*R.J. Sobey, S.A. Hughes *r *Coastal Engineering 36 (1999) 1736*

24

Squaring now and using in the *f*1 equation establishes a single implicit algebraic

equation in the local wave number *k*:

2

r *g *cosh *k*Z h q *z * P .

*gk*

*F *Z k . s Z uabs y *U*a . Z uabs y *U*a .

o

o

s 0.

Z 17.

y

*p*d bs cosh *k*Z h q *z * UV .

o

tanh *kh*

Some special cases of this equation are very revealing. When *z * P s *z * UV , Eq. Z17.

becomes:

2

r*g*

*kh*

1

X

*u*abs y *U*a

o

*u*abs y *U*a

o

*F *Z k. s

s 0.

Z 18.

y

Z

.Z

.

*p*d bs

o

*gh*

tanh *kh*

The function *kh*rtanh *kh *is 1 at *kh *s 0 Z` wave shallow' water., and approaches *kh*

asymptotically from above for large *kh *Z` wave deep' water.. There are solutions only

for:

2

r*g*

1

*u*abs y *U*a

o

*u*abs y *U*a

o

) 1.

Z 19.

Z

.Z

.

*p*d bs

o

*gh*

Solutions do not exist under quite a wide range of conditions. This is a potentially fatal

impediment.

It may be possible to reformulate the locally linear problem to avoid these difficul-

ties. Those time steps where there is no solution might be accommodated by the solution

of Eq. Z14. in the least-squares sense rather than an exact solution. It may also be

advisable to use neighboring observations in the local solutions. From the experience of

the Sobey Z1992. locally nonlinear solution from water surface traces, both of these

measures might help in dealing with observational error bands. All of these, however,

increase the complexity of the locally linear formulation and negate the sole advantage

of a linear formulation, simplicity. A locally nonlinear analysis is viable and is

introduced in Section 4.

**4. A local nonlinear analysis**

Nonlinear and irregular waves follow a field equation, the Laplace equation:

E 2f

E 2f

s 0,

Z 20.

q

E *z*2

E *x*a E *x*a

together with kinematic and dynamic free surface boundary conditions:

Eh

Eh

X

s 0 at *z *s h ,

*f*K s*w*y

y *u*a

Z 21.

E*t*

E *x*a

Ef

1

Z ub ub q *w * 2 . q *g*h s 0 at *z *s h ,

*f*D s

Z 22.

q

E*t*

2

and a kinematic bottom boundary condition on a sloping bed:

E*h*

*w *q *u*a

s 0 at *z *s y*h*.

Z 23.

E *x*a