Definitions for wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 27

rather than on an actual count of the total number of wave crests in each frequency bin,
unless stated otherwise.

2.6 Dispersion relationship

The dispersion relationship, that is the relationship between the temporal (i.e. wave period/
frequency) and spatial (i.e. wavelength/wavenumber) scales is a general property of surface
water waves rather than a wave-breaking characteristic. It has to be introduced among
definitions here, however, because it will frequently be mentioned and used throughout
this topic.
In a general case of a weakly nonlinear deep-water unidirectional modulated wave train,
it can be written as

gk 1

2

1
2 a 2 k 2

a xx
8 k 2 a

2

ω = + + (2.14)

(e.g. Yuen & Lake , 1982 ). In deriving (2.14) , the steepness ak (1.1) was assumed

to be a small parameter, and therefore subsequent perturbation terms of higher orders of

=

are not shown. The first term on the right characterises the linear frequency dispersion
of surface waves, and the second term describes nonlinear correction to the dispersion
due to finite amplitude. The last term in (2.14) , where a xx is the second derivative of
wave amplitude a with respect to spatial coordinate x , comes from assuming that wave
frequency, wavenumber and amplitude are slowly varying functions in space and time. It
is only relevant for nonlinearly modulated wave trains and is usually omitted.
For wave trains of constant frequency, wavenumber and amplitude, the phase speed c is

g
k

1

2 a 2 k 2

T = k = 1

c = + . (2.15)

When waves can be treated as linear, that is their steepness (1.1) is small

= ak → 0 , (2.16)

then the linear dispersion relationship is simply

2

ω = gk (2.17)

and the linear phase speed is

g
k =

g
ω .

c = (2.18)

As seen in (2.18) , even linear waves with different frequencies/wavenumbers propagate
with different phase speeds. For nonlinear waves, phase speed (2.15) additionally depends
on wave amplitude/steepness. These properties of surface waves are broadly used in wave-
breaking experimental techniques to achieve frequency or amplitude focusing of waves of
different scales at a particular point in space/time, in order to make these waves break.