circle: then the centre (E) of the tracing circle (i.e. the mid-height of the trochoid) will be above the level line (1) Now R is known when the length

by a distance equal to

2.2

2R

of the wave is known: also r is given for any depth by the above approximate rule. Consequently, the reader will have in his hands the means of drawing the series of trochoidal subsurfaces for any wave that may be chosen.

Columns of particles which are vertical in still water become curved during the wave passage; in Fig. 59, a series of such vertical lines is drawn (see the fine lines a, b, c, d, &c.) ; during the wave transit these lines assume the positions shown by the strong lines (a, b, c, d, &c.) curving towards the wave crest at their upper ends, but still continuing to inclose between any two the same particles as were inclosed by the two corresponding lines in still water, The rectangular spaces inclosed by these vertical lines (a, b, c, d, &c.) and the level lines (0, 1, 2, &c.) produced are changed during the motion into rhomboidal-shaped figures, but remain unchanged in area. Very often the motions of these originally vertical columns of particles have been compared to those occurring in a corn-field, where the stalks sway to and fro, and a wave form travels across the top of the growing corn. while there are points of resemblance between the two cases, there is also this important difference-the corn-stalks are of constant length, whereas the originally vertical columns become elongated in the neighbourhood of the wave crests, and shortened near the wave hollows.

But

These are the chief features in the internal structure of a trochoidal wave, and in the following chapter they will be again referred to in order to explain the action of waves upon ships. It is necessary, however, at once to draw attention to the fact that the conditions and direction of fluid pressure in a wave must differ greatly from those for still water. Each particle in the wave, moving at uniform speed in a circular orbit, will be subjected to the action of centrifugal force as well as the force of gravity; and the resultant

FIG 60.

of these two forces must be found in order to determine the direction and magnitude of the pressure on that particle. This may be simply done as shown in Fig. 60 for a surface particle in a wave. Let BED be the orbit of the particle; A its centre; and B the position of the particle in its orbit at any time. Join the centre of the orbit A with B; then the centrifugal force acts along the radius AB, and the length AB may be supposed to represent it. Through A draw AC vertically, and make it equal to the radius (R) of the rolling circle; then AC will represent the force of gravity on the same scale as AB does that of centrifugal force. Join BC, and it will represent in magnitude and direction the resultant of the two forces acting on the particle. Now it is an established property of a fluid that its free surface will place itself at right angles to the resultant force impressed upon it. For instance, take the simple case of a rectangular box (shown in Fig. 61) con

FIG 61.

taining water, which is made to move along a smooth horizontal plane by the continued application of a force F; then we shall have uniformly accelerated motion, equal increments of velocity being added in successive units of time.* In order to compare this force with that of gravity,

* See remarks on this subject at page 106 of Chapter IV.

if ƒ is the velocity added per second of time, and W is the weight of the box and water, we should have,

Now it is well known that under the assumed circumstances of motion the surface of the water in the box will no longer remain level, but will attain some definite slope such as AB in Fig. 61; and it is easy to ascertain the angle of slope. Through any point G draw GH vertical to represent the weight W, and GK horizontal to represent the force F; join HK, and it will represent the resultant of the two forces, the water surface AB placing itself perpendicular to the line, on the principle mentioned above.*

Reverting to Fig. 60, the resultant pressure shown by BC must be normal to that part of the trochoidal surface PQ where the particle B is situated. Similarly, for the position B1, CB, will represent the resultant force; PQ1, drawn perpendicularly to CB,, being a tangent to the trochoid at B,. Conversely, for any point on any trochoidal surface in a wave, the direction of the fluid pressure must lie along the normal to that surface. Hence it follows that wave motion involves constant changes in the magnitude and direction of the fluid pressure for any trochoidal surface; these changes of direction partaking of the character of a regular oscillation keeping time with the wave motion. At the wave hollow the fluid pressure acts along a vertical line; as its point of application proceeds along the curve, its direction becomes more and more inclined to the vertical, until it reaches a maximum inclination at the point of inflexion of the trochoid; thence onwards towards the crest the direction of the normal pressure is constantly decreasing until at the crest it is once more vertical. If a small raft

* If a be the angle of inclination of the surface to the horizon: then F.

tan a = W

floats on the wave (as shown in Fig. 62), it will at every instant place its mast in the direction of the resultant fluid pressure, and in the diagram several positions of the raft are indicated to the left of the wave crest. These motions of the direction of the normal to the trochoid may be compared with those of a pendulum, performing an oscillation from an angle equal to the maximum inclination of the normal on one side of the vertical to an equal angle on the other side, and completing a single swing during a period equal to half the wave period.

The maximum slope of the wave to the horizon occurs at a point somewhat nearer the crest than the hollow, but no great error is assumed in supposing it to be at mid-height in ocean waves of common occurrence where the radius of the tracing arm (or half-height of the wave) is about onetwentieth of the length. For this maximum slope, we have

served by writing the circular measure of the angle instead of the sine; hence ordinarily we may say,

Take, as an example, a wave for which the dimensions were actually determined in the Pacific, 180 feet long and 7 feet high:

Maximum slope = 180° x

7 180

= 7° (nearly).

The variation in the direction of the normal was in this case equivalent to an oscillation of a pendulum swinging 7 degrees on either side of the vertical once in every halfperiod of the wave-some 3 seconds. These constant and rapid variations in the direction of the fluid pressure in wave water constitute the chief distinction between it and still water, where the resultant pressure on any floating body always acts in one direction, viz. the vertical.

But it is also necessary to notice that in wave water the intensity as well as the direction of the fluid pressure varies from point to point. Reverting to Fig. 60, and remembering that lines such as BC represent the pressure in magnitude as well as direction, we can at once compare the extremes of the variation in intensity. In the upper half of the orbit of a particle, centrifugal force acts against gravity, and reduces the weight of the particle; this reduction reaches a maximum at the wave crest, when the resultant is represented by CE (R-r). In the lower half of the orbit, gravity and centrifugal force act together, producing a virtual increase in the weight of each particle; the maximum increase being at the wave hollow, where the resultant is represented by CD (R+r). If a little float accompanies the wave motion, it may be treated as if it were a particle in the wave, and its apparent weight will undergo similar variations. In a ship, heaving up and down on waves very large as compared with herself, the same kind of variations will occur, though