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S.A. Hughes / Coastal Engineering 51 (2004) 10671084

tabulated by Williams (1985) and expressed by Sobey

4.2. Extended linear wave theory

(1998) as the rational approximation

By assuming expressions for linear theory wave

x2Hlimit

a1r a2r2 a3r3

kinematics are valid above the still water level, it is

cotanh

13

1 b1r b2r2

g

possible to derive a somewhat more accurate estimate

of maximum depth-integrated wave momentum flux

where r=x2h/g, a1=0.7879, a2=2.0064, a3=0.0962,

at the wave crest. This technique has been referred to

b1=3.2924, b2=0.2645, and co=1.0575. Sobey noted

as extended linear theory or one-and-a-half-order

the above expression has a maximum error of 0.0014

wave theory.

over range of Williams' table. Williams' (1985)

Substituting Eqs. (8) and (9) for p d and u ,

tabulation of limit waves is more accurate than the

respectively, in Eq. (7), integrating from z=h to

traditional limit steepness given by

z=a (wave crest), applying the dispersion relation

x2=gk tanh kh, and making use of the identity sinh kh

Hlimit

0:142 tanhkh

14

cosh kh=1/2 sinh 2kh as before yields

L

qga sinhkh aŠ qga2

Eq. (14) overestimates limiting steepness for long

MFmax

coshkh

k

2

waves and underestimates limiting steepness for short

!

waves.

sinh2kh aŠ 2kh a

17

The relative contribution of the velocity term (qu2)

sinh2kh

to the total depth-integrated wave momentum flux

varies between about 5% for low-amplitude long-

Dividing Eq. (17) by (qgh2) and substituting a=H/2

period waves to nearly 30% for waves approaching

gives the nondimensional form of the maximum

limiting steepness. Linear theory estimates of max-

depth-integrated wave momentum flux parameter for

imum depth-integrated wave momentum flux are

extended linear theory, i.e.,

lower than actual because the momentum flux above

the still water level is neglected.

2

H sinhkh H =2Š 1 H

MF

1

As the wave period increases, and the wave length

qgh2

kh coshkh

2

h

8 h

becomes very long (shallow water waves), the wave

max

!

number approaches zero, and Eq. (12) approaches a

sinh2kh H =2Š 2kh H =2

limiting value for the wave momentum flux parameter

sinh2kh

given by

18

2

MF

1

H

1

H

The asymptotic long-wave limit of Eq. (18) as

qgh2

2

h

4

h

max

kY0 is given by

for very short waves

15

2!

MF

1

H

1 H

This limit is evident on the ordinate axis of Fig. 2.

qgh2

2

h

4 h

Similarly, Eq. (12) approaches an asymptotic form for

max

!

1 H

very short period waves given by

1

2 h

2

1

MF

1

H

h

1 H

for very long waves

19

qgh2 max 8p2 h

gT 2

8 h

for very short waves

16

which is seen to be an extension of the linear theory

long-wave limit.

However, this deepwater limit is of little interest when

considering nearshore coastal processes or coastal

the wave momentum flux parameter as a function of

relative depth h/gT2. The limiting wave steepness, as

structures.

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