3.4 Elimination of the "Garden Sprinkler Effect"
55
1
S(wk,f3z) = :2.:: ajb;S(wk+;,f3l+j),
(3.11b)
i,j=-1
where aj, b; are the interpolation coefficients: a_ 1 = a 1 = b_ 1 = b1 = 1/24;
a0 = b0 = 11/12. The averaged energy within this interval contains not
only the component S(wk, {31) but the spectra values of the neighbouring
components as well.
In a similar way the wave energy balance equation (3.1) can be transformed to describe the mean energy evolution taking into account the frequency Llw and the angular resolution .6.{3. This is achieved using an integral
operator of the type (3.11a) in (3.1).
The traditional expansion of trigonometric functions valid for small values
1.6.!31 < 1 can be estimated as:
cosf3z±l = cos(f3z ± .6.{3) = cosf3z(1- (.6.{3) 2 /2)
=f sinf3z.6.f3(1- (.6.{3) 2 /6) + 0((.6.{3) 4 );
sinf3z±1 = sin(f3z ± .6.{3) = sin/31(1- (.6.{3) 2 /2)
± cosf3z.6.f3(1- (.6.{3) 2 /6) + 0((.6.{3) 4 ).
(3.12a)
(3.12b)
Taking into account that the group velocity in (3.1)-(3.4) is inversely proportional to the frequency Cg "' 1/w, the following expansion can be applied:
_ 1
_ 1
Llw
Llw
Llw
Llw
[
( ) 2 ( ) 3 ( ) 4]
W;± 1 = W; 1 =f -z; + -z; =f -z; + 0 -z;
(3.12c)
Thus, (3.1) is written as:
- + - -
+ ____: ___: _ _ __;_ _ ___; : . . . . . . . . . : . .
as [ 1 a( 0(1 + s /2 + o) cos cpS) a( ~(1 + s /2 + o)s)
at
cos 'P
ocp
ofJ
8(,@(1- s/2 + o)S)l [Cg a ( f3as sinf3 as) a,6 8 2 Sl
+
8{3
+ E R 8{3 cos Otp - cos 'P f){) + 8{3 8{32
Cgw a [( . as cosf3 as
a
- )]
- 0 - - smf3- + ----- tancp-(cosf3S)
R ow
ocp cos 'P ofJ
fJ {3
+ O(s 2 ) + 0(8 2 ) = 0,
(3.13)
where E = (.6.{3) 2 /12 « 1, o = (Llwjw) 2 /12 « 1.
Deriving (3.13), it is assumed that E"' o. The first two terms (the second
is in the first square brackets) in (3.13) are similar to the appropriate terms
of the main wave energy balance equation (3.1). In addition they contain
the correction terms of order 0 and s, describing changes in the propagation
velocity of spectral components connected with the frequency and angular