Mechanics of Fluids

478 Flow with a free surface
and thus the wave amplitude increases – sometimes quite dramatically. This
is why large waves are often found at headlands at the sides of a bay; and it
is an important consideration in the design of harbours.
When the depth becomes little more than the amplitude, the Airy theory
ceases to hold and the wave profile is increasingly distorted. The crests
become sharper and the troughs flatter. Moreover, the velocity of propagation
of the upper part of the profile is greater than that of the lower part;
consequently the crests curl forwards and finally break. Breaking usually
occurs when a ≏ 3
4 h.
10.13.9 Standing waves
If two trains of waves of the same amplitude, wavelength and period, but
travelling in opposite directions, are combined, the result is a set of standing
(i.e. stationary) waves. This can happen when a series of waves is reflected
by a fixed solid boundary perpendicular to the direction of propagation: the
two individual wave trains are then formed of the incident waves travelling
with velocity c and the reflected waves with velocity −c. For example, if the
individual waves are small-amplitude sine waves, the equation of the free
surface for the resulting standing wave is
η = a sin{m(x − ct)} + a sin{m(x + ct)} = 2a sin mx cos mct A (10.66)
That is, at any instant the free surface is a sine curve but its amplitude
2a cos mct varies continuously with time. The values of x that give η = 0
are independent of t and thus the wave profile does not travel over the
surface; instead it simply rises and falls as indicated by the dotted lines in
Fig. 10.48. The positions at which η is always zero are called nodes and
those of maximum vertical motion are called antinodes. Equation 10.66
shows that the wavelength λ = 2π/m is the same as for the original waves;
so too is the period λ/c.
From eqn 10.47 A =−ca cosech{m(h + a)} where a again denotes the
maximum value of η. Substituting this into eqn 10.55 we see that the horizontal
velocity of a particle in a single moving wave (not necessarily of small
amplitude) is
mca cosech � m(h + a) � cosh � m(h + z) � sin{m(x − ct)}
If we add to this the value for a wave of equal amplitude travelling with
velocity −c we obtain the horizontal velocity of a particle in a standing
wave:
mca cosech � m(h + a) � cosh � m(h + z) �
×[sin {m(x − ct)} − sin{m(x + ct)}]
=−2mca cosech{m(h + a)}cosh{m(h + z)}cos mx sin mct
Similarly the vertical velocity may be shown to be
−2mca cosech � m(h + a) � sinh � m � h + z �� sin mx sin mct