124
4 How to Determine Wave Parameters
solitary profile and small amplitude wave theory should not be used for very
small water depths.
Since the Korteweg and de Vries time, it was thought that the collision of
two solitary waves would result in a strong interaction and eventually end in
their destruction. Then, in 1965 the computer simulation of the collision of
the two solitary waves performed by Zabusky and Kruskal (1965) showed that
after collision the waves still retain their shapes and propagation velocities.
Due to somewhat elementary particle-like behaviour of these waves, Zabusky
and Kruskal coined the word soliton to describe them. The examples of physical soliton solutions are the water waves, ion-acoustic waves and magnetohydrodynamic waves in plasma, pressure waves in liquid-gas bubbles mixtures,
propagating of sound waves through the crystal lattice, and photon packets in
low-temperature crystal (Osborne and Burch, 1980; Massel, 1989).
In Chap. 6 we will show that internal waves occurring within subsurface layers
of the stratified waters sometimes behave as solitons.
4.2.4 Wave Shoaling and Refraction
In previous sections we have dealt only with waves propagating on water of
constant depth. However, the sea bottom very rarely is horizontal. In the deep
ocean, when water depth is larger than half of the wavelength, depth changes
(even by a few hundreds of meters) do not influence the surface waves. On the
other hand, in the coastal zone the shoaling water depth effects the phase speed
of waves (Eq. 4.16), and as was demonstrated in Fig. 4.5, the phase velocity
increases with the local water depth. Therefore, parts of the wave crest lying
over deeper water travel faster than the parts of the same crest lying over
shallow water. In the course of propagation, such a wave front gradually turns
towards the shallow water (see Fig. 4.13).
Let us consider a wave crest at some time instant, t, and two points, A and B,
located on that crest, at water depths h3 and h < h3, respectively. After some
time, llt, point A moves for distance iA along ray 1, while point B moves some
distance, iB, along ray 2. The term wave ray has been taken from optics. Rays
are the lines which can be drawn perpendicularly to the wave crests and which
indicate the direction of wave movement. Because water depth at point A is
larger than depth at point B, the velocity of point A is greater than velocity
of point B. Therefore, the distance LA will be greater than the distance iB ,
and the wave crest which joins points A and B gradually becomes parallel to
the shoreline. This is in agreement with the common observation on beaches
that the crests end up almost parallel to the shoreline, even when they are
approaching the coast at an oblique angle from the sea.
This phenomenon is known as refraction. Refraction of waves over decreasing water depth can be described by a relationship similar to Snel's law, which
describes refraction of light rays through materials of different refractive indices
(Willebrord Snel van Royen was a Dutch scientist who lived 1580-1626).
4 How to Determine Wave Parameters
solitary profile and small amplitude wave theory should not be used for very
small water depths.
Since the Korteweg and de Vries time, it was thought that the collision of
two solitary waves would result in a strong interaction and eventually end in
their destruction. Then, in 1965 the computer simulation of the collision of
the two solitary waves performed by Zabusky and Kruskal (1965) showed that
after collision the waves still retain their shapes and propagation velocities.
Due to somewhat elementary particle-like behaviour of these waves, Zabusky
and Kruskal coined the word soliton to describe them. The examples of physical soliton solutions are the water waves, ion-acoustic waves and magnetohydrodynamic waves in plasma, pressure waves in liquid-gas bubbles mixtures,
propagating of sound waves through the crystal lattice, and photon packets in
low-temperature crystal (Osborne and Burch, 1980; Massel, 1989).
In Chap. 6 we will show that internal waves occurring within subsurface layers
of the stratified waters sometimes behave as solitons.
4.2.4 Wave Shoaling and Refraction
In previous sections we have dealt only with waves propagating on water of
constant depth. However, the sea bottom very rarely is horizontal. In the deep
ocean, when water depth is larger than half of the wavelength, depth changes
(even by a few hundreds of meters) do not influence the surface waves. On the
other hand, in the coastal zone the shoaling water depth effects the phase speed
of waves (Eq. 4.16), and as was demonstrated in Fig. 4.5, the phase velocity
increases with the local water depth. Therefore, parts of the wave crest lying
over deeper water travel faster than the parts of the same crest lying over
shallow water. In the course of propagation, such a wave front gradually turns
towards the shallow water (see Fig. 4.13).
Let us consider a wave crest at some time instant, t, and two points, A and B,
located on that crest, at water depths h3 and h < h3, respectively. After some
time, llt, point A moves for distance iA along ray 1, while point B moves some
distance, iB, along ray 2. The term wave ray has been taken from optics. Rays
are the lines which can be drawn perpendicularly to the wave crests and which
indicate the direction of wave movement. Because water depth at point A is
larger than depth at point B, the velocity of point A is greater than velocity
of point B. Therefore, the distance LA will be greater than the distance iB ,
and the wave crest which joins points A and B gradually becomes parallel to
the shoreline. This is in agreement with the common observation on beaches
that the crests end up almost parallel to the shoreline, even when they are
approaching the coast at an oblique angle from the sea.
This phenomenon is known as refraction. Refraction of waves over decreasing water depth can be described by a relationship similar to Snel's law, which
describes refraction of light rays through materials of different refractive indices
(Willebrord Snel van Royen was a Dutch scientist who lived 1580-1626).