Extreme elevations and slopes ofinteracting Kadomtsev-Petviashvili

solitons in shallow water

Tarmo Soomere, Jüri EngelbrechtMarine Systems InstituteInstitute of Cybernetics

Tallinn University of Technology

Linear and nonlinear equationsfor surface waves

Classical surface waves:solutions to the Laplace equation

0=++ zzyyxx φφφ

Nonlinear surface wave equations:Korteweg-de Vries (KdV) equation (1+1D)

Kadomtsev-Petviashvili equation (weak 2+1D)( ) 036 =+++ yyxxxxxt ηηηηη

06 =++ xxxxt ηηηη

KdV Solitons KP

Each single waveis a KdV soliton

Stem waves along vertical wall due to random wave incidenceMase, Memita, Yuhi, Kitano, Coastal Engineering 44 (2002)339–350

If a large-amplitude wave reflectsfrom a wall: Mach stem

Near the wall the two wavecrests merge into one structure- this is a truly nonlinear effect

If there is wall: Mach stem

Soliton 1

Soliton 2

Common crest(Mach stem)

If there is no wall: phase shift

A part of solitons merge

Soliton 1

Soliton 2 Common crest

The central part may be particularly high

Soliton 1

Soliton 2 Common crest

Wave height increases drastically:up to four times

L=???

The extension of the high hump:geometry as a first approximation

First approximation: ‘geometric’length of the area of high elevationsInteraction pattern: defined by

heights and directions of waves( )111 , lk=κr

( )222 ,lk=κr

( )( )221

2

221

2

12 kkkkA

+−−−=

λλPhase shift

parameter2

2

1

1

kl

kl −=λ

Equal amplitudesolitons 1

1212

lnlAL =

42

2

12~

kllA−

=

Wide wave hump ~ resonance lk ≈2

Length of the high hump:important only in near-resonant case

k/kres

A12, L12l1=l2 fixed, k1=k2=k

10.5 0.750

1

2

3

4

A12

L12

A12<0

1

1212

lnlAL =

42

2

12~

kllA−

=

The extension of the particularly high hump:not always follows the geometry

Soliton 1

Soliton 2 Common crest

Interaction soliton

Area where elevation exceeds sum ofamplitudes of incoming solitons =

= Area where interaction soliton exceedsthe amplitude of incoming solitons

(a proof exists today only for a1=a2)

OK, wave height increases up to fourtimes, but the slope of water surface?

Look here!

The central part of the structure isparticularly narrow

Single soliton

Single KdV/KPsoliton rescaled tothe height of theinteraction patternInteraction

pattern

The slope may EIGHT TIMESexceed the slope of single solitons

Soomere,Engelbrecht,Wave Motion2005

Interaction of solitons of unequalamplitudes I: amplification is weaker

Amplitude amplification for unequalamplitude solitons

( )21)max( aaMampl +=

Interaction of solitons of unequalamplitudes II: extensive bending of

the larger soliton

Waves maycome fromanother /

unexpecteddirection

Global maximum of surface elevation?

located at theinteractioncentre for A>1and equalamplitudesolitons (asimple proofexists)

k1=0.600 k2=0.300 l1=0.200 l2=-0.200 A_12= 4.789k1=0.660 k2=0.300 l1=0.200 l2=-0.200 A_12= 43.325k1=0.666 k2=0.300 l1=0.200 l2=-0.200 A_12=428.582k1=0.667 k2=0.300 l1=0.200 l2=-0.200 A_12=42806.703

All extreme points == singularity points of isolines

Soomere,Phys. Lett A,2004

Are interacting Kadomtsev-Petviashvili solitons realistic

on the water surface?

Long surface waves in shallow areas

Photo by L.Ilison

More long surface wavesBendingLarge

amplitude

Phase shift

Photo by L.Ilison

Fast ferries:source oflong-crestedsoliton-likewaves

Ship wave heights are moderate

08:00 10:00 12:00 14:00 16:00 18:00 0

0.2

0.4

0.6

0.8

1

1.2

Max

imum

wav

e he

ihgt

, m

AutoExpress/2

AutoExpress/2 SuperSeaCat Nordic/Baltic Jet

Nordic/Baltic Jet AutoExpress/2

SuperSeaCat Nordic/Baltic Jet AutoExpress/2

Nordic/Baltic Jet

Aegna, 14.04.2002

Blue: long components T>8s Red: all components

Ships may produce solitons

Three-dimensional nonlinear solitary waves inshallow water generated by an advancingdisturbance, YILE LI AND PAUL D. SCLAVOUNOS,J.Fluid Mech. (2002), vol. 470, pp. 383-410. 2002

“Negative” shockwave

Transient shipwaves inshallow areasare cnoidal orvery close tosolitarysolutions ofKdV.

Sine wave

Water surface

Cnoidal waves

0 5 0 1 0 0

-6 0

-4 0

-2 0

0

2 0

4 0

6 0

T im e , s

Su

fac

e e

leva

tion

, c

m

Long ship waves are highly nonlinear

Blue==recorded

waves

cnoidal wave approximation

sine wave approximation

Long waves approachingshallow area are:

•highly nonlinear

•and have shapes close to solitonsolutions of the KdV equation

Facit:

• There are three states ina human’s life:

(Anacharsis)

alivedeadon sea