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October, 1953
HOULT : WATER EFFECTS
87
Fig. 1 . Diagram of a Trochoid.
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Fig. 2. Generation of a Trochoid.
locus of a point on a radius of a circle which rolls along a given straight line (Fig. 1).
The geometrical construction of the wave shape is seen in Fig. 2 which shows the generation of a trochoid by the Point P on the radius of the circle which rolls along the underside of the line AB. In practice it is formed by the movement of the water itself, the individual particles travelling in circles, which become smaller the deeper the particles.
The paths of three particles of water, C, D and E, in the wave which is travelling in the direction of the arrow are shown in Fig. 3. It will be seen how the phases of C and E are the same, while D is of the opposite phase. It is also seen how the orbits of the particles below C decrease below the surface : the diameter at the surface being a while the diameter at a depth d is a'. Approximately we can say that there is no wave motion occurring below a depth d, and that it is confined to the surface layer of the water only.
The distance between the nearest points at which the wave-motion possesses the same phase is known as the wave-length and is denoted by A. When the wave has travelled from C to E, all the particles have made one orbital travel. The time for this to occur is known as the period and is denoted by T. The wave moves at a certain velocity which we denote by V. Then
A = V.T.
A, (1)
It can also be shown that when d
nL J J
a ~ e2w ~'~~ 535
that is, at a depth equal to CE the wave amplitude a is only 1/535 of the surface value, a, and may in fact be regarded as non-existent for most purposes. The significance of this is that the depth of the water does not affect the wave-motion, provided it is of the order of the wave-length on the surface. It should not be less than A/4.
It should be noted that the above considerations apply to a true trochoidal motion. Water waves resemble this closely but not exactly. In the extreme case, the maximum height wave obtainable from a trochoid is a cycloid, the locus of a point on the circumference of a circle which rolls on a given straight line. Fig. 4 shows the generation of a cycloid. This shape is never obtained in water, the nearest being one in which the angle a = 120° with a sharp ridge to the wave. Then the maximum value for the ratio
d
A
wave height . 1 wave length ' 7
If it is attempted to produce waves higher than this, the crests break (as in the formation of " white horses " at sea) and limit the height. Fig. 5 shows the sharpest crest which can be obtained in a water wave in deep water. In practice, waves may be as large as 600 ft. long and 1/7 of this in height