1

exp *a *cos(θ - θ p )

(C3)

2π*I*0 (*a*)

where *I*0 is the modified Bessel function of the first kind and *a *is a parameter

describing the degree of directional spreading with *a *→ ∞ representing a

unidirectional wave field.

The wrapped-normal distribution was suggested by Mardia (1972) and is

given by*:*

1

1 N

1

+ ∑ exp - ( *j*σθ )2 cos *j*(θ - θ p )

(C4)

2π

π j =1

2

Figure C1 shows a plot of the distributions for the three different spreading func-

tion formulations corresponding to a standard deviation σθ of 25.5 deg. The

associated spreading indices are *s *= 2 for the cosine-power function and *a *= 5.55

for the circular-normal distribution. Thirty components (*N *= 30) were used for the

wrapped-normal distribution. The cosine-normal and wrapped-normal distribu-

tions are slightly narrower than the cosine-power function although the differences

can be considered to be minimal.

0.02

Cosine Power (s = 2)

Circular Normal (a = 5.55)

Wrapped Normal (σθ= 25.5o)

0.015

0.01

0.005

0

-90

-60

-30

0

30

60

90

θ (deg)

Figure C1. Comparison of the cosine-power, circular-normal and wrapped-normal

distributions with a standard deviation of 25.5 deg

C2

Appendix C Directional Wave Spreading Functions

Integrated Publishing, Inc. |