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Proceedings of the Poyal Society of Edinburgh. [Sess. 
by Professor Lamb in his investigation of Ship- Waves ( Hydrodynamics , 
§ 253), but applied to the Fourier trains which constitute any wave- 
disturbance. When this view is accepted, the difficulties raised by Lord 
Kelvin are removed, as it is consistent with the dynamical theory given by 
Osborne Reynolds and Lord Rayleigh and with calculated results shown in 
Lord Kelvin’s diagrams, which are reproduced below for the sake of illus- 
tration. The whole investigation may be useful in drawing attention to 
the manner in which group-velocity is concerned in the modification of an 
initially regular group of waves, or of any disturbance initially confined to 
a finite portion of a dispersive medium ; and in showing thereby that the 
idea of group-velocity contains the explanation of the modus operandi of 
dispersion. 
§ 7. Following Lord Kelvin’s paper above referred to, let us consider, 
as being fundamental, the case of an infinitely intense disturbance confined 
to a point of the dispersive medium which is taken as the origin of co- 
ordinates. Let V be the wave-velocity of an infinite train of waves of 
period 27 r/kV and wave-length X ; then we have k = 27 r/A, and, since the 
medium is such that the wave-velocity varies with the wave-length, we 
have also V =f(k). The period and the velocity corresponding to A are 
each functions of k ; and for convenience in what follows we refer to period 
F(&) simply as period k. According to Fourier’s theorem, the displace- 
ment f at point x and time t is given by the equation 
+ oo 
£= — I dk cos k{x- W} . . . . (1). 
27 rj 
-oo 
This means that the initial disturbance may be regarded as due to the 
superposition of the effects of an infinite number of trains of regular waves 
of equal amplitudes, all of which agree in phase at the origin. Not only is 
the total number of trains infinite, but the number of trains whose wave- 
periods lie between k and k + 3k is also infinite, no matter how small Sk 
may be. At the origin the displacement is sensible ; at all other points of 
the medium, on account of disagreement in phase, the various trains 
interfere and produce zero displacement. At any time after the commence- 
ment of motion, the effect at any point is got by summing the effects due 
to all the trains, supposing each to have travelled in the interval a distance 
corresponding to its wave-velocity. By applying this we can get an idea 
of the manner in which the initial disturbance is propagated, reasoning as 
follows. 
§ 8. Since all the trains initially agree in phase at phase zero, and since 
each train moves with a velocity corresponding to its wave-length, it is 
