IL NUOVO CIMENTO VOL. 6 C, N. 2 Marzo-Aprile 1983

Cylindrical Solitons in Shallow Water of Variable Depth (').

P. CARBONA~0

Istituto di _~isica dell' Universitd . Palermo, Italia

R. FLOm.S and P. P ANTANO

Dipariimento di Mat~*matica, Universit~ della Calabria - Arcavacata, Cosenza, Italia

(ricevuto il 4 Marzo 1983)

Summary. - - The propagation and the interaction of cylindrical soli- tons in shallow vater of variable depth are studied. Starting from the cylindrically symmetric version of the equations describing long waves in a beach, a Korteweg-de Vries equation type ~ + 6 ~ + ~ ~ - - P(~)~ is derived. Since no exact analytical solution has been found to date for this equation, some remarkable cases in which the equation takes up a trac- table form are analysed. Finally the interaction between cylindrical imploding and expanding waves is considered and the phase shifts caused by the head-on collision are given.

PACS. 92.10. - Physics of the ocean.

l . - Inu-oductlon.

The physics of nonl inear p ropaga t ion is i ndeb ted to the shal low-water

waves for the first recorded observa t ion of soli tons since J . S . RUSSELL SaW

in 1834 the ~ grea t t r ans l a t ion wave ~ t r ave l l ing in a channel . I t is well known

t h a t th is phenomenon has been theore t i ca l ly exp la ined as a special solution

of the Kor teweg-de Vries (KdV) equat ion. However , when one considers waves

p ropaga t i ng in shal low wate r of va r i ab le dep th , the equa t ion governing such

(') This work was supported by CNR-GNFM.

133

134 P. CARBONAR0, R. FLORIS and P. PANTAN0

waves differs f rom the K d u equat ion due to the variabi l i ty of the coefficients (1) which affects the s t ructure of the localized waves (2). Fur the r modifications of the KdV equat ion are produced when the propagat ion is no longer one- directional (3). Our aim is to invest igate the shallow-water wave propagat ion when both these effects are acting. In this first approach we t r ea t the prob- lem in respect of a cylindrical symmetry .

In sect. 2 we shall derive an equat ion governing the propagat ion of cylin- drical diverging waves. In general the resulting equat ion is too complicate to be solved exact ly with the technique of the spectral t ransform ~nd only approximate informat ion can be obtained about the evolution of the initial data. In some ease, however, when the variable depth has a par t icular spatial dependence, it is possible, by means of a t ransformat ion, to reduce this equat ion to a K d u equat ion with constant coefficients. In some sense, physically, this occurs when the two geometrical effects, slope Of the b o t t o m and two-dimen- sionality of propagation, balance each other. We show tha t in correspondence of this critical slope we have a soliton with vary ing ampli tude and velocity.

In sect. 3 we a t t e m p t to cast light also on the more general case when the for tuna te above-ment ioned balance does not occur. We suppose tha t the slope of the bo t tom is slightly different f rom the critical slope, and we look for solutions describing solitons with slowly vary ing characteristics. This pro- cedure enables us to find out more about the effects of two-dimensional prop- agation on an uneven bot tom.

In sect. 4, assuming the existence o f bo th imploding and expanding cylindrical waves, we s tudy thei r mutua l interact ion following an elegant method developed some years ago by O:KAWA and YAXI~A (4). Exper imenta l da ta on the head-on collision of solitons are available in the l i terature, and the t ime delay caused by t h e interact ion has been measured in different physical contexts (5.e). In part icular , experiments about the collision between one- dimensional shallow-water solitons travell ing in opposite directions have been aimed at explaining some features observed in Jupi ter ' s a tmosphere (6). A more realistic approach, however~ would take into account the multidimension- a l i ty of the propagation. I t seems, therefore~ interest ing to us to investigate~ f rom a theoret ical point of view, in what way the phase shifts suffered during the interact ion are affected by the geometrical ingredients w i t h which we are concerned.

(1) T. KAKUTANI: J. Phys. Soc. Jpn., 3@, 815 (1971); R. S. JOHNSON: J. Fluid Mech., 60, 813 (1973). (a) C. J. KN:CY~RB0O~.R and A. C. N]~W~LI,: J..Fluid Mech., 98, 803 (1980). (8) P. PR~AD and R. R~.VI~rl)RAU: J. Inst. Math. AppZ., 20, 9 (1977). (4) M. OIKAWA and J. YAJIMA: J..Phys. Soe. Jpn., 34, 1093 {1973). (5) Y. NAlrAMURA, M. 0OYAMA and T. 0GI~O: .Phys. Rev. Lea., 45, 1565 (1980). (6) T. MXXWORTH•: J..Fluid Mech.,76, 177 (1976).

CYLINDRICAL SOLITONS IN S~IIT.u W A I ~ R OF VARIABLE DEPTH 135

2. - Cylindrical s o l i t o n ~

We consider the cylindrically symmetr ic version of the equations describing long waves on a beach (~)

(1)

~7 + (H + h),~ ~ (rv) + ~r

6 ~$~r ~-~(rv)

where v is the radial flow velocity, H ~- h is the water depth with the H un- per tu rbed value, r is the radial co-ordinate and t the time. We now expand (") v and h in power series of a small per turbat ion p a r a m e ~ r e around an unper- tu rbed s ta te t ha t we suppose homogeneous and constant (for simplicity he = v0 = 0), i.e.

(2) v = sv~ A- ~ v , a t- . . . , h = ~h~ + ~'h, A- . . . .

Next we look for solutions h~, vt, the expansive wave, by supposing

(3)

where

v, = v,($1, ~ ) , h, = h,($~, ~ ) ,

(4) --t) / j ~ ,

We also suppose tha t H has a slow dependence by r, i.e. H ---- H(~). Then, by inser t i rg eq. (2) in (1), by using (3) and (4) and equating to zero

the coefficients of the obtained series in e, we have, a t first order,

(5) v~ = V ~ : ~ , h~ = H ~ I ,

where gl = ~x(~x, 7) is a function which will be determined in a nex t step. At the second order we obtain, by the compatibi l i ty condition for ~v,/O~l

and Oh~/O~, tha t the evolution of gx is governed by the following modified cylin- drical KdV equat ion:

(7) D. H. P~.~.~Gm~E: J..Fluid Mech., 27, 815 (1967). (s) T. TANIUTI and C. C. W~.I: J. Phys. ~o~. J p , . , 24, 941 (1968); N. ASANO and H. 0NO: J. Phys. So~. Jpn., 31, 1830 �9

1 ~ P, CARBONARO, R . FLORIS and P, PANTANO

tha t differs by t ha t obtained by KA~u~ANI and J o n ~ s o ~ (1) for the presence of the geometrical t e rm �89

Equat ion (6) can be wri t ten in a more convenient form by using the trans- formation z l ----- ~H~, ~ ~ ~ f V ~ d ~ / , obtaining

(7) ~ ~ ~*Y + + = -

where

(s) 9 4 + \ J V Y / "

In general eq. (7) is not integrable, because hi ther to i t has not been pos- sible to find a t ransformat ion t ha t reduces (7) or (6) to a KdV equation with constant coefficients, and some informat ion can be obtained only numerical ly or by per turba t ion methods (9). Bu t for the par t icular slope H oc 3" (lo), /,(~) _~ ---- (1/~)(2p -{- ~) and then, if p = 0 p --~ �88 (7) becomes the so-cailed cylindrical or spherical KdV equation. These equations have been deduced by MAxo~ and V~ECEI~I (~1) in s tudying the propagat ion of cylindrical or spherical ion- acoustic waves in a eollisionless plasma. They predicted numerically, for the first t ime, t ha t these kinds of solitons modify their shape and behind them de- velop a , r e s idue , due to irreversible processes.

I f p ---- 0, eq. (7) can be t ransformed into a K d V equation with constant coefficients (~ms). Bu t af ter the t ransformat ion a localized initial pulse becomes an unloealized initial da tum for the KdV equat ion with constant coefficients. The direct-inverse scat ter ing for the associated SchrSdinger equat ion with this unlocalized potent ia l was solved by CAT~OGERO and DE~ASPERIS (~4).

For the slope corresponding to p ~ -- �88 F(~) vanishes and eq. (7) is the classical KdV equat ion with constant coefficients which exhibits the well-known soliton solution. Accordingly we have an explicit solution of eq. (6) in terms of the original s t re tched variables $ and

(9) )] ~1(~, 7) = ~H(~/) seeh 8 ~ , - - e H V / - ~ / ,

(o) K . I . K~ MAN and E. M. MASLOV: SOY. Phys. JETP, 46, 281 (1977); D. J. KAUP, and A. C. NwW~LL: Proe. R. Sac. London, Set. A, 361, 413 (1978); K. Ko and H. It. K ~ . : Phys. l~ev. LESS., 40, 233 (1978). (lo) j . M. BAI~CK~R and G. B. WmTHAM: Commun. Pure AppZ. Ma~h., 33, 447 (1980). (1~) S. M~xoN and J. VI~CELLI: Phys. Rev. Lett., 32, 4 (1974); Phys. Fluids, 17, 1614 (1974). (18) R. HrROTA: Phys. Lett. A, 71, 393 (1979). (ls) T. BRUGARINO and P. PANTANO: Phys. Lett. A, 80, 223 (1980). (1~) F. C~LOa~-RO and A. Dv.GASP~.RIS: Lett. lfuavo Cimento, 23, 143, 150 (1978).

C Y L I N D R I C A L SOLITONS IN SHAT.LOW W A T E R OF VARIABLE D E P T H 137

where

[ 3 \ - , I , H ( , ) = "

This expression represents a soliton with vaxiable ampl i tude and velocity and constant width (see fig. 1). I t can be also noted t h a t the relationships between ampli tude, velocity and width axe no longer retained.

In the space (r, t) eq. (9) takes the form

where

L ~ \ F ( r ) ~ '

�9 H ( r ) = ( 3 e m r ) -~/~ ]' and F(r ) = - - e ~-~ oH(r) ,

indicating tha t in the real space the soliton ampli tude varies as the unper- turbed depth H(r), the width is of the order e -~/2 and the wave velocity is given by the linear wave velocity V~ff(r) modified by the factor F(r) .

The well-known multisoliton solution (15) of (7), when p = -- �88 in the (~, ~) space takes the form

~, = H(~) ~ log [det (I + C)],

where I is the ident i ty matr ix, while C a matr ix whose generic element c~. is given by

[ ] c~. - - k~ + k. exp (kin + k.)~, + 4 (k~ + k~)~"/' .

3. - Perturbed cylindrical solitons.

In order to invest igate in more detai l the effects of geometrical terms on soliton propagation, we now consider t h a t the r.h.s, of eq. (8) is proport ional to ~, with I~1<< 1, i.e. F(~) = ~/~, which corresponds to H oc v ~/~+~/~. Follow- ing (o.xe), we seek a solution of eq. (7) representing a soliton with slowly varying characteristics

(lo) ~(~1, v) = 2~2 ~ sech2~2(~a -- ~ ) ,

(15) R. HmOTA: .Phys. ~gev. LeSt., 27, 1192 (1971). (ae) S. L~a.BOVTCH and J. D. RAND~T.L: Phys. Fluids, 14, 2559 (1971); J. Fluid Mech., 58, 481 (1973).

1 3 8 P . C A R B O N A R O , R . F L O R I S and P . P A N T A N O

I H

A

i

J fi

f If

o0

O

O

o

I

w-4

GYLINDRICAL SOLITONS IN 8RAu WATER OF VARIABLE D~.PTH 139

where 8~]8v = 4~=; subst i tu t ing (10) into (7) and using the energy conser- va t ion law, we obta in

8z 3

then in tegra t ing

, *

The ampl i tude .s and the veloci ty ~/v of the soliton v a r y as v-~/8 and thus one has an increase or a damping of ampl i tude and veloci ty if O is, respect ively, less or more t han zero.

We note now tha t , a t the leading order, the energy conservat ion law

is verified, bu t nei ther the expression of conservat ion of mass

+ c a + c a

nor of the first m o m e n t u m

+r +ca +o0

are satisfied. This p rob lem is solved b y adding ano ther t e r m % of order

to the solution (10), i.e.

= 2Q ~ sech= Q(~I - - $1) + ~ .

The presence of this second t e r m was found numer ica l ly b y JOHNSON (1), LEra0WICH and RA~DAL (16) and was explained theoret ical ly by K~uP and

~EWELL~ KARP~AN and M~SLOV (9). In our case

~=/(ar)~l = 01 [ ] 0

where sgn ( x ) = 1 for x > 0 and sgn ( x ) = - - 1 for x < 0 .

140 P. CARBONARO~ R. FLORIS a n d P. PANTANO

For $ ~ $ ~ 0, the t e r m in b racke t s vanishes and for 0 ~ ~ ~ $ one has

for $ <~ 0

+, +, r ' (2 ,/81

Thus, be tween $ ---- 0 and the posit ion of sol i tary wave ~, there appears a shelf of height

0 1 0 3 1

which t r anspor t s a mass equal to 0/3~2v. The conservat ion laws can be now verified following the same procedure as in (17).

The following comments are wor th making abou t the fact t h a t the shelf is raised or depressed according to whether ~ is less or more t han zero. In the source t e r m /'(w) = (2/w)(p ~ �88 p gives the slope of the b o t t o m (H oc w~), while the num ber �88 is re la ted to the cylindrical geome t ry of the wave front . Our pe r tu rba t ion approach consists of posing p : -- �88 ~ 0/2, where the small quan t i ty J is a measure of the deviat ion f rom the s i t ua t ion /~ = 0. This leads one to conclude t h a t the presence of a raised or depressed shelf depends upon whether the dominat ing effects are those produced by the uneven b o t t o m (0 < 0) or those re la ted to the geomet ry of the wave f ront (0 ~ 0).

I t has been shown t h a t the two combined effects due to the var iable depth and geometr ical spreading in general affect the shape of the cylindrical solitons and moreover lead, when p ~ - - �88 to the presence of a shelf, while no shelf appears when p = - �88 This la t te r effect is a t t r ibu ted to a perfect balance between geometr ical t e rms re la ted to var iable depth and two-dimensional f ront wave.

4. - In teract ion o f cy l indrical so l i tons .

Following a procedure analogous to t h a t in sect. 1, we introduce the n e w

co-ordinates ~2 = ell2(fdr/v ~ -{- t) and y----eS/2r describing now the imploding

wave, and we pose v, = v~(~, 7), h , -~ h~(~2, 7). At first order we obta in

vl : - - ~/-Hz~,(~l, ~]) and h 1 = Hz~a(~, 7) ,

(17) T. T ~ I u T I and K. I~OZAKX: J..Phys. 8oc. Jpn., 40, 573 (1976) .

CYLINDRICAL SOLITONS IN SHALLOW WATER OF VARIABLE I)lZPTII 141

while at second order we have

8~q 3 1 8~2 ~ 8 ~ , 5 ~ , _ _ _ _ 8 H 1 ~ 2 = 0

That can be written in terms of ~, ~s and z as

8~ 8~ 8s~ 8-4 + 6~ 8~, 8~; = - - F(~)~,

that is equal to eq. (7) in sect. I except for the minus sign ahead of the third derivative. Also this equation for H oc r -,I' has the solitonlike solution

n, 3 k16]

and, for increasing t, the amplitude increases as ~-,I, and the velocity as ~-u,. In order to study the interaction of imploding and expanding waves, we

define (9 the new co-ordinates

r = e ' ' ' ( f dr - t ) \ J v ~ § s~',p,Cr, 0 ,

= ( f dr + ,)), k3v~

and we suppose that v, = v~(},, ~,, z) and h~ = h~(},, ~ , v). The functions ~, and ~ are two phases introduced to take into account the

mutual interaction between waves travelling in opposite directions. The usual perturbation procedure yields at the lowest stage of approximation

v'~Sv' 8h, 8v, 8h, - 88 + ~ + ~ + ~ = 0 , (11)

H Sv~ 8h, _ 8% v~Sh* N - v ' - ~ + ~ + ~ = o .

From (11) we see that the unknown variables v, and h, can be expressed as

(12) v~ = ' ~ (~z, -- n%), h, = H(:z, -I- =,),

where ~, and ~, are two arbitrary differentiable functions of ~,, ~, and ~ to be

142 P. CARBONA.RO,: R. FLORIS and P. PANTANO

determined in the next stage. A condition for eqs. (12) to verify eqs. (11) is that z~ depend only on ~ and ~.

In the next stage of approximation, by using (12), and by expressing also v, and h. in terms of two new functions 0"~(~1, ~, 7) and az(~l , ~2, ~), i.e. v, : = V ~ (a~ -- a,) and h, ~- H(al ~ a,), we obtain a linear inhomogeneous algebraic system in the unknowns 8a~/8~, 8a~/8~i, with nonvanishing determinant. Easy calculations yield

aal 1 [ 8n, 8x, Hs~a , 1 1 aH ~_~] -

5 ~ 1 8 H 1 ~ ] 4 H 8 ~ + 2 '

If we conjecture that the phases (Pl and ~02 depend only on amplitudes of interacting waves, then, following (9, we suppose that ~1 and ~0z satisfy

i.e.

0~1 g z _ 0 and ~ 0 4 4 '

where 01 and 02 are two arbitrary functions to be determined by imposing the initial conditions.

Finally, from the nonsecularity condition for al and as (i.e. by imposing the boundedness of al and a~) we obtain the equations governing the functions Zl and ~ :

(13) 8~1 3 7{: 1 8;7I: 1 ~ ~3~r[ 1 5 ~'1~ 1 ~ J ~ 1 '~1 ~ "-~ ~ f - ~ l ~- 6 8 ~ "-~ 4J~ ~] J [ - . ~ - ' ~ O~

(14) 8~___~ 3 ~ 8~2 V ~ 8 ~ 5 ~2 8H 1 ~2 = 0 e "

Equations (13) and (14) are the same obtained separately for expanding and imploding waves. In our approach the only effects of interaction are con- tained in the dependence of ~ on ~ .

C Y L I N D R I C A L SOLITONS IN SHALLOW W A T E R OF VARIABLE D E P T H l t 3

Now through the t r ans format ions z~.s----[Hyr r and ~ : ~ f ~ d w , eqs. (13) and (14) become

where ~ - ~ • ,

9 1 OH {2v'-~ d~ ~-' + ~, ~ / '

w i t h / ' ( z ) vanish ing for H oc z-1/4. In this case we have soliton solutions tha t , in t e rms of the radia l veloci ty v~ and water dep th hi (at the first order of ap- proximat ion) , are

where

vl = �89 seeh 2 ~ --{- c2 sech s I2~],

hz = �89 seeh ' Q~ -- e2 sech 2 ~22],

[ 3 \ - . l . R I l l = ~ ,~d~} ' = - ~

The phase shifts due to the m u t ua l in teract ion can be ex t ima ted as

for ~j --~ c~

(i , j -----1,2; i C j ) . I n our case

A~z------~ H(~) , A~,,= 3 H(~) .

I n the real space (r, t) we have

A~I ---- - - ~

We note t h a t in this case, a t var iance with previous theoret ical results in one-dimensional p ropaga t ion (4), the phase shifts depend on the spat ial co-ordinate.

�9 RIASSUNTO

L'oggetto di questo lavoro ~ 1o studio della propagazione e delrinterazione di solitoni cilindrici in acque superficiali di profondit~ variabile. A tal fine, partendo dalla versione cilindrica delle equazioni descriventi le onde lunghe suuna spiaggia, si deriva un'equa-

144 P. CARBONARO, R. FLORIS a n d P. PANTANO

z ione del t i po di K o r t e w e g - d o Vr ies n o n o m o g e n e a ~r ~ - 6 ~ - } - ~ = - - F ( v ) ~ . Po ieh~ in genera le la so luz ione a n a l i t i e a d i t a l e equaz ione n o n ~ n o t a , si p r e n d o n o i n e s a m e a l cun i casi n o t e v o l i in cui l ' e q u a z i o n e ~ f a c i l m e n t e t r a t t a b i l e . In f ine , si con- s ide ra l ' i n t e r a z i o n e t r a onde c i l ind r i che d i v e r g e n t i e conve rgen t i , d e t e r m i n a n d o le dif- f e renze di fase p r o d o t t e d u r a n t e la colHsione f ron ta l e .

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