Post on 17-Oct-2020
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