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Long-wave propagation in water with bottom discontinuity
Int. 1. Non-Lmear .Urchmcs. Printed in Great Bntam
@xO-7466?,92 ss.00 + .al Pergamon Prcs.5pk
Vol. 21. No. L pp. 197-201. 1992
LONG-WAVE PROPAGATION IN WATER WITH BOTTOM DISCONTINUITY K. V. PRAMOD and M. J. VEDAN Department of Mathematics and Statistics, Cochin University of Science and Technology, Cochin 682022, India (Receked Abstract-Long-wave
KdV (Korteweg4e
9 August 1989; in revisedform
23 May 1990)
propagation in water is studied, when there is a sudden change in depth. A Vries) type equation is obtained. A solution for this equation is also given.
1. INTRODUCTION
de Vries [l] derived their equation from the Euler equations as an approximation to the Navier-Stokes equations assuming that waves being considered have an amplitude which is small and a wavelength which is large compared to the undisturbed depth. Additionally they recognised the importance of assuming that the Stokes number is neither too large nor too small. These choices of parameters make the effects of nonlinearity and dispersion balance, which renders the existence of solitary-wave solutions possible for the equation. Thus the crucial step in the derivation of the KdV equation is the choice of appropriate parameters. Johnson [2] has shown that by paying particular attention to the proper choice of parameters, four types of KdV equations can be derived in the context of water waves. The scaling required in the derivation of these equations suggests the existence of a transformation between these equations, which has been confirmed. The unidirectional propagation of a shallow-water solitary wave in a canal of slowly varying depth is well-described by a perturbed KdV equation. Johnson’s equation [3] also belongs to a class of such KdV equations. Kaup and Newell [4] have discussed the motion of a soliton solution of the KdV equation under various perturbations. The effect in the context of a weakly non-linear water wave propagating in a relatively shallow basin where the depth is changing slowly is also discussed. It is to be noted that this study is based on the model of a perturbed KdV equation. Here we derive a KdV-type equation for shallow-water waves when there is a sudden jump in the bottom level. A solution of this equation is also obtained. Korteweg
and
2. DERIVATION
OF THE EQUATION
For convenience we denote the dimensional variables by primes. Let h’ be the depth of water below the equilibrium level as shown in Fig. 1. Then we have h’ =
ho h, - d
for x’ < xb for x’ 2 xb’
(2.1)
Thus we define h’ = ho - dH(x’
- XL)
where H(x’ - xb) is the Heaviside function which is zero for x’ < XLand unity for x’ 2 XL. Since the flow is irrotational there exists a velocity potential 4’ which we define as $,
=f,(x,)
_
(Y'
+
2!
h’)*-___ a’_f’(x’)+ C_v’ + U4 a*_f’(x’) axI* 197
4!
dX’4_“”
(2.2)
K.
198
V.
PRAMOD
and M. J. VEDAN
wrface
bo-an ..
Fig. 1. Definition sketch.
We substitute for h’ from equation (2.1) and non-dimensionahse
y’ = h,y
E’ = ho&,
.ub = lx,,
x’ = Ix,
the variabfes by
(2.3) where c0 = fi Thus we get
and I and a are the characteristic wavelength and amplitude, respectively. 11,=f_
(J + I)2 - Z&(L’+ l)H(x - .x0) B.&x 2! + (J f 1)4 - 4&(Y+ l)‘H(.u - .YO)p*J xxx - ... 4!
(2.4)
where Now the velocity components
in the x- and y-directions are
u = $, =fx - (y + 1)2 - 2s’;:
f)H(x - x0) pL-,
+ E(.Y+ 1)Q.x - xrJ)B_Lr + O(P2)
(2.5)
and c = c;b,= [ - (y + 1) + EH(X - x0)] pfxx + 0(/P)
(2.6)
where 6(x - x0) = H’(x - x0), is the Dirac delta function. Thus we find that
B g + ; = 0 + o(p).
(2.7)
Therefore, the equation of continuity is satisfied if terms of 0(f12) are neglected. On the free surface, y’ = $, the boundary condition to be satisfied is Sfj’
d=,,+U$
I Zrl’
(2.8)
We define 4’ = a~, so that on the free surface, y = zq, where z = a/ho. In terms of the dimensionless variables the free-surface condition (2.8) becomes (2.9) Substituting for u and u from equations (2.5) and (2.6), equation (2.9) reduces to [ - frq + 1) + &H(X - .u,)]fx* + (rS + I)’
- 3&(ZV+ I)ZH(x - x,) 6
x B_Lx = Vf + &4x + O(@VP’).
(2.10)
Long-wave
propagation
199
in water
As in the derivation of the KdV equation, neglecting terms containing equation (2.10) becomes
z2, /?* and a/?,
tlr + Cl - sH(x - %)lW, + 4w)x + c- is + + BH(x - x0)1 wxx.x= 0
(2.11)
where w =fx. Now in the long-wave approximation
we have the equation of motion
u;, + liu;. + vlu;* = - g&
(2.12)
u,+stuu,+I-au,= B
(2.13)
which reduces to -qx.
Substituting for u and u from equations (2.5) and (2.6) and neglecting, as we did above, terms in z2, /I2 and aP we cast equation (2.13) in the form w, + tlX+ aww, - +B wXXI+ .$[H(x
(2.14)
- xlJw,,lx = 0.
Equations (2.11) and (2.14) are the governing equations for the wave motion. Neglecting terms in E, a and /I we get a solution of equations (2.11) and (2.14) to the lowest order: nJ = tl,
(2.15)
9, + tlX= 0.
Now we look for a solution of the form w = q + &A+ r/I + DC + &aD + @E.
(2.16)
Substituting equation (2.16) in to equations (2.11) and (2.14) we get + a[& + (v2Ll + PCC, - i~,l
tit + G + &CA,- Wx - 0~3 - H(x - ~oP,l
+ ~BCCX- H(x - x,)C, - $4,x
+ E~CD,+ WI),
+ iH(x - QLxxl
= 0
(2.17)
and tlr + tlX+ &AX+ a[& + VLI + BCG - h,,l
+ EaCD,+ WM
+ EB{E, - +A,,, + CH(x - xohl,I
= 0.
(2.18)
Since qX+ qr = 0 to the lowest order, we shall replace the t-derivatives by the negative of the x-derivatives. Then the two equations (2.17) and (2.18) are consistent if A, = tH(x - x,)~, B* = +I% CX = f&XX D, = - aH(x - xo)tlrx and Ex = - itH(x
- x&L,,
+ G(x - x&L,
+ fW
- x&x.
Then we have equation (2.17) as qr + [l - iEH(X - xg) + &&@‘(X - x0hl + hm
+ Ch + +Wx
- huh,, + W - W-w
- xo)lw, - x~~L,,
= 0 (2.19)
where we have taken A@,) = trt(x,). Here 6(x - ~0) = -&H(x - x0) is the Dirac delta function and the prime denotes differentiation with respect to x. For x < x0 we have H(x - x0) = 0, therefore we get from equation (2.19) tit + r!X+ %WX + ML,,
= 0
(2.20)
200
K. V.
and M. J.
PRAMOD
VEDAN
and for x > x0 we get, from equation (2.19) tit + (l’- &)tlX + ($2 + ac+PL
+ (# - M)L,
(2.21)
= 0.
Equation (2.20) is identical with the classical KdV equation for constant depth and equation (2.21) also reduces to this equation as E + 0.
3.
SOLUTION
OF
EQUATION
(2.19)
Since the coefficients of equation (2.19) contain generalised functions, its solution is difficult. But we shall obtain a solution by treating 6(x - x0) and 8(x - x0) as limits of classical functions [S]. We have n 6(x - x0) = lim n-z n[l + n2(x - x0)2]
(3.1)
and 6’(x - x0) = lim n-33
- 2n’(x - x0) n[l + n2(x - xo)2]2’
(3.2)
Substituting from equations (3.1) and (3.2) into equation (2.19) we get 1 -+f(x-x,)+rEB
&+
12
i +S@
- 2n3(x - x0) rlX+ c+ + ?ezH(x x[l + n2(x - xo)2]2 I
n + co - i&BHb n[l + f?(X - x0)2] rlXx
x0)1llxxx -+0
as n-
x0)1wx ,x.
(3.3)
We look for a local solution, rl = ‘II + H(x -
xo)(tl,
-
(3.4)
Sl)
of equation (3.3) in a neighbourhood (x0 - p, x0 + p) of x0, where qI and q2 are solutions of equations (2.20) and (2.21) in - cc < x < 30 . For any test function $, we define
=-
=
a Cat + H(x I -(o
s
co _9 v&NWdx
+
%2
-
xo)h2
+
v~)lW’(x)dx
-
(~2
-
vl)(xoM(xo)
~lM+Wd~~
I x0
(3.5) where [(t/2 - t11)6(x - xo)l(JI) =
(rt2
-
v,)(xoM(xo)
and m (tl2 I IO
%l, -q,,($) =
-
slLCW(x)dx.
Similarly we get tlxxw
=
tllrrW
-
[(t/2
-
r11)@
-
x0)1(1(/‘)- %J2-,,,w
(3.6)
+
Ch2
-
rtl)b(X
-
~o)lW’”
(3.7)
and %A$) =
‘I1 x.x.x ($1
+
@,qz
-&u.
Long-wave propagation in water
201
We can give a meaning to the operator qqXas follows: rtrt,($) = rl[%X + (112- rltP(x - x0) + @Ccsl -Ir,,l(lcI)
Let rl/ be a smooth function with support in (.x0 - P, x0 + P). We assume that q1 (x0) = ~2(xo) and the first two derivatives (with respect to .x) of qt and q2 vanish at x0. Substituting from equations (3.5), (3.6), (3.7) and (3.8) into equation (3.3) and taking the limit as n -, CL:we get
by equations (2.20) and (2.21) for all test functions $ with support in (x0 - p, x0 + ,n).Thus 9 is a solution of equation (3.3). 4. DISCUSSION
Here we have derived a KdV-type equation for the propagation of long waves, when there is a sudden change in depth due to a step-like discontinuity at the bottom. Non-linearity and dispersion are measured by the same parameters as in the classical KdV equation. The additional effect of change in depth is taken into account by a parameter E, which is a measure of change in depth, by retaining terms linear in this parameter. We find that as E -+ 0, the equation reduces to the classical KdV equation. The coefficients in the equation show singularity at the shelf. An additional feature is a diffusion term as in Burger’s equation. We have also obtained a solution for this equation. The previous studies by Johnson [33, Grimshaw [6], [73 and Leibovich and Randall [8] have given rise to perturbed KdV equations, which have been anafysed by Kaup and Newell [43 and Knickerbocker and Newell [93 and it has been shown that a KdV equation under perturbation need not satisfy conservation of mass though energy is conserved, while the classical KdV equation which is completely integrable has an infinite number of conserved quantities. Using inverse scattering transform technique (ET) Kaup and Newell (41 argue that in the case of the perturbed KdV equation a continuous spectrum is excited due to interaction between the soliton and the perturbation, and the reflection coefficient will have a Dirac delta function behaviour. Due to this, conservation of mass is obtained only by the formation of a shelf (since the depth below the mean level is constant). This formation of a shelf may be accounted for by a change in the depth of the undisturbed water, and in this sense a perturbed KdV equation may be a model equation. However, unlike a perturbed KdV equation, further studies of equation (2.19) can shed more light on long-wave propagation on an uneven bottom. REFERENCES 1. D. G. Korteweg and de Vries. On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. Mug. 39, 422-443 (1895). 2. R. S. Johnson, Water waves and Korteweg-de Vries equations. J. Ffuid Mech. 97, 701-719 (1980). 3. R. S. Johnson, On the development ofsolitary wave moving over an uneven bottom. Camb. Phil. Sm. Proc. 73,
183-203 (1973). 4. D. J. Kaup and A. C. Newell, Solitons as particles, oscillators and in slowly changing media: a singular perturbation theory. Proc. R. Sot. Land. A361, 413-146 (1978). 5. D. H. Griffel, Applied Funcrionol Analysis. Ellis Horwood, Chichester (1981). 6. R. Grimshaw, The solitary wave in water of variable depth. J. Fluid Mech. 42, 639-656 (1970). 7. R. Grimshaw, The solitary wave in water of variable deoth. Part 2. J. Fluid Mech. 46.611622 (1971). 8. S. Leibovich and J. D. Randall, Amplification and decay of long nonlinear waves. J. Ftuid Mech: Ss, G81-493
(1973). 9. C. J. Knickerbocker and A. C. Newell, Shelves and the Kortewegae (1980).
V&s equation. J. Fluid Me&
98,803-818
@xO-7466?,92 ss.00 + .al Pergamon Prcs.5pk
Vol. 21. No. L pp. 197-201. 1992
LONG-WAVE PROPAGATION IN WATER WITH BOTTOM DISCONTINUITY K. V. PRAMOD and M. J. VEDAN Department of Mathematics and Statistics, Cochin University of Science and Technology, Cochin 682022, India (Receked Abstract-Long-wave
KdV (Korteweg4e
9 August 1989; in revisedform
23 May 1990)
propagation in water is studied, when there is a sudden change in depth. A Vries) type equation is obtained. A solution for this equation is also given.
1. INTRODUCTION
de Vries [l] derived their equation from the Euler equations as an approximation to the Navier-Stokes equations assuming that waves being considered have an amplitude which is small and a wavelength which is large compared to the undisturbed depth. Additionally they recognised the importance of assuming that the Stokes number is neither too large nor too small. These choices of parameters make the effects of nonlinearity and dispersion balance, which renders the existence of solitary-wave solutions possible for the equation. Thus the crucial step in the derivation of the KdV equation is the choice of appropriate parameters. Johnson [2] has shown that by paying particular attention to the proper choice of parameters, four types of KdV equations can be derived in the context of water waves. The scaling required in the derivation of these equations suggests the existence of a transformation between these equations, which has been confirmed. The unidirectional propagation of a shallow-water solitary wave in a canal of slowly varying depth is well-described by a perturbed KdV equation. Johnson’s equation [3] also belongs to a class of such KdV equations. Kaup and Newell [4] have discussed the motion of a soliton solution of the KdV equation under various perturbations. The effect in the context of a weakly non-linear water wave propagating in a relatively shallow basin where the depth is changing slowly is also discussed. It is to be noted that this study is based on the model of a perturbed KdV equation. Here we derive a KdV-type equation for shallow-water waves when there is a sudden jump in the bottom level. A solution of this equation is also obtained. Korteweg
and
2. DERIVATION
OF THE EQUATION
For convenience we denote the dimensional variables by primes. Let h’ be the depth of water below the equilibrium level as shown in Fig. 1. Then we have h’ =
ho h, - d
for x’ < xb for x’ 2 xb’
(2.1)
Thus we define h’ = ho - dH(x’
- XL)
where H(x’ - xb) is the Heaviside function which is zero for x’ < XLand unity for x’ 2 XL. Since the flow is irrotational there exists a velocity potential 4’ which we define as $,
=f,(x,)
_
(Y'
+
2!
h’)*-___ a’_f’(x’)+ C_v’ + U4 a*_f’(x’) axI* 197
4!
dX’4_“”
(2.2)
K.
198
V.
PRAMOD
and M. J. VEDAN
wrface
bo-an ..
Fig. 1. Definition sketch.
We substitute for h’ from equation (2.1) and non-dimensionahse
y’ = h,y
E’ = ho&,
.ub = lx,,
x’ = Ix,
the variabfes by
(2.3) where c0 = fi Thus we get
and I and a are the characteristic wavelength and amplitude, respectively. 11,=f_
(J + I)2 - Z&(L’+ l)H(x - .x0) B.&x 2! + (J f 1)4 - 4&(Y+ l)‘H(.u - .YO)p*J xxx - ... 4!
(2.4)
where Now the velocity components
in the x- and y-directions are
u = $, =fx - (y + 1)2 - 2s’;:
f)H(x - x0) pL-,
+ E(.Y+ 1)Q.x - xrJ)B_Lr + O(P2)
(2.5)
and c = c;b,= [ - (y + 1) + EH(X - x0)] pfxx + 0(/P)
(2.6)
where 6(x - x0) = H’(x - x0), is the Dirac delta function. Thus we find that
B g + ; = 0 + o(p).
(2.7)
Therefore, the equation of continuity is satisfied if terms of 0(f12) are neglected. On the free surface, y’ = $, the boundary condition to be satisfied is Sfj’
d=,,+U$
I Zrl’
(2.8)
We define 4’ = a~, so that on the free surface, y = zq, where z = a/ho. In terms of the dimensionless variables the free-surface condition (2.8) becomes (2.9) Substituting for u and u from equations (2.5) and (2.6), equation (2.9) reduces to [ - frq + 1) + &H(X - .u,)]fx* + (rS + I)’
- 3&(ZV+ I)ZH(x - x,) 6
x B_Lx = Vf + &4x + O(@VP’).
(2.10)
Long-wave
propagation
199
in water
As in the derivation of the KdV equation, neglecting terms containing equation (2.10) becomes
z2, /?* and a/?,
tlr + Cl - sH(x - %)lW, + 4w)x + c- is + + BH(x - x0)1 wxx.x= 0
(2.11)
where w =fx. Now in the long-wave approximation
we have the equation of motion
u;, + liu;. + vlu;* = - g&
(2.12)
u,+stuu,+I-au,= B
(2.13)
which reduces to -qx.
Substituting for u and u from equations (2.5) and (2.6) and neglecting, as we did above, terms in z2, /I2 and aP we cast equation (2.13) in the form w, + tlX+ aww, - +B wXXI+ .$[H(x
(2.14)
- xlJw,,lx = 0.
Equations (2.11) and (2.14) are the governing equations for the wave motion. Neglecting terms in E, a and /I we get a solution of equations (2.11) and (2.14) to the lowest order: nJ = tl,
(2.15)
9, + tlX= 0.
Now we look for a solution of the form w = q + &A+ r/I + DC + &aD + @E.
(2.16)
Substituting equation (2.16) in to equations (2.11) and (2.14) we get + a[& + (v2Ll + PCC, - i~,l
tit + G + &CA,- Wx - 0~3 - H(x - ~oP,l
+ ~BCCX- H(x - x,)C, - $4,x
+ E~CD,+ WI),
+ iH(x - QLxxl
= 0
(2.17)
and tlr + tlX+ &AX+ a[& + VLI + BCG - h,,l
+ EaCD,+ WM
+ EB{E, - +A,,, + CH(x - xohl,I
= 0.
(2.18)
Since qX+ qr = 0 to the lowest order, we shall replace the t-derivatives by the negative of the x-derivatives. Then the two equations (2.17) and (2.18) are consistent if A, = tH(x - x,)~, B* = +I% CX = f&XX D, = - aH(x - xo)tlrx and Ex = - itH(x
- x&L,,
+ G(x - x&L,
+ fW
- x&x.
Then we have equation (2.17) as qr + [l - iEH(X - xg) + &&@‘(X - x0hl + hm
+ Ch + +Wx
- huh,, + W - W-w
- xo)lw, - x~~L,,
= 0 (2.19)
where we have taken A@,) = trt(x,). Here 6(x - ~0) = -&H(x - x0) is the Dirac delta function and the prime denotes differentiation with respect to x. For x < x0 we have H(x - x0) = 0, therefore we get from equation (2.19) tit + r!X+ %WX + ML,,
= 0
(2.20)
200
K. V.
and M. J.
PRAMOD
VEDAN
and for x > x0 we get, from equation (2.19) tit + (l’- &)tlX + ($2 + ac+PL
+ (# - M)L,
(2.21)
= 0.
Equation (2.20) is identical with the classical KdV equation for constant depth and equation (2.21) also reduces to this equation as E + 0.
3.
SOLUTION
OF
EQUATION
(2.19)
Since the coefficients of equation (2.19) contain generalised functions, its solution is difficult. But we shall obtain a solution by treating 6(x - x0) and 8(x - x0) as limits of classical functions [S]. We have n 6(x - x0) = lim n-z n[l + n2(x - x0)2]
(3.1)
and 6’(x - x0) = lim n-33
- 2n’(x - x0) n[l + n2(x - xo)2]2’
(3.2)
Substituting from equations (3.1) and (3.2) into equation (2.19) we get 1 -+f(x-x,)+rEB
&+
12
i +S@
- 2n3(x - x0) rlX+ c+ + ?ezH(x x[l + n2(x - xo)2]2 I
n + co - i&BHb n[l + f?(X - x0)2] rlXx
x0)1llxxx -+0
as n-
x0)1wx ,x.
(3.3)
We look for a local solution, rl = ‘II + H(x -
xo)(tl,
-
(3.4)
Sl)
of equation (3.3) in a neighbourhood (x0 - p, x0 + p) of x0, where qI and q2 are solutions of equations (2.20) and (2.21) in - cc < x < 30 . For any test function $, we define
=-
=
a Cat + H(x I -(o
s
co _9 v&NWdx
+
%2
-
xo)h2
+
v~)lW’(x)dx
-
(~2
-
vl)(xoM(xo)
~lM+Wd~~
I x0
(3.5) where [(t/2 - t11)6(x - xo)l(JI) =
(rt2
-
v,)(xoM(xo)
and m (tl2 I IO
%l, -q,,($) =
-
slLCW(x)dx.
Similarly we get tlxxw
=
tllrrW
-
[(t/2
-
r11)@
-
x0)1(1(/‘)- %J2-,,,w
(3.6)
+
Ch2
-
rtl)b(X
-
~o)lW’”
(3.7)
and %A$) =
‘I1 x.x.x ($1
+
@,qz
-&u.
Long-wave propagation in water
201
We can give a meaning to the operator qqXas follows: rtrt,($) = rl[%X + (112- rltP(x - x0) + @Ccsl -Ir,,l(lcI)
Let rl/ be a smooth function with support in (.x0 - P, x0 + P). We assume that q1 (x0) = ~2(xo) and the first two derivatives (with respect to .x) of qt and q2 vanish at x0. Substituting from equations (3.5), (3.6), (3.7) and (3.8) into equation (3.3) and taking the limit as n -, CL:we get
by equations (2.20) and (2.21) for all test functions $ with support in (x0 - p, x0 + ,n).Thus 9 is a solution of equation (3.3). 4. DISCUSSION
Here we have derived a KdV-type equation for the propagation of long waves, when there is a sudden change in depth due to a step-like discontinuity at the bottom. Non-linearity and dispersion are measured by the same parameters as in the classical KdV equation. The additional effect of change in depth is taken into account by a parameter E, which is a measure of change in depth, by retaining terms linear in this parameter. We find that as E -+ 0, the equation reduces to the classical KdV equation. The coefficients in the equation show singularity at the shelf. An additional feature is a diffusion term as in Burger’s equation. We have also obtained a solution for this equation. The previous studies by Johnson [33, Grimshaw [6], [73 and Leibovich and Randall [8] have given rise to perturbed KdV equations, which have been anafysed by Kaup and Newell [43 and Knickerbocker and Newell [93 and it has been shown that a KdV equation under perturbation need not satisfy conservation of mass though energy is conserved, while the classical KdV equation which is completely integrable has an infinite number of conserved quantities. Using inverse scattering transform technique (ET) Kaup and Newell (41 argue that in the case of the perturbed KdV equation a continuous spectrum is excited due to interaction between the soliton and the perturbation, and the reflection coefficient will have a Dirac delta function behaviour. Due to this, conservation of mass is obtained only by the formation of a shelf (since the depth below the mean level is constant). This formation of a shelf may be accounted for by a change in the depth of the undisturbed water, and in this sense a perturbed KdV equation may be a model equation. However, unlike a perturbed KdV equation, further studies of equation (2.19) can shed more light on long-wave propagation on an uneven bottom. REFERENCES 1. D. G. Korteweg and de Vries. On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. Mug. 39, 422-443 (1895). 2. R. S. Johnson, Water waves and Korteweg-de Vries equations. J. Ffuid Mech. 97, 701-719 (1980). 3. R. S. Johnson, On the development ofsolitary wave moving over an uneven bottom. Camb. Phil. Sm. Proc. 73,
183-203 (1973). 4. D. J. Kaup and A. C. Newell, Solitons as particles, oscillators and in slowly changing media: a singular perturbation theory. Proc. R. Sot. Land. A361, 413-146 (1978). 5. D. H. Griffel, Applied Funcrionol Analysis. Ellis Horwood, Chichester (1981). 6. R. Grimshaw, The solitary wave in water of variable depth. J. Fluid Mech. 42, 639-656 (1970). 7. R. Grimshaw, The solitary wave in water of variable deoth. Part 2. J. Fluid Mech. 46.611622 (1971). 8. S. Leibovich and J. D. Randall, Amplification and decay of long nonlinear waves. J. Ftuid Mech: Ss, G81-493
(1973). 9. C. J. Knickerbocker and A. C. Newell, Shelves and the Kortewegae (1980).
V&s equation. J. Fluid Me&
98,803-818