Wave Motion.
1. The trochoidal theory of wave motion, as applied to the surface of a body of water, assumes that each particle in the surface of a wave describes a vertical circle with uniform velocity.
In the accompanying diagram let ABP denote the vertical circle described by the particle P. This particle is acted upon by three forces; W, its weight, F, the centrifugal force, and R, the resistance of the surrounding particles of water. Hence we have, to denoting the constant angular velocity,
a/b = W/F = Mg/Mw2b whence a = g/w2
a is therefore constant, and MP intersects the vertical line through C in the fixed horizontal line MN. It is further evident that the resistance R acting along the line q is normal to the surface. To obtain the equation of the curve APD, we have
tan? = bsin?/a-bcos? = (dy/d?)/(dx/d?)
but dy = bsin?d?, whence y = c-bcos?; ? = 0 gives y = a-b ∴ c = a
Again:
dx = (a-bcos?)d? ∴ x = a?-bsin?+c’;
but ? = 0 gives x = 0 ∴ c’ = 0
hence we have for the equations of the curve
x = a?-bsin? and y = a-bcos?
These are the equations of a trochoid, a being the radius of the generating circle, and MN the line upon which it rolls. The curve is a prolate cycloid when a > b; in this case the wave presents an unbroken surface; but when a < b the curve is a curtate cycloid, and the wave breaks.
2. From the values of dy and dx given in the preceding article we have
ds/d?=√[(dy/d?)2+(dx/d?)2]
=√(a2+b2-2abcos?)=q
Hence ds=qd?
M is in fact the instantaneous centre of rotation of the generating circle. If a denotes the angular velocity of this circle about M, we have for the velocity with which the centre C moves
V=a?
And, denoting the length of a wave from crest to crest or from hollow to hollow by ?, we have
?=2?a
Whence
V=??/2?;
that is, for a given value of a, the velocity of a wave is directly proportional to its length.
It has been ascertained by observation that a deep-sea wave 200 feet in length has a velocity of 19 knots per hour, and that a wave 400 feet in length has a velocity of 27 knots per hour.
The Resistance of Ships.
3. The following method of computing the resistance of ships is due to Professor W. J. M. Rankine, and although more recent investigators have advanced our knowledge of the subject, and in fact have furnished a somewhat different formula for this purpose, the method of Professor Rankine is still in use, and has been far too influential in all recent investigations to be ignored.
4. "A stream line is the line, whether straight or curved, that is traced by a particle in a current of fluid." Let us suppose a body to be constructed all of whose longitudinal sections are stream-lines, and suppose this body to be completely submerged in a frictionless fluid. It is not difficult to show that this body when once set in motion will move on indefinitely without loss of velocity. In case, however, the body is only partly submerged, its motion will tend to create waves which represent a loss of energy, and a small expenditure of force will be necessary to maintain its velocity. If we further suppose the fluid to be frictional and slightly viscous like water, an additional expenditure of force will be necessary to overcome the friction of the particles of fluid on the surface of the body, and to compensate for the loss of energy represented by the eddies caused by the adhesion of the fluid.
Of these three sources of resistance, friction is by far the most important. In fact, in certain well constructed ships, it has been ascertained that other sources of resistance may be neglected without material error. If, however, the curves forming the water-lines of a ship are not continuous and well proportioned, the loss of energy expended in making waves may be very great.
5. The first theory advanced for the construction of ship's waterlines from mechanical principles was that of Mr. J. Scott Russell. His plan was to divide the water-line into three parts, called the entrance, the middle body, and the run. The entrance consisted of two sinusoids; the middle body of two straight lines parallel to the keel, and the run .was composed of two symmetrical prolate cycloids. The length of the run was two-thirds the entrance, and it was assumed that there must be a fixed proportion between the length of the entrance and run, and that of a wave whose velocity equaled that which the ship was expected to attain.
There is no reason to suppose that these are lines of least resistance. In fact there are many other stream-lines, differing essentially from these, which would doubtless serve equally well. The influence of this theory has however been of considerable service to the science of naval architecture.
6. The determination of the engine-power necessary to drive a ship constructed with trochoidal lines suggested the following investigation. The preliminary statement is given in the words of Professor Rankine.
"Conceive that the trough between two consecutive crests of the trochoidal surface of a series of waves is occupied, for a breadth which may be denoted by z, by a solid body with a trochoidal surface, exactly fitting the wave-surface; that the solid body moves forward with a uniform velocity equal to that of the propagation of the waves, so as to continue always to fit the wave-surface; and that there is friction between the solid and the contiguous liquid particles, according to the law which experiment has shown to be at least approximately true, viz., varying as the surface of contact, and as the square of the velocity of sliding.
"Conceive, further, that each particle of the liquid has that pressure applied to it which is required in order to keep its motion sensibly the same as if there were no friction; the solid body must of course be urged forwards by a pressure equal and opposite to the resultant of all the before-mentioned pressures.
"The action, amongst the liquid particles, of pressures sufficient to overcome the friction will .disturb to a certain extent the motions of the liquid particles, and the figures of the surfaces of uniform pressure; but it will be assumed that those disturbances are small enough to be neglected, for the purposes of the present inquiry.
The smallness of the pressures producing such disturbances, and consequently the smallness of those disturbances themselves, may be inferred from the fact, that the friction of a current of water over a surface of painted iron of given area.is equal to the weight of a layer of water covering the same area, and of a thickness which is only about 0.0036 of the height due to the velocity of the current."
7. That is, denoting by P the frictional resistance, by p the weight of a cubic foot of water, and by f the coefficient of friction, we have
P=pf/2g v2=pf/2g(ds/dt)2
This resistance acts along the surface of the wave, and is therefore only a component of the force to be overcome in moving the ship. Denoting by so the inclination of the wave-surface to the horizon, by zds an element of this surface, and by R the required resistance, we have
Horizontal resistance = Psec?;
and, since the resistance of friction is directly proportional to the area,
R=pf/2g?(ds/dt)2sec?zds
in which
ds/dt=q?, sec?=q?/a?=q/a, and ds=qs?:
∴ R=fpza2/2ga ?2?0 q4d?=fpza2/2ga ?2?0 (a2+b2-2abcos?)2d?
And, taking out the factor a4,
R = fpza3?2/2g ?2?0 [1+(b2/a2)-2(b/a)cos?]2d?
Expanding, we obtain the integrals
?2?0 (1+b2/a2)2d? = (1+b2/a2)22?, ?2?0 cos?d? = 0,
and 4(b2/a2) ?2?0 cos2?d? = 16(b2/a2) ??/20 cos2 ?d? = 4(b2/a2)?
Whence R=fpza3?2/2g[1+4(b2/a2)+b4/a4)2? or, putting b/a=sinß, 2?a=?= a wave length, and V=a?,
R = (fpV2/2g)?z(1+4sin2ß+sin4ß)
In applying this formula ? is taken as the length of the ship on the plane of flotation and z as the mean immersed girth. To compute the value of (1+4sin2ß+sin4ß), the sine of greatest obliquity is determined by measurement in each water-line, and the mean of the squares of these sines is substituted for sin2ß and the mean of the fourth powers for sin4ß.
The factor ?z(1+4sin2ß+sin4ß) is called the augmented surface and is denoted by S; hence
R=(fpV2/2g)S
The method of applying this formula to the determination of the probable engine-power necessary to drive a ship at a given speed is explained in Wilson's Ship-Building, pp. 125, 126 and 127. This formula has given remarkably correct results in certain cases of ships whose lines were not trochoidal. When the lines of a ship are fine, the term sin4ß may be omitted in the expression for the augmented surface.