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
z67
(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 NSWequations 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
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~
~
9.13
References

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~
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=lm
(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 NewtonRaphson 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 NewtonRaphson 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 )