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12 Large amplitude long waves U >> 1: The nonlinear shallow water equations (NSW)

12 Large amplitude long waves U >> 1: The nonlinear shallow water equations (NSW)

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Introduction t o nearshore hydrodynamics

is still << 1, this implies that U = 6 / p >> 1. In this case, 6 is not small

and the resulting equations will therefore be strongly nonlinear. Therefore,

in the equations found in Chapter 7 for this case, we only need to include

terms 0 ( 1 ) to get equations with the first approximation to the nonlinear

terms. This results in


+ 64,rlz










= 6q




Following the same procedure as before, we substitute the solution for 4

to get for the kinematic condition. In 1DH 4 for varying depth from (9.8.2)

is given by

4 = 4o - p 2 ( z + h)hz40z- TP 2( z + h)24ozz+ 0b4)


This implies that




which shows that at z



=- U o z

= 67 we

4026 rlz

- (z


get for t,he last two terms in the kinematic



+ h ) 4ozz

+ Srl)40Z)z




-4z = ((h


The kinematic condition then becomes

+ ((677+ h M O x ) ,



Similarly, we get for the dynamic condition


+ 4ot + $1v 4 0 z ) 2



Again, we can express these two equations in terms of the bottom velocity

uo to get

+ ( ( d r l + h)uo)z = O h 2 )

72:+ UOt + UOUOz = 0 ( P 2 )


However, since the approximation for 4 used in deriving these equations

is just +(z, z , t ) = #o(z,

t ) the assumed velocity is constant over the depth

and consequently uo = U.


Large amplitude long waves U

>> 1: N S W


In the general 2DH version and dimensional form the equations become



qt Vh(u(h T I ) ) = 0

fit + u ' v h f i + g Vhq = 0



This is the set of equations that is called the Nonlinear Shallow

Water (NSW) equations. They have interesting features that are discussed briefly in Section 9.11. Note that the Boussinesq waves described in

earlier parts of this chapter include the NSW equations as a subset.

The deformation toward breaking of a wave propagating according to

the nonlinear shallow water equations.

Fig. 9.12.1

It may also be mentioned that these equations are often extended by

adding the effect of a bottom friction by replacing the 0 on the RHS of

(9.12.9) with a term of the form

illill/h, where f is the friction factor

(see Chapter 10).

The NSW-equations have no solutions for waves of constant form. Analysis of the above derivation shows that they correspond to hydrostatic pressue relative to the instantaneous (i.e. local) surface level and as mentioned

to constant velocity over depth. Using the method of characteristics (see

e.g. Abbott and Basco, 1989) it is found that in waves propagating according to these equations each part of the wave will propagate with a speed

of c = u d m , i.e. a c corresponding to the local depth of water

and including the local particle velocity. This means that higher parts of

a wave will move faster than lower parts, and overtake the lower parts as

shown in Fig. 9.12.1, and eventually the waves break. Of course the basic assumptions behind the equations have broken down long before this

happens, because the characteristic horizontal length is no longer large in

comparison to the water depth.

As with linear shallow water waves the phase velocity for these waves

is independent of the wave period so they are only amplitude dispersive.





lntroduction t o nearshore hydrodynamics

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high-order Boussinsq models. J. Fluid Mech., 399, 319-333.

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Benjamin, T. B., J. L. Bona, and J . J. Mahony (1972). Model equations

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Brocchini, M., M. Drago, and L. Iovenitti (1992). The modelling of short

waves in shallow waters. Comparison of numerical models based on

Boussuinesq and Serre euqations. ASCE Proc. 23rd Int. Conf. Coastal

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Brocchini, M. and M. Landrini (2004). Water waves for engineers.

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Chaplin, J. D. (1978). Developments of stream function theory. Rep.

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Chen, Q., P. A. Madsen, H. A. Schaffer, and D. R. Basco (1998). Wavecurrent interaction based on an enhanced Boussinesq approach. Coastal

Engrg., 33, 11 -39.

Cox, D. T., N. Kobayashi, and A. Wurjanto (1992). Irregular wave transformation processes in surf and swash zones. ASCE Proc. 23rd Int.

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Cox, D. T., N. Kobayashi, and D. L. Kriebel (1994). Numerical model

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water waves. Coastal Engineering, 13, 357-378.

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Chapter 9


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McGraw-Hill, New York.

Fenton, J. D. (1979). A high order cnoidal wave theory. J. Fluid Mech.,

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Gobbi, M. F. and J. T. Kirby (1999). Wave evolution over submerged

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Gobbi, M. F., J. T. Kirby, and G. Wei (2000). A fully nonlinear Boussinesq

model for surface waves. Part 2. Extension to O(kh)*. J. Fluid Mech.,


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products. Academic Press, New York.

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Kennedy, A. B., Q. Chen, J. T. Kirby, and R. A. Dalrymple (2000).

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1D. J. Waterway, Port, Coastal and Ocean Engineering, 126, 206-214

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Kobayashi, N., G. S. DeSilva, and K. D. Watson (1989). Wave transformation and swash oscillation on gentle and steep slopes. J. Geophys.

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Introduction t o nearshore hydrodynamics

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Appendix 9A: Elliptic integrals and functions

This appendix gives a very brief review of the definitions of elliptic

functions and integrals used in the derivations in Chapter 6.5.

Incomplete elliptic integrals and functions

The elliptic functions used here are of the Jacobian type. The central

definition is that of F ( $ , m ) which is called an incomplete elliptic integral

of the first kind. F is defined by


m is called the parameter of F , $ the amplitude.

Appendix 9B: Derivation of Eq. (9.5.38)

The inverse function of F with respect to

sary restrictions of monotony so that


4 is defined within the neces-

4 = arccos(m(F, m ) )


cos q5 = cn(F,rn)



which defines the elliptic function cn. this is written

C O S =

~ cn


(9 .A.4)

= F ( 4 ,m)


by the simplifying definition


One also encounters incomplete elliptic integrals of the second

kind defined by


and since 4 = +(ul m ) , E ( 4 ,m) is usually written E ( u ,m).

These two forms of elliptic integrals represent the standard forms to

which all other elliptic integrals can be reduced.

By various auxiliary definitions, the basic forms may be written in many

different forms (see e.g., Abramowitz and Stegun (1964), Chapter 17). The

reduction of arbitrary elliptic integrals to standard forms is well described

by Erdelyi et al. (1953), vol. 11.

Complete elliptic integrals

, integrals are called complete elliptic integrals.

When 4 = ~ / 2 the

We define

K ( m ) = F(Ir/2,m )


E(m) = E ( K ( m )m)



where (9.A.5) has been used.

Appendix 9B: Derivation of Eq. (9.5.38)

This appendix gives the derivation of equation (9.5.38).

Introduction t o nearshore hydrodynamics


Substituting (9.5.36) into (9.5.37) yields

L1 1’

qd38 =


+ Hcn2 2K6} d6 = 0


where we have also used (9.5.39). This can also be written

cn2u du = 0


From Abramowitz and Stegun (1964) (A & S in the following) equations

(16.25) and (16.26) we have

C ~ ’ Udu =



-E(a,m) - -a







Hence, (9.B.2) becomes


+( E (2K, m)





2mlK) = o


A & S eq. (17.4.4) yields

E(2K,m) = 2E(m)


which brings (9.B.5) on the form (after multiplication by mK)




mlK) = 0


As m is given by (9.5.30), we have for ml

772 - 773

ml = 1 - rll - 772 -~




(9 .B3)

771 -773

Substituting this into (9.B.7) together with (9.5.27) for H and (9.5.30)

for m and rearranging terms finally yields

which is eq. (9.5.38).

Appendix 9C: Numerical solution for cnoidal properties


Appendix 9C: Numerical solution for cnoidal properties

This appendix outlines simple numerical algorithms for obtaining the

parameter m and for solving for the wave parameters when the waves are

specified by H , h, and T .

Numerical solution of U = 16/3 m K 2

Solution of the equation



U = - m K2(m)


for given values of U provides the value of m in the elliptic functions and

integrals in the cnoidal wave solution. Here 0 5 m < 1. For practical cases

we normally have U > 10 which means the relevant values of m is close to

1, and K ( m ) is large.

(9.C.1) is a transcendental equation for m, and a Newton-Raphson iterative solution will give m with only a few iterations. The problem is

evaluation of K(m). To obtain a convenient formulation of the problem

we introduce the complimentary parameter ml = 1 - m for the elliptic

functions and write (9.C.l) as


u=( l -m1) K2(m1)



It turns out that in the range of solutions in question K(m1) can with

sufficient accuracy be approximated by (see Abramowitz and Stegun, 1964,

p 592, (17.3.35))

K(m1)= [ ao+al ml+a2 m2]-[bo+bl m1+b2 m2 ] In ml+c(ml) (9.c.3)


uo = 1.3862944





bl = 0.1213478


b2 = 0.0288729



a2 =

bo = 0.5


and E < 3 . lop5.

Writing (9.C.2) as


f(m1)= 1 - m l - 16 K 2

(9.C.5 )


Introduction t o nearshore hydrodynamics

We seek values of ml for which f(m1)= 0. The Newton-Raphson iterative

formula for ml becomes


where the derivative f ’ is obtained directly by differentiation of (9.C.5)

3 U dK

f’(ml)= -- - 8 K 3 dml




Initial estimate for ml

Using a suitable initial estimate ml (9.C.5) will rapidly converge toward

the solution for ml. An initial estimate for ml can be obtained from (9.C.2)

by substituting (9.C.3) for K and assuming ml << 1. This gives


U - -[ a0 - bo 1nm1I2





For U > 30 (9.C.10) will even function as a quite accurate direct solution

for ml.

With ml known K is determined from (9.C.3).

Calculation of L / h

Since in most cases the wave is specified by H , h and T it is necessary

to first to calculate L l h in order to determine U . For L l h we have







(9.C.1 2 )

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12 Large amplitude long waves U >> 1: The nonlinear shallow water equations (NSW)

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