420 W A V E equation of continuity, we may proceed to further tests and restric tions of it. Substitution leads to so that Y AS n * + Bg **, where A and B are arbitrary constants. (a) If the deptli of the water be unlimited, the value of A must be zero, for otherwise we should be dealing with disturbances which increase, without limit, as we go farther down. Hence, in this case, a particular integral of the equation, corresponding to a disturbance which can exist by itself, is 0=Bg-#cos (mt -nx). We will now avail ourselves of the supposition under which, as we have seen, disturbances are necessarily superposable, i.e., assume terms in B 2 to be negligible. The ordinary kinetic equation (HYDROMECHANICS) then becomes = gy + mBg - "" sin (mt - nx) . If we differentiate this expression with regard to t, and apply the result to the surface only, where v is constant, and remember that dy fdd> . i -s- = I - - , we nave simply dt dyj = -ng + m 2 , which is the condition that no water crosses the bounding surface. This determines n, without ambiguity, when m is given, and thus gives a relation between the period of a wave and its length, or between the period or the length and the speed of propagation. For we may write 1-iT mt - ii:c = (vt - x) . where A is the wave-length, and v the speed ; so that Thus we have and the longer waves move faster, even when the vertical displace ment is small in both. This is quite different from the result for sound-waves. The components of the velocity of the particle of water whose mean position is x, y are I -r- ) Bug ~"y sin (mt - nx) , parallel to x . dx J
- ind
- B/tg ~"* cos (mt - nx) , parallel to y . Hence the path is a circle whose radius is B-g-*, oi-B -g-*. m g At the surface this is Bm/<7, as in fact we see at once by the equa tion for the pressure, which gives for the form of the surface C = gy + m B sin (mt - nx) . Each surface-particle is at the highest point of its circular path, and moving forwards, when the crest of the wave passes it. When the trough passes, it is at the lowest point of its circle and moving backwards. The radii of the circles diminish in geometrical pro gression at depths increasing in arithmetical progression. The factor is g- ^= g 2n J A ) S o that at a depth of one wave-length only the disturbance is reduced to g~ 2ir or about 1/535 of its surface- value. From the investigation above we see that Atlantic rollers, of a wave-length of (say) 300 feet, travel at the rate of about 40 feet per second, or 27 miles an hour. But, even if they be of 40 feet height from trough to crest (which is probably an exaggerated estimate), the utmost disturbance of a water particle at a depth of 300 feet is not quite half an inch from its mean position. This shows, in a very striking manner, what a mere surface-effect is in this way due to winds, and how the depths of the ocean are practically undisturbed by such causes. This investigation has been carried to a second, and even to a third, approximation by Stokes, with the result that the form of a section of the surface is no longer the curve of sines, in which the crests and troughs are equal. The crests are steeper and higher, and the troughs wider and shallower, than the first approximation shows. Also the forward horizontal motion of each particle under the crest is no longer quite compensated for by its backward motion under the trough, so that what sailors call the " heave of the sea " is explained. The water is per- i manently displaced forwards by each succeeding wave. But this effect, like the whole disturbance, is greatest in the surface-layer and diminishes rapidly for each lower layer. The third approximation shows that the speed of the waves is greater than that above assigned, by a term depending on the square of the ratio of the height to the length of a wave. (6) When the depth of the water is limited, we cannot make the simplification adopted in the last investigation. If h be the depth of the water, our condition is that the vertical motion vanishes at that depth, and the relation between m and n is now m* = ng(t" h - g-"*)/(g A + g""*). If h be regarded as infinite this gives as before m- = tig . If, on the other hand, h be small compared with the wave-length, the equation approximates to in^ = n" <jfi , or i?*=gh; and we have the formula for long waves again. Thus the expres sion above includes both extremes, though, so far as long waves go, it limits them to harmonic forms of section. The surface-section is still the curve of sines, but the paths of the individual particles are now ellipses whose major axes are horizontal. Both axes decrease with great rapidity for particles considered at gradually increasing depths ; but the minor axes diminish faster than the major, so that the particles at the bottom oscillate in horizontal lines. (8) Ripples. Stokes in 1848 pointed out that the surface-tension of a liquid should be taken account of in finding the pressure at the free surface, but this seems not to have been done till 1871, when W. Thomson discussed its consequences. If T be the surface-tension, and r the radius of curvature of the (cylindrical) surface, in the case of oscillatory waves, the pressure at the free surface must be considered as differing from that in the air by the quantity T/r (CAPILLARY ACTION). As T/gp is usually a small quantity, this, term will be negligible unless r is very small. If the waves be oscillatory, this means that their lengths must be very short, so that the depth of the fluid may be treated as infinite in comparison. The curvature is practically d?r)/dx 2 , because TJ/A is small ; so that the term Td?rf/dx s must be introduced into the kinetic equa tion along with p. The result is that 3 T m- = nq H -- - , mp , (} 2* T or p2_|_ + _ _. Zir A p Thus the speed is, in all cases, increased by the surface-tension ; and the more so the shorter is the wave-length. Hence, as the speed increases indefinitely with increase of wave-length when gravity alone acts, and also increases indefinitely the shorter the wave when surface-tension alone acts, there must be a minimum speed, for some definite wave-length, when both causes are at work. It is easily seen that v- is a minimum when and that the corresponding value of v* is _ In the case of water the value of A is about OS inch, and v is 76 in feet-seconds, nearly. This slowest-moving oscillatory wave may therefore be regarded as the limit between waves proper and ripples. That ripples run faster the shorter they are is easily seen by watching the apparently rigid pattern of them which precedes a body moving uniformly through still water. The more rapid the motion the closer do the ripple-ridges approach one another. Excellent examples of ripples are produced by applying the stem of a vibrating tuning-fork to one side of a large rectangular box full of liquid. From the pitch of the note, and the wave-length of the ripples, we can make (by the use of the above formula) an approximate determination of surface-tension, a quantity
somewhat difficult to measure by statical processes.