a wheel, when the wheel is made to run along a level piece of ground at the foot of the wall; but when thus described, it would be inverted from the position shown in Fig. 57.

To determine a point on the trochoid is very simple. As the rolling circle advances, a point on its circumference (say 3) comes into contact with the corresponding point of the directrix-line QR; the centre of the circles must at that instant be (S) vertically below the point of contact (3), and the angle through which the circular disc and the tracing arm OP have both turned is given by Q03. The angle POc, on the original position of the circles, equals Q03; through S draw Sc2 parallel to Oc, and make Sc, equal to Oc; then c2 is a point on the trochoid. Or the same result may be reached by drawing cc, horizontal, finding its intersection (c) with the vertical line S3, and then making cc3 equal to cc. In algebraical language, this may be simply expressed. Take Q Take Q as the origin of co-ordinates, QR for axis of abscissæ (x).

Then

angle QO3 = 0,

and æ, y co-ordinates of point c2 on trochoid.

x = C1C2 = C1C3 +C_C3

= a + b sin 0;

y = c1Q = OQ+Oc1

=a+b cos 0.

The tracing arm (OP) may, for wave motion, have any value not greater than the radius of the rolling circle (OQ). If OP equals OQ, and the tracing point lies on the circumference of the rolling circle, the curve traced is termed a cycloid, and corresponds to a wave on the point of breaking. The curve RTR, in Fig. 57, shows a cycloid, and it will be noticed that the crest is a sharp ridge or line (at R), while the hollow is a very flat curve.

A few definitions must now be given of terms that will be frequently used hereafter. The length of wave is its measure

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.

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,

L 7 CDR = = -= 22

π

* This diagram we borrow from Mr. Froude's paper on "Wave Motion" in the Transactions of the

L (nearly).

Institution of Naval Architects for 1862; it was one of the first constructed, and is therefore reproduced.

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

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