447 
1908-9.] On Group- Velocity and Propagation of Waves 
tation of group- velocity given by Lord Rayleigh ( Scientific Papers, vol. i. 
p. 540) has been presented with greater generality by Professor Lamb, who 
gives an interesting application of the theory to the wave-system arising from 
an impulse applied at a single point of a water surface.* Three separate 
presentations of the kinematical group-velocity are given by Professor Lamb. 
The first is contained in the argument from the interference of two infinite 
trains of waves of equal amplitudes and nearly equal wave-lengths, referred 
to in § 2 above. Here the groups considered are mutually supporting, the 
shape and motion of each depending on the presence of the others, and we 
cannot regard this case of group-velocity as providing a satisfactory explana- 
tion suitable to all cases. The more general treatment of the matter given 
in the two later demonstrations entirely completes the investigation ana- 
lytically. The second and third presentations are, however, in a sense 
distinct, one giving a phase relation between the Fourier trains into which 
any group of waves can be analysed, the other referring to the actual wave- 
disturbance ; and the question arises as to whether the demonstrations 
cannot be correlated with each other in such a way as to show that the same 
fundamental idea underlies all three. In neither of the two later demon- 
strations is the “ group ” a regular procession of waves such as that required 
by Osborne Reynolds and Lord Rayleigh in their dynamical theorems, and 
in neither is it clear that a “ group ” of waves has any definite group-velocity ; 
that is, if we understand by “ group ” “ a long succession of waves in which 
the distance between successive crests and the amplitude vary very slightly.” j- 
The two accounts of the matter agree if we understand “ group-velocity ” 
to mean the velocity of a point which moves so as to always coincide 
with the point of the wave-system where a particular wave-length singled 
out for observation is to be found. This is indeed the definition of group- 
velocity adopted by Professor Lamb ; but fuller investigation seems to be 
necessary to make clear the fundamental relation between the arguments 
used by Stokes, Osborne Reynolds, and Lord Rayleigh, and those given by 
Professor Lamb. 
§ 6. The object of this paper is to present the idea of group- velocity in 
the way in which it is used by Lord Kelvin in his paper of 1887, “ On the 
Waves produced by a Single Impulse in Water of any Depth, or in a 
Dispersive Medium,” J but with greater fulness, in order to show that Lord 
Kelvin’s paper provides the explanation that is required. The group- 
velocity of this paper is essentially the principle of “ stationary phase ” used 
* Lamb, Hydrodynamics , §§ 234-238. 
t Lamb, “On Group-Velocity,” Proc. Lond. Math. Soc., vol. i., 1903-4. 
I Sir William Thomson, Phil. Mag., March 1887. 
