View Full Version : solitons and water waves
alistair
Aug30-04, 03:25 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I understand that water waves called solitons keep the\nsame amplitude for a much longer period of time than\nmost water waves do,even after colliding with other masses,\nand that this stability is due to a non-linear\nterm in the Kortewege de Vries equation.But what is the physical\norigin of the non-linear term - what property of the water molecules\ncauses it? And why are solitons much rarer than water waves that dissipate\ntheir energy far more quickly?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I understand that water waves called solitons keep the
same amplitude for a much longer period of time than
most water waves do,even after colliding with other masses,
and that this stability is due to a non-linear
term in the Kortewege de Vries equation.But what is the physical
origin of the non-linear term - what property of the water molecules
causes it? And why are solitons much rarer than water waves that dissipate
their energy far more quickly?
rge11x@netscape.net
Aug31-04, 03:40 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>alistair wrote:\n> I understand that water waves called solitons keep the\n> same amplitude for a much longer period of time than\n> most water waves do,even after colliding with other masses,\n> and that this stability is due to a non-linear\n> term in the Kortewege de Vries equation.But what is the physical\n> origin of the non-linear term - what property of the water molecules\n> causes it? And why are solitons much rarer than water waves that\ndissipate\n> their energy far more quickly?\n\nIt is nonlinear because the boundary condition of the differential\nequation is the wave itself, ie., it is a surface wave. The soliton\nresults as the balance between the interaction of dispersiveness and\nnonlinearity of this wave. It is not related to anything "molecular"\nand it is quite common occuring in any shallow channel.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>alistair wrote:
> I understand that water waves called solitons keep the
> same amplitude for a much longer period of time than
> most water waves do,even after colliding with other masses,
> and that this stability is due to a non-linear
> term in the Kortewege de Vries equation.But what is the physical
> origin of the non-linear term - what property of the water molecules
> causes it? And why are solitons much rarer than water waves that
dissipate
> their energy far more quickly?
It is nonlinear because the boundary condition of the differential
equation is the wave itself, ie., it is a surface wave. The soliton
results as the balance between the interaction of dispersiveness and
nonlinearity of this wave. It is not related to anything "molecular"
and it is quite common occuring in any shallow channel.
alistair
Sep1-04, 04:22 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>rge11x@netscape.net :\n\n>It is nonlinear because the boundary condition of the differential\n>equation is the wave itself, ie., it is a surface wave. The soliton\n>results as the balance between the interaction of dispersiveness and\n>nonlinearity of this wave.\n\nAlistair:\n\nBut what is it that makes this surface wave different\nfrom other surface waves - it must be physically different\nin some way?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>rge11x@netscape.net :
>It is nonlinear because the boundary condition of the differential
>equation is the wave itself, ie., it is a surface wave. The soliton
>results as the balance between the interaction of dispersiveness and
>nonlinearity of this wave.
Alistair:
But what is it that makes this surface wave different
from other surface waves - it must be physically different
in some way?
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>An excellent description of this (and lots of other) subject is in :\nG.B Whitham : Linear and Nonlinear Waves (1974)\nSee especially Chapter 13.\n\nAs a side note: surface waves in the context of EM are fundamentally\ndifferent form water waves. An EM example is the Goubau wave that is\npropagated along a cylindrical metallic rod covered with a dielectric.\nHere you have a homogeneous boundary that guides and acts as an open\nwaveguide along which a "surface" propagates. The surface itself is not\n"actively" part of the wave, it does not change as the EM wave moves\nby. For the water wave, the interface between the fluids (air/water) is\nwhere the wave motion occurs, and the boundary condition is changing in\ntime as the wave moves on.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>An excellent description of this (and lots of other) subject is in :
G.B Whitham : Linear and Nonlinear Waves (1974)
See especially Chapter 13.
As a side note: surface waves in the context of EM are fundamentally
different form water waves. An EM example is the Goubau wave that is
propagated along a cylindrical metallic rod covered with a dielectric.
Here you have a homogeneous boundary that guides and acts as an open
waveguide along which a "surface" propagates. The surface itself is not
"actively" part of the wave, it does not change as the EM wave moves
by. For the water wave, the interface between the fluids (air/water) is
where the wave motion occurs, and the boundary condition is changing in
time as the wave moves on.
alistair
Sep4-04, 03:05 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>rge11x@netscape.net wrote in message news:<ch0f0m\\$bts@odah37.prod.google.com>...\n> alistair wrote:\n> > I understand that water waves called solitons keep the\n> > same amplitude for a much longer period of time than\n> > most water waves do,even after colliding with other masses,\n> > and that this stability is due to a non-linear\n> > term in the Kortewege de Vries equation.But what is the physical\n> > origin of the non-linear term - what property of the water molecules\n> > causes it? And why are solitons much rarer than water waves that\n> dissipate\n> > their energy far more quickly?\n>\n> It is nonlinear because the boundary condition of the differential\n> equation is the wave itself, ie., it is a surface wave. The soliton\n> results as the balance between the interaction of dispersiveness and\n> nonlinearity of this wave. It is not related to anything "molecular"\n> and it is quite common occuring in any shallow channel.\n\nAlistair:\n\nThe non-linearity arises because water waves can only\nbe added linearly a certain number of times:\n\nhttp://216.239.59.104/search?q=cache:YCds9Xtiz4oJ:www.physicscentral.com/people/people-02-6.html+%22rogue+waves%22+%22schrodinger+equation%2 2&hl=en&ie=UTF-8\n\n\nThis link about the work of Al Osbourne, an expert in wave mechanics\nand solitons, mentions the non-linearity and says that Osbourne has\nadapted the Schrodinger equation to account for how water waves borrow\nenergy from one another to become giant "rogue" waves.Other theories\npredict fewer\nrogue waves than have actually been observed.Giant solitons\nhave been discovered off the coast of Sumatra under the sea,\ntravelling between layers of water at different temperatures.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>rge11x@netscape.net wrote in message news:<ch0f0m$bts@odah37.prod.google.com>...
> alistair wrote:
> > I understand that water waves called solitons keep the
> > same amplitude for a much longer period of time than
> > most water waves do,even after colliding with other masses,
> > and that this stability is due to a non-linear
> > term in the Kortewege de Vries equation.But what is the physical
> > origin of the non-linear term - what property of the water molecules
> > causes it? And why are solitons much rarer than water waves that
> dissipate
> > their energy far more quickly?
>
> It is nonlinear because the boundary condition of the differential
> equation is the wave itself, ie., it is a surface wave. The soliton
> results as the balance between the interaction of dispersiveness and
> nonlinearity of this wave. It is not related to anything "molecular"
> and it is quite common occuring in any shallow channel.
Alistair:
The non-linearity arises because water waves can only
be added linearly a certain number of times:
http://216.239.59.104/search?q=cache:YCds9Xtiz4oJ:www.physicscentral.com/people/people-02-6.html+%22rogue+waves%22+%22schrodinger+equation%2 2&hl=en&ie=UTF-8
This link about the work of Al Osbourne, an expert in wave mechanics
and solitons, mentions the non-linearity and says that Osbourne has
adapted the Schrodinger equation to account for how water waves borrow
energy from one another to become giant "rogue" waves.Other theories
predict fewer
rogue waves than have actually been observed.Giant solitons
have been discovered off the coast of Sumatra under the sea,
travelling between layers of water at different temperatures.
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I have no idea what you mean by "can only be added linearly a certain\nnumber of times". There are solitons in what is called the nonlinear\n"Schroedinger equation" but those are optical solitons and not related\nto water waves. As far as I know, Prof. Joseph of Johns Hopkins was the\nfirst to have discovered solitary waves and two-solitons in the\nthermocline and gave explicit solution for them by solving analytically\nthe Benjamin-Ono equation, which is a nonlinear differential-integral\nequation describing the wave at the interface of a two-fluid system of\nfinite depth(the Korteweg-deVries of shallow depth is nonlinear\ndifferential but without an integral kernel). Later, all the\nmulti-soliton solutions of the BO equation were given by many others.\nThese two articles of Joseph are:\n\nR I Joseph : Solitary waves in a finite depth fluid\nJournal of Physics A: Mathematical and General\nVolume 10, Number 12, December 1977\n\nR I Joseph and R Egri : Multi-soliton solutions in a finite depth fluid\nJournal of Physics A: Mathematical and General\nVolume 11, Number 5, May 1978\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I have no idea what you mean by "can only be added linearly a certain
number of times". There are solitons in what is called the nonlinear
"Schroedinger equation" but those are optical solitons and not related
to water waves. As far as I know, Prof. Joseph of Johns Hopkins was the
first to have discovered solitary waves and two-solitons in the
thermocline and gave explicit solution for them by solving analytically
the Benjamin-Ono equation, which is a nonlinear differential-integral
equation describing the wave at the interface of a two-fluid system of
finite depth(the Korteweg-deVries of shallow depth is nonlinear
differential but without an integral kernel). Later, all the
multi-soliton solutions of the BO equation were given by many others.
These two articles of Joseph are:
R I Joseph : Solitary waves in a finite depth fluid
Journal of Physics A: Mathematical and General
Volume 10, Number 12, December 1977
R I Joseph and R Egri : Multi-soliton solutions in a finite depth fluid
Journal of Physics A: Mathematical and General
Volume 11, Number 5, May 1978
Maarten van Reeuwijk
Sep9-04, 03:58 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>> But what is it that makes this surface wave different\n> from other surface waves - it must be physically different\n> in some way?\n\nThe only physical difference is that solitons come in solitude and surface\nwaves don\'t. Therefore, the essential difference is in the generation\nmechanism.\n\nSurface waves are normally generated by wind, bottom geometry, currents or\nships, and involve a continuous generation mechanism. In Russell\'s\ndiscovery of solitons, the soliton was generated by a stopping ship (a\nsingular event). Furthermore, solitons are nearly-hydrostatic (only WEAKLY\nnon-linear, i.e. the propagation speed is c=sqrt(h+zeta_0) and the\nhorizontal velocity distributed uniformly throughout the vertical), so the\ngenerator must be able to generate a shock throughout the entire water\ncolumn. A ship is quite a good generator, occupying a large cross-section\nof the channel and moving a significant amount of water. Mechanisms like\nwind do not excite the whole water column and thus do not generate\nsolitons. If you would put a ship in too deep water or too large a channel\nit wouldn\'t be able to produce solitons either.\n\nHTH,\n\nMaarten\n\n--\n================================================ ===================\nMaarten van Reeuwijk Thermal and Fluids Sciences\nPhd student dept. of Multiscale Physics\nwww.ws.tn.tudelft.nl Delft University of Technology\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>> But what is it that makes this surface wave different
> from other surface waves - it must be physically different
> in some way?
The only physical difference is that solitons come in solitude and surface
waves don't. Therefore, the essential difference is in the generation
mechanism.
Surface waves are normally generated by wind, bottom geometry, currents or
ships, and involve a continuous generation mechanism. In Russell's
discovery of solitons, the soliton was generated by a stopping ship (a
singular event). Furthermore, solitons are nearly-hydrostatic (only WEAKLY
non-linear, i.e. the propagation speed is c=\sqrt(h+\zeta_0) and the
horizontal velocity distributed uniformly throughout the vertical), so the
generator must be able to generate a shock throughout the entire water
column. A ship is quite a good generator, occupying a large cross-section
of the channel and moving a significant amount of water. Mechanisms like
wind do not excite the whole water column and thus do not generate
solitons. If you would put a ship in too deep water or too large a channel
it wouldn't be able to produce solitons either.
HTH,
Maarten
--
================================================== =================
Maarten van Reeuwijk Thermal and Fluids Sciences
Phd student dept. of Multiscale Physics
www.ws.tn.tudelft.nl Delft University of Technology
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