Mechanics of Fluids

468 Flow with a free surface
For waves not meeting these conditions the analysis is much more
complicated, and in what follows we shall therefore restrict ourselves almost
entirely to Airy waves. (The corresponding equations are those marked A.)
Although application of this theory may lead to some error in practice, this
is often tolerable in comparison with uncertainties in measurements.
The velocity c is known as the phase velocity because points of the same
phase (i.e. of equal η) move at this velocity, regardless of the shape of the
wave. Wave propagation is described as dispersive when the phase velocity
depends on the wavelength λ. This is because if a wave of general shape
were split into components of different wavelengths the components would
move at different velocities and thus become separated, that is, dispersed.
It is helpful to examine some specific results that can be deduced from the
generalized eqn 10.54.
Equation 10.54 shows that the effect of surface tension on the phase
velocity is negligible if
2πγ
ϱλ
≪ gλ
2π
� �1/2 γ
that is, if λ ≫ 2π
ϱg
For water this last quantity is about 17 mm, and so we can safely disregard
the effect of surface tension on ocean waves, for example.
Deep water waves are often regarded as those for which h >λ/2. Then
tanh(2πh/λ) differs from unity by less than 0.004 and we can take
c 2 = 2πγ
ϱλ
+ gλ
2π
(10.54a)
This has a minimum value when λ = 2π(γ/ϱg 1/2 ) (≏ 17 mm for a water);
then cmin = (4gλ/ϱ) 1/4 (≏ 0.23 m · s −1 for water).
10.13.2 Gravity waves
Waves whose properties are primarily determined by gravitational effects
are referred to as gravity waves. Surface tension effects can then be ignored
and eqn 10.54 reduces to
c 2 = gλ
2π tanh
� �
2hπ
(10.54b)
λ
For long waves, in which λ ≫ h, tanh(2πh/λ) → 2πh/λ. We then obtain
c = (gh) 1/2
(10.54c)
as for the single surge wave (Section 10.8). Instead of the description long
wave, the term shallow-water wave is frequently used for waves which satisfy
the condition λ ≫ h. However the term long wave is to be preferred,
because it more accurately reflects the fact that waves of long wavelength are
included, irrespective of whether the water is considered deep or shallow.