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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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452



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

rlt



+ 64,rlz



-



1



-dz

P2



z



=0



1

rl+4t+-6(4z)2=0

2



= 6q



z-67



(9.12.1)

(9.12.2)



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)



(9.12.3)



This implies that

1

-



P2



which shows that at z

condition



4z



=- U o z



= 67 we



4026 rlz



- (z



(9.12.4)



get for t,he last two terms in the kinematic

1



-



+ h ) 4ozz



+ Srl)40Z)z



(9.12.5)



0(P2>



(9.12.6)



-4z = ((h



P2



The kinematic condition then becomes



+ ((677+ h M O x ) ,



%



=



Similarly, we get for the dynamic condition

77



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



=0(P2)



(9.12.7)



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 )



rlt



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.



9.12



Large amplitude long waves U



>> 1: N S W



453



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



(9.12.8)

(9.12.9)



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.



--if



+



454



9.13



lntroduction t o nearshore hydrodynamics



References - Chapter 9



Abbott, M. B. and D. R. Basco (1989). Computational fluid dynamics.

Longman Scientific and Technical. 425 pp.

Abramowitz, M. and I. A. Stegun (1964). Handbook of mathematical

functions. Dover Publications, New York.

Agnon, Y., P. A. Madsen, and H. A. Schaffer (1999). A new approach to

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

Baker, G. S. Jr. and P. Graves-Morris (1980). Pad6 approximants, Cambridge University Press, 745 pp.

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

for long waves in non-linear dispersive systems. Phil. Trans. Roy. SOC.

Lond., A, 272, 47 78.

Boussinesq, J. (1872). Theorie des onde et des resous qui se propagent le

long d'un canal rectangulaire horizontal, en communiquant au liquide

contenu dans ce canal des vitesses sensiblement pareilles de la surface

au fond. Journal de Math. Pures et Appl., Deuxieme Serie, 17, 55-108.

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

Engrg., 76-88.

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

Springer Verlag, Berlin (Approx 250 pp, in preparation)

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

MCE/2/78, Dept. Civ. Engrg., Univ. Liverpool, UK.

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.

Conf. Coastal Engrg., Chap 10, 156 169

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

verification using Supertank data in surf and swash zones. ASCE Proc.

Coastal Dynamics '94, Barcelona Spain.

Deigaard, R. and J. Freds@e(1989). Shear stress distribution in dissipative

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

Dingemans, M. W. (1997). Water wave propagation over unenven bottoms. World Scientific, Singapore, 967 pp.

~



~



9.13



References



-



Chapter 9



455



Erdklyi, A. (editor) (1953). Higher transcendental functions, Vol 11.

McGraw-Hill, New York.

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

94,1, 129-161.

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

sills: test of a high order Boussinesq model. Coastal Engrg., 37,57-96.

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.,

405,181-210.

Gradshteyn, I. S. and I. M. Ryzhik (1965). Tables of integrals, series and

products. Academic Press, New York.

Hibberd, S. and D. H. Peregrine (1979). Surf and runup on a beach: a

uniform bore. J. Fluid Mech. 95,323-345.

Karambas, T. and C. Koutitas (1992). A breaking wave propagation

model based on the Boussinesq equations. Coastal Engineering, 18,

1-19.

Kaihatu, J. M. (2003). Frequency domain wave models in the nearshore

and surfzones. In Advances in Coastal Modeling, (V. C. Lakhan ed.),

Elsevier Science, 43-72.

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

Boussinesq modeling of wave transformation, breaking and runup. I:

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

Kirby, J. T. (1997). Nonlinear dispersive long waves in water of variable

depth. In: Gravity waves in water of finite depth (J. N. Hunt, ed.)

Advances in Fluid Mechanics, 10, 55-125. Comp. Mech. Publications.

Kirby, J. T. (2002). Boussinesq models and appications to nearshore wave

propagation surf zonne processes and wave induced currents. In: Advances in nearhsore modelling (Lakhan ed.)

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

Res., 94, C1, 951-966.

Kobayashi, N. and Wurjanto, A. (1992). Irregular wave setup and runup

on beaches. A X E J. Waterw., Port, Coast, and Ocean Engrg., 118,

368-386.

Korteweg, D. J. and G. DeVries (1895). On the change of form of long

waves advancing in a canal, and on a new type of long stationary waves.

Phil. Mag., Ser. 5, 39, 422 - 443.



456



Introduction t o nearshore hydrodynamics



Madsen, P. A., R. Murray, and 0. R. Sorensen (1991). A new form of the

Boussinesq equations with improved linear dispersion charactieristics.

Coast. Engrg. 15, 371-388.

Madsen, P. A. and 0. R. Smrensen (1992). A new form of the Boussinesq

equations with improved linear dispersion characteristics. Part 2. A

slowly-varying bathymetry. Coast. Engrg. 18, 183-204.

Madsen, P. A., 0. R. Smensen and H. A. Schaffer (1997a) Surf zone dynamics simulated by a Boussinesq type model.Part I. Model description

and cross-shore motion of regular waves. Coast. Engrg, 32, 255-288.

Madsen, P. A., 0. R. Smensen and H. A. Schaffer (1997b) Surf zone

dynamics simulated by a Boussinesq type model.Part 11. surf beat and

swash oscillations for wave groups and irregular waves. Coast. Engrg,

32, 289-319.

Madsen, P. A. and H. A. Schaffer (1998). Higher order Boussinesq equations for surface waves: derivation and analysis. Phil. Trans. R. SOC.

Lond. A 356, 3123-3184.

Madsen, P. A. and H. A. Schaffer (1999). A review of Boussinesq-type

equations for surface gravity waves. Advances in Coastal Engineering,

5, 1-93.

Madsen, P. A., H. B. Bingham, and H. Liu (2002). A new Boussinesq

method for fully nonlinear waves from shallow to deep water. J. Fluid

Mech., 462, 1-30.

Madsen, P. A., H. B. Bingham, and H. A. Schaffer (2003). Boussinesqtype formulations for fully nonlinear and extremly dispersive water waves. derivation and analysis. Proc. Roy. SOC.Lond. A, 459,

1075-1104.

Madsen, P. A. and Y. Agnon (2003). Accuracy and convergence of velocity

formulations for water waves in the framework of Boussinesq theory. J.

Fluid Mech. 477,285-319.

Masch, F. D. and R. L. Wiegel (1961). Cnoidal waves, tables of functions.

Council of Wave Research, The Engineering Foundation, Univ. Calif.,

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Milne-Thomson, L. M. (1968). Theoretical hydrodynamics, 5th ed.

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Scientific, Singapore.

Nwogu, 0. (1993). Alternative form 0s Boussinesq equations for nearshore

wave propagation. J. Waterway, Port Coastal and Ocean Engrg., ASCE,

119, (6) 618-638.



9.13



References - Chapter 9



457



Packwood, A. and D. H. Peregrine (1980). The propagation of solitary

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Numerical recipes, Camebridge University Press.

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model for waves breaking in shallow water. Coastal Engrg., 29, 185202.

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Boussinesq-type equations. Coast. Engrg, 26 1 14.

Serre, P. F. (1953). Contribution c i l’ktude des kcoulement permanents et

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regions. Den Private Engineering Fund, Technical University of Denmark.

Svendsen, I. A., J. Veeramony, J. Bakunin, and J. T. Kirby (2000). Analysis of the flow in weakly turbulent hydraulic jumps. J . Fluid Mech.

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horizontal nearshore circulations.

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waves: Comparison with experiments. Proc 2Gh Intl. Conf. Coastal

Engrg.. 258-271.

-



458



Introduction to nearshore hydrodynamics



Veeramony, J. and I. A. Svendsen (2000) The flow in surf zone waves.

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nonlinear Boussinesq model for surface waves. Part I. Highly nonlinear

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Zelt, J. A. (1991). The runup of nonbreaking and breaking solitary waves.

Coastal Engineering, 15, 205 246

~



~



Appendix

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



(9.A.l)

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



459



4 is defined within the neces-



4 = arccos(m(F, m ) )



(9.A.2)



cos q5 = cn(F,rn)



(9.A.3)



or



which defines the elliptic function cn. this is written

C O S =

~ cn



u



(9 .A.4)



= F ( 4 ,m)



(9.A.5)



by the simplifying definition

u



One also encounters incomplete elliptic integrals of the second

kind defined by



(9.A.6)

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 )



(9.A.7)



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

,



(9.A.8)



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



460



Substituting (9.5.36) into (9.5.37) yields



L1 1’

qd38 =



{q2



+ Hcn2 2K6} d6 = 0



(9.B.1)



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



cn2u du = 0



(9.B.2)



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

(16.25) and (16.26) we have

C ~ ’ Udu =



1

ml

-E(a,m) - -a

m

m



(9.B.3)



where



ml=l-m



(9.B.4)



Hence, (9.B.2) becomes

U



+( E (2K, m)

2mK

11



772



-



2mlK) = o



(9.B.5)



A & S eq. (17.4.4) yields



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



(9.B.6)



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



mK72



+H(E



-



mlK) = 0



(9.B.7)



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

772 - 773

ml = 1 - rll - 772 -~



771



-



773



(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



461



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

16

3



U = - m K2(m)



(9.C.1)



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



16

u=( l -m1) K2(m1)

3



(9.C.2)



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)

where

uo = 1.3862944



;



0.1119723

0.0725296



:



bl = 0.1213478



;



b2 = 0.0288729



a1



1



a2 =



bo = 0.5



(9.C.4)



and E < 3 . lop5.

Writing (9.C.2) as



3u

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



(9.C.5 )



462



Introduction t o nearshore hydrodynamics



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

formula for ml becomes



(9.C.6)

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



3 U dK

f’(ml)= -- - 8 K 3 dml



1



(9.C.7)



and



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

16

U - -[ a0 - bo 1nm1I2

3



(9.C.9)



or



(9.C.10)

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



L

h=T&$



h

(l+ilA)



(9.C.11)



where



(9.C.1 2 )



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