Horace Lamb

“ Thus in the wave method the actual motion of the string is considered as the resultant of two wave motions, neither of which is of itself, and without the other, consistent with the condition that the ends of the string are fixed. Each of the wave motions is periodic with a wave-length equal to twice the distance between the fixed points, and the one set of waves is the reverse of the other in respect of displacement and velocity and direction of propagation ”

“ The longer waves travel faster than the shorter, so that each wave is continually being drawn out in length, and its velocity of propagation therefore continually increases as it advances. If we fix our attention on a particular point of the surface, the level there will rise and fall with increasing rapidity and increasing amplitude. ”

“ The velocity c being constant there is no tangential acceleration, and the tension P is accordingly uniform. The resultant of the tensions on the two ends of an element δs is Pδs/R, in the direction of the normal, where R denotes the radius of curvature. This will be exactly sufficient to produce the normal acceleration c2R in the mass ρδς, provided c2=P/ρ. Under this condition the tube, which now exerts no pressure on the string, may be abolished, and we have a free stationary wave on a moving string. ”

“ The result is in general that in the final stages the surface will be marked by a series of groups of waves of diminishing amplitude separated by bands of comparatively smooth water. The fact that the wave-velocity of a simple-harmonic train varies with the wave-length has an analogy in optics, in the propagation of light in a dispersive medium. In both cases we have a contrast with the simpler phenomena of waves on a tense string or of light waves m vacuo, and the notion of " group-velocity, " as distinguished from wave-velocity, comes to be important. ”

“ Fourier’s theorem asserts that any periodic function of a single variable period p, which does not become infinite at any phase, can be expanded in the form of a series consisting of a constant term, together with a double series of terms, one set involving cosines and the other sines of multiples of the phase. ”

“ If the medium be crystalline the velocity of propagation of plane waves will depend also on the aspect of the wave-front. For any given direction of the wave-normal there are in the most general case three distinct velocities of wave-propagation, each with its own direction of particle-vibration. ”

“ If a wave of any form travelling in the positive direction meets another travelling in the opposite direction, the form of which is such that the lines joining corresponding points of the two waves are all bisected in a fixed point in the line of the string, then the point of the string corresponding to this point will remain fixed, while the two waves pass it in opposite directions. ”

“ The region occupied by the disturbance at any instant I is therefore delimited by the envelope of a family of spheres of radius cl described with the points of the original boundary as centres. One remarkable point about waves diverging in three dimensions remains to be noticed. It easily appears from (3) that the 'alue of the integral fsdt at any point P, taken over the whole time of transit of a wave, is independent of the position of P, and therefore equal to zero, as is seen by taking P at an infinite distance from the original seat of disturbance. ”

“ This is an instance of a familiar result in the theory of vibrations, viz. that in a forced oscillation of a body under a periodic force the phase is opposite to that of the force if the imposed frequency exceed that of the corresponding free vibration (see Mechanics) . ”

“ It is hardly worth while to attempt an account here of the singularities of this surface, or of the simplifications which occur for various types of crystalline symmetry, as the subject has lost much of its physical interest now that the elastic-solid theory of light is practically abandoned. ”

“ If initially we have zero velocity everywhere, but a uniform condensation So throughout a spherical space of radius a, it is found that we have ultimately a diverging wave in the form of a spherical shell of thickness 2a, and that the value of j within this shell varies from isoa/r at the anterior face to —^Soa/r at the interior face, r denoting the mean radius of the shell. ”

“ If it were not on account of the waste of time, instead of having a curve drawn by the action of the tide, and the curve afterwards acted on by the machine, the time axis of the machine itself might be driven by a clock, and the tide itself might work the second variable of the machine, but this would involve the constant presence of an expensive machine at every tidal station. ”

“ If attention is concentrated on a particular wave, this is seen to progress through the group, gradually dying out as it approaches the front, whilst its former place in the group is occupied in succession by other waves which have come for¥ard from the rear. ”

“ The mathematical test of conjugacy is that the energy of the system arising from two of the harmonics existing together is equal to the sum of the energy arising from the two harmonics taken separately. ”

“ The orbits of the particles, in this type, are accurately circular, being defined by the equations x=a- -k -^e'sink{a-ct) , y = b — k -^e'cosk (, a-ct) ,
(i) where (a, b) is the mean position of the particle, k = 2Trl; and the wave-velocity is ^ = V (g/*) = V (gX/2^) . -.
(2) xxvin 8 The lines of equal pressure, among which is included of course the surface-profile, are trochoidal curves. The extreme form of waveprofiie is the cycloid, with the cusps turned upwards. The mathe-FiG. 6. matica! ”

“ The theor) ; was further e;mean depth. There is, however, a variation of amplitude; the constancy of the energy, combined with the equation of continuity, require that the elevation 17 in any particular part of the wave should vary as b-ih-i, where 6 is the breadth of the water surface and h is the mean depth. ”

“ Fig. 4. — iSinS-, wave-length X at points to the right of O is uniform, being that proper to a wave- velocity c, viz. X = 27rcVg. The disturbance is therefore followed by a train of waves of approximately simple harmonic profile, of the length indicated. An approximate calculation shows that, e.xcept in the immediate neighbourhood of the source of disturbance, the surface-elevation is given by 2P0. er "=^1?='"!?. • • • where x is now measured from O, and Poi^fpdx) represents the integral of the disturbing surface pressure over the (infinitely small) breadth of the band on which it acts. ”