ment (in feet usually) from crest to crest, or hollow to hollow -QR in Fig. 57 would be the half-length. The height of a wave is reckoned (in feet usually) from hollow to crest; thus in Fig. 57, for the trochoidal wave, the height would be Ph-twice the tracing arm. The period of

a wave is the time (usually in seconds) its crest or hollow occupies in traversing a distance equal to its own length; and the velocity (in feet per second) will, of course, be obtained by finding the quotient of the length divided by the period, and would commonly be determined by noting the speed of advance of the wave crest.

Accepting the condition, that the profile of an ocean wave is a trochoid, the motion of the particles of water

FIG.58.

P

Direction of Advance

The

in the wave requires to be noticed, and it is here the explanation is found of the rapid advance of the wave form, while individual particles have little or no advance. trochoidal theory teaches that every particle revolves with uniform speed in a circular orbit (situated in a vertical plane which is perpendicular to the wave ridge), and completes a revolution during the period in which the wave advances through its own length. In Fig. 58, suppose P, P, P, &c. to be particles on the upper surface, their orbits being the equal circles shown: then for this position of the wave the radii of the orbits are indicated by OP, OP, &c. The arrow below the wave profile indicates that it is advancing from right to left; the short arrows on the circular orbits show that at the wave crest the particle is moving in the same direction as the wave is advancing in, while at the

L

hollow the particle is moving in the opposite direction. It need hardly be stated again that for these surface particles the diameter of the orbits equals the height of the wave. Now suppose all the tracing arms OP, OP, &c. to turn through the equal angles POP, POp, &c.: then the points P, p, p, &c. must be corresponding positions of particles on the surface formerly situated at P, P, &c. The curve drawn through p, p, p, &c. will be a trochoid identical in form with P, P, P, &c., only it will have its crest and hollow further to the left; and this is a motion of advance in the wave form produced by simple revolution of the tracing arms and particles (P).* The motion of the particles in the direction of advance is limited by the diameter of their orbits, and they sway to and fro about the centres of the orbits. Hence it becomes obvious why a log dropped overboard, as described above, does not travel away on the wave upon which it falls, but simply sways backward and forward. One other point respecting the orbital motion of the particles is noteworthy. This motion may be regarded at every instant as the resultant of two motions—one vertical, the other horizontal-except in four positions, viz. : (1) when the particle is on the wave crest; (2) when it is in the wave hollow; (3) when it is at mid-height on one side of its orbit; (4) when it is at the corresponding position on the other side. On the crest or hollow the particle instantaneously moves horizontally, and has no vertical motion. At mid-height it moves vertically, and has no horizontal motion. Its maximum horizontal velocity

will be at the crest or hollow; its maximum vertical velocity at mid-height. Hence uniform motion along the circular orbit is accompanied by accelerations and retardations of the component velocities in the horizontal and vertical directions. The particles which lie upon the trochoidal upper surface

* It is possible to construct a very simple apparatus by which the simultaneous revolution of a series of particles will produce the

apparent motion of advance; and in lectures delivered at the Royal Naval College such an apparatus was used by the Author.

of the wave are situated in the level surface of the water when at rest. The disturbance caused by the passage of the wave must extend far below the surface, affecting a great mass of water. But at some depth, supposing the depth of the sea to be very great, the disturbance will have practically ceased: that is to say, still, undisturbed water may be conceived as underlying the water forming the wave; and reckoning downwards from the surface, the extent of disturbance must decrease according to some law. The trochoidal theory expresses the law of decrease, and enables the whole of the internal structure of a wave to be illustrated in the manner shown in Fig. 59.* On the right-hand side of the line AD the horizontal lines marked 0, 1, 2, 3, &c. show the positions in still water of a series of particles which during the wave transit assume the trochoidal forms numbered respectively 0, 1, 2, 3, &c. to the left of AD. For still water every unit of area in the same horizontal plane has to sustain the same pressure: hence a horizontal plane would be termed a surface or subsurface of "equal pressure," when the water is at rest. As the wave passes, the trochoidal surface corresponding to that horizontal plane will continue to be a subsurface of equal pressure; and the particles lying between any two planes (say 6 and 7) in still water will, in the wave, be found lying between the corresponding trochoidal surfaces (6 and 7).

In Fig. 59, it will be noticed that the level of the stillwater surface (0) is supposed changed to a cycloidal wave (0), the construction of which has already been explained; this is the limiting height the wave could reach without breaking. The half-length of the wave AB being called L, the radius (CD) of the orbits of the surface particles will be given by the equation,

All the trochoidal subsurfaces have the same length as the cycloidal surface, and consequently they are all generated by the motion of a rolling circle of radius R; but their

FIG.59.

tracing arms-measuring half the heights from hollow to crest-rapidly decrease with the depth (as shown by the dotted circles), the trochoids becoming flatter and flatter in

consequence. The crests and hollows of all the subsurfaces are vertically below the crest and hollow of the upper wave profile. The heights of these subsurfaces diminish in a geometrical progression, while the depth increases in arithmetical progression; and the following approximate rule is very nearly correct. The orbits and velocities of the particles of water are diminished by one-half, for each additional depth below the mid-height of the surface wave equal to one-ninth of a wave length.* For example

Depths in fractions of a wave length below

the mid-height of the surface wave Proportionate velocities and diameters

0, J, Z, j, f, &c.

1, 1, 1, 1, 1e, &c.

Take an ocean storm wave 600 feet long and 40 feet high from hollow to crest: at a depth of 200 feet below the surface (of length), the subsurface trochoid would have a height of about 5 feet; at a depth of 400 feet (8 of length) the height of the trochoid-measuring the diameter of the orbits of the particles there-would be about 7 or 8 inches only; and the curvature would be practically insensible on the length of 600 feet. This rule is sufficient for practical purposes, and we need not give the exact exponential formula expressing the variation in the radii of the orbits with the depth.

The

It will be noticed also in Fig. 59 that the centres of the tracing circles corresponding to any trochoidal surface lie above the still-water level of the corresponding horizontal plane. Take the horizontal plane (1), for instance. height of the centre of the tracing circle for the corresponding trochoid (1) is marked E, EF being the radius; and the point E is some distance above the level of the horizontal line 1. Suppose r to be the radius of the orbits for the trochoid under consideration, and R the radius of the rolling

See page 70 of Shipbuilding, Theoretical and Practical, edited by the late Professor Rankine: who,

with Mr. Froude, has done much to develop the trochoidal theory.