatelt at -a="o,

~

^{use the}

boundaey' cc>ndfti6n' :

q~=H~i+\;_0 ~ the'.

^{value ..}

x~11

needed to colllpute

. \

th-"e'i"lght hand: side of

'< J • 4 >

is !otlna

^by

iirieii.:r·

·exti~poiation i~oni

. x~·Jt ..

^{ahd' '}

kj/~ /:At:

a rooving vertical' wall, located at a~(m+~)lia

we use the condition

u~~=(un)wall"

ifh~· Ische~. ci~flned"tiy-

(3. 2)'

thr~u9h.

(3:4> is

--~~'b~i.tt~d

...

t~·

t}le same tests as ··the· scheme used in (PG) with equally good, or

--·7 -

the comJUqn point ~-.cat

a ,

^{'whe·re 9 denotes the} inclination angle Qf t4e tiop'6.n9, baa-th (see',fi9ur:e 1 )·. By patchi·ng the non-lihe~r -

,.,,

so,lp,ti;ona o£ Cartie11.: l& Gr-een11pan 'to: 1:,,J::heaJ:- off-shore

-Sdh.kfbni

i t i~1;pqs.,,ib~e·, t;o a~touttt nor nonlinear effe-cts' near' the !fre.ely

lffbV:..

,inq. -eboreU:n~ wttere they' o'tWiol!Bl.y>aPe 'ih't>s~·· 'ifa1)ottanl~~- tri~lulfi.bh' of these nonlinearities will however ·not change the computed

v~lues ~W the_. ·l'l\axim~ run-up -·heights a-e long as· the patching c.Qnd.it.;iOQ,s are line.arized. We ther•fol'e·· :use> linea':tized ·· thedry< iri b<;>~·.;regions .• A ·•wr:veYt.•C1>f'" n2L~ '~p- oalC4J'lati6ns bas.a' on''

• ' J ~ f . '

(4.3)

w~r f:h;1d Jhf! bopn~Ed.:Q~~r Js}'lore s9l1r1t~pn of ¹ ~~'! l ~4-. 2J expt"e.sseti by t}l~ .. -~~S@~~ n~J?G!~?~ /~~r.Zce~.o ~r~~i:; .. :

.. ,: ~-.

·. · .• i:, , i

The requirement that ^Tl and u are continuous at ·x=cot_e qe"."'

... r' -. .'- ^{r".c '{~ . • l '} "' "

termines the phase of the reflected wave and the e?tpreipsion fo:r

C.-''"": ^{' ; ·:.,}

the run-up height becomes:

'· ';-., ... ..

~1 .'.

J ₁(2w)cos·wt) /\

where

~w

^{. .} cot; 9. We denote 'the maxilnurtr run-Up height by R.

The- .a;a.tio .between ..

·a -

obta-ined

from· (1'.

5} -and A is depleted in

fi~ a

as ft1netien of

Jw/~

which ts-

the

ll!ngt.h

~f

the incoming . wave re.lati ve ·to the len<gth of the sloping bottom. The ratio R/A ia a monotonic increasing_ function of

~

with stationary

inflex~

ion point corresponding,

to

efgenfrequen61es ~£cir -the, nearshoi:e

~TJi'=¹ ~(:xd ^sin fty-wt)

.. ,,~.: ^.. O{·w²co~·.e·x)· ''.fdr · 'x<faSt ·a

·::.·· ---.: ... ·- !-- :· ';_~·-.~ -

- _ . . -IK2-w2x

'lr:•'? ··~ ·' • .. · .. .,. :;:. ^{for x>cot}

a

\ .. ,

where" y is 'the : it:longshore r, eo&rdina te . and Q

t·s) .

is a bounded' . solution Qf the. equatian

. .. d

^dQ

(1-K 2 s)Q + ~(s ~) = n '· · · ds

a.

^¥

{'

whe~'\ile ·have adepted the riotae'ions of ·Abramowitz and ~fte'~un

.

( 1965 )1;; '.Pat.el\iti·9 '.fil:i'. :x=i:~ '0

'then cj\tns

the '<ffepetr'sion. rela'tion :.

quencies for eabh . ^{K· •.} 'i'he Small~St Value

df

' ~ I . :

A

(&)

(4 •. 8)

. . . ~ .'' " l •

any· ettfje' wave; ·'bf 'th~ form · (4 ~·1'-~ ·'is

C=·!:t'. SJ.

Subharrrib'n'ic res'onance

may·'tltus''

;~

4fxpetted

to

occur ^{fot (.)}

^{c'ot ·a-~s}

.

and 'synchron'ol.lst''"

reisonan·~ £or

.w· cot

0~2·.-,5.: 'EVt!rt'thdti~ dfspe~sion ~hd nOrilinecff effeo.ts'. .a-1•e '-h~TeC-t.ed ~iri the"aoove equation th:e' results

ihdica'ttts '

that' ed94f· .we.ve~r:e«eit.at:ibn·~ wfl·l-' be' O'r

mlnoF

imPortanee

for'.mo$t

-·---~ ---·-.---, --~~--~--.-,-.---..-

5 • NVMJiSJU,CAL RE:SUL!l'S. . ·

, .... App:l-!ettiar·of"equationa·'"('!T1 )b1!! .fl')--a@llaMs M1ste'fi~g"c)t

the second derivative of the bottom.function· h~ Hence the bottom

- 11 -

· For :.q;a.1uet5 Of A ·'close tb' 6 the'·backwasl1 becomes so cloie t.G

breakittg t'hat ·a·~~riodle. stat·e cannot' be reach;ed ~ If tbe ilmplitii:de of the incoming wave is decreased, or ·the' arigle: ·cif: the sYope'

fa ·

increased, this limit is lowered as demonstrated by the results

... _;.. l . . (-,

depicted in figure 8. Figure 6 shows time-series for the stronger

noti~'li--hear eiiff~

'·

:Jtfb<O~.

l

,~,

tm.3"0'\

At·

a•deit' · 9, .. _, ·•' Rear'ly steady

'J:'*!ri ..

odic M6tri6h Is readied'

as

f'or'. A.=o.

o·s

~· it'He~ Bh~rtff'ine f\ldti~¹;

'·fa··:ow

the' other·· hand' eubj~cted to relati'1ely

'rapta

f1:uctua.t'.i.otil! "'1i«~h' inten~i~ displays_ nb a~rEfnt t!~-cay in t1rae. · The&~ f lttoi·ttat'i·dns are 1!6b 'Si.OV:. t.d·be··.ftsootat<Etd with 'the ·grid·;S:&~& u.rora 'atrel

ar.- ·

m0st. pronc>unc;~ ··in the ~CkWa~h. ·~ersen· C'l'gsn"'¹ob1S~tvedvgertertl'i..

ti.Ott ¹<21f short .pe:17ifoaicr:wa:veaPat :th& end of the ::b8cltwa&h pe·riod·~:t sol:i.~fty ~-·irt··nUntelr.f<!a'l ;·caleula.·t.ioWs ..

rn·

~he preae!it':ftidd~· the ra'fke:·o'f.·.0f'f11li0Jte radia'lt.ion°"-tif.'the;&e

wavie rtey;·'be''too:s1ow~:for a

steadiy ·.state ''soluti"'1;: tic be readhed wi-t:hin a ~•«'iitsn'able i'n~ra""' tion time •. Implementation of dissipative effects may remove.'-thf&·

p~~.;· ^{i ,_.} '' .J;

.·.' In al.'l \simula·t1:-t>ne . ..tih-e nime·' incrEiMent 'is chosen' as art· 'int·e- gral fraction·, ny , l/N;~ ·'·of. the' •pe:r1o<1· of .t;1fe· ·in1?0ming wave. Fast Fourier trans.form applied to the horizontal dispacements

owr· ;a:··

"

--~-- -"'." ~~-f -...-,----_ ---;---: ,. ,-'

----~-~~

. ^{'· .}

~ . - ^{... ..}

_ _,_· ___ " :L_, .. .;,;._._._ ,_____,;___ ··--c__:'._ _ _ _ _ _ ...,_ _ _ _____:, __ :_

where the amplitudes A are real and positive, and

o

^denotes

phases. When a periodic state is reached only the amplitudes with integral second subscript should be noticable. In figure 7 we have depicted some spectra for. 0=20, A=0.05 0 and K=3. These spectra clearly correspond to nearly periodic motion even though some

.

^.

'BUJl\En"ical · int<egration of ·t.he · Bouasin:e~q equations· ihows that

for ·incoMiilg cn6£dal waves with mbderate amplitudes .(A<O.l) a neatly periodic Jbbtiori'

fs

fie'ached in the shore regi·on 'which enables predictions of run-up heights. For higher amplitudes generat.ion of short periodic oscillations at the shoreline rnakes reliable run-up estimations impossible. Short incoming waves

combined with gentle slopes lead to very large run-up and backwash and the numerical solutions collapse. In these cases the waves are probably close to breaking. The present theory is however not suitable for developing any breaking criterion because of the limitations of the long wave assumption. For the same reason no results are reported for .incoming waves shorter than five times the maximum water depth.

Contrary to the run-up heights of sinusoidal waves de'termined

by analytical solutions of the linearized hydrostatic equations, the run-up height of cnoidal waves is not a strictly decreasing function of the wavelength. The results in figure 2 show that the second harmonic of the cnoidal wave gives rise to extremaes even in the linear hydrostatic approximation. Inclusion of nonlinearity and dispersion does not alter this irregular behaviour but reduces the maximum run-up height considerably (up to30%).

In the limit of infinitively long waves the .run-up heights of··

cnoidal wavei:; equal to· the run-up heights of s-olitary waves of the same wave height. With exception for very short waves the maximum run-up heights of cnoidal waves are smaller than for solitary waves. The reduction is at most 20-30% for•the cases investigated

in the present report.

- 15 -

APPENDIX

A

The cnoida1 and solit'ary wave solutions

A perm<S.nent wave form solution has the form

T) = y ( • ) I

U

= u ( • ) I • : X + £t (A 1 )

where 11 is t'he surface elevation and

c

is the propagation speed ·Of the- wave· whieh i·s to be determined as a part Of the solution. To: achieve solutions of the form

fAl)

the depth has

to

be constant arid (2. 1).._'(2. 2) simplifies to

an

autonomous set of ordiria-ry differential equations in

+·

The continuity equation reads:

(C+U)Y' = -(l+Y)U' (A2)

and ID<S.Y immediately be integrated to

C. +U (A3)

where IC is a constant of integration. Performing the appropriate ID<S.nipulations the equation of motion is integrated twice to give the· first order equation:

Y' ² ==

L

IC

2 ^ca 1 ^-H>C~ 2 ^-H>Ca 3 ^-a>

^{- 3} c2 ^q(Y) (A4)·

where H•l+Y and a1 ^a2^a3^{=·1C2 •} The two remaining degrees of freedom for the zeroes a₁,a2 . and ^a3 correspond to the constants of the two. integra-t.ions. Phase-pla-ne analysis of (A-4) reveals easily the existence of two types of bounded solutions: the solitary waves and the periodic cnoidal waves. Additional conditions are needed to get unique solutions from (A4). For a solitary wave we demand

..

where E is the elliptic integral of second kind as defined by Abramowitz and Stegun.

A JDOving rigid wall is abtays situate.d at the same Lagrangian point. Therefore the surfa~e elevation ^y

w at a wavemaker generating cnoidal waves is ^~ound from the relation:

dY w

dt

=

^.LTt(a

at ..

w ^,t)

⁼

^Dn

x=x w

=

(c-u )Y' w w

x=x w

(A10)

where D is as defined below equation (2.2) and the subscript w . refers to quantities evaluated at the wall. '"1.e fourier expansion (2.9) is most easily obtained by applying a, standard perturbation

pr~ecure to the differensiated version of (A4). For large values of >../A the cnoidal wave train becomes a series of solitary ~aves

and asymptotic

anal~sis

leads

to~

6 • A -

tc

./2A"{3 • 'l'he second term has to be accounted for when comparing the behaviour of cnoidal waves to the behaviour of solitary waves.

- 19 -

REFERENCJiiS

Abramowitz, M. & Stegun, I.A., 1965. Handbook of mathematical functions. Dover Publications, New York.

Arntsen, '1J.A., 1978. Theoretical and experimental study of wave run-up on relatively steep slopes. Cand.real. thesis.,

Univettsity e·f,·oslo. (Written in Norwe9ian).

Carrier, G.F., J.966• Gravity waves on water of variable depth.

J~Fluid Mech. 24, 641-659.

Carrier, G.F. &.G~eenspan, H.P., 1958. W•ter waves of finite amplitude on a sloping beach. J.Fluid Mech. 4, 97-"109.

Gjevik, B. " Pedersen, G., 1981. Run-up of long waves on an inclined plane. Preprint Ser. no. 2, Oept .. of Maths,

Univer~.ity of Oslo.

Guza, R.T. and Davis, R.E., 1974. Excitation of Edge Waves by Wave'•· Incident on. a Beach. J .Geophys .Research 79, 1285 ... 1291.

Guza, R.T. & Inman, D.L., 1975. Edge waves and beach cusps.

J.Geophys.Research 80, 2997-3012.

Hall, J~v. & Wat:tfl, GcoM•, 1953. LaboratoryinYestigation of the vertic.al rise of solitary waves on impermeable slopes.

u.s.

Army, Corps of Engrs, Beach Erosion Board, Tech.Memo no. 33.

Hibberd,

s.

& Pergrine, D.H., 1979. Surf.and run-up on a beach:

a uniform bore. J.Fluid Mech. 95, 323-345.

Kim, S.K., Liu, P.L-F & Ligget, J.A., 1983. Boundary integral equation solutions for solitary wave generation, propagation and run-up. Coastal Engineering 7, 299-317.

FIGURE CAPTIONS

Fi~re.l. Definition sketch of geometry for linear hydrostatic calculations.

Figure 2. The ratio R/A as function of >.. for 0=20 .• ⁰ (i) : Cnoidal shaped wave, A=0.05

(ii): Sinusoidal wave.

FigUre 3. Definition sketch of the computational domain on which.

run-up calculations for cnoidal. waves are perforD\ed. TJ(x,O) is the initial surface .elevation and the fat line. at a=L.

indicates the wavemaker.

Figure 4. Time stories of the horizontal displacement .of particltt~

. ·.' ~· .'[.' 'f •· - _ , •

for A=0.05 and 0=20 • The scaling factor for the til'.116 axis· ⁰ is the period, T, and the vertical axi• is 111calec1 by Ft,•A>../2 K.

~ corresponds to the ma·ximum horizontal particle displace- ment of a linear sinusoidal wave with wavelength >.. and amplitude A. The graphs are numbered as follows: (i) 1'.=8, a=O (shoreline), (ii) >..=8, a=2.76, .(iii) >..=14, a•O, (;i.vJ

>..=14, a=2.BO. For (i) and (ii)

we

have T=877, ~=0.064 and for (iii) and (iv) T=l 4. 39, ~=O. 111. The displacements in (iii) are small because the particle is situated .close to a node for the first harmonic. We note that a negative dis- plCljcement of the shoreline corresponds to run-up.

Figure 5. Run-up of a cnoidal wave with .A=O.O!?, >..=24 compared - - - - -

--- -~- - - - ^--~

-~- ---~--

-- --- -- - --~

to-tlie-ruri=up-of

a solitary wave with amplitude 0.1. The time of maximum run-up of the solitary wave, tm' has been chosen

· to coincide with a maximum for the cnoidal wave. To make the · time stories comparable we have accounted for the.non zero

·"

..

...

I"")

-

^Q)

cc j...j

;::!

x ^t:J.

~ -~ ~

.>. II

..

2

-2

( i) x/~

(ii) 0.5 x

++

-o.

^5.

Figure 5.

t/T

t-t

_m

"

al 1-4 t;\ ::J

-rl r.... ^-.-l

)(

co

0

co

0

~

0

x

)(

0

..

...

d

)(

-

•.-l

·.-l

M

....

:::>

)(

)(

co

0

co

0

...

0

x

.. !'t

R/A R/A

+ (v) (vi)

+

4. +

4 • ⁺

+

+ + ^{+ +} +

3. + 3. _{+ + + +} ⁺

10 20 10 20

R/A R/A

+ (vii) (viii)

+ ₊ +

+ + 5. ₊

+ + 5.

+ ⁺

4 . _4. ⁺

3. _3.

10 20 ₁₀ 20

Figure

.ab.