399 Dr. T. H. Havelock. 
the few simple groups that are important, and hence to isolate the chief 
regular features, if any, in the phenomena. 
In certain of the following sections well-known results appear; the aim 
has been to develop these from the present point of view, and so illustrate 
the dependence of the phenomena upon the character of the velocity function. 
In the other sections it is hoped that progress has been made in the theory 
of the propagation of an arbitrary initial group of waves, and also of the 
character of the wave pattern diverging from a point impulse travelling on 
the surface. 
§ 2. Definition of Simple Group. 
We have to consider the transmission of disturbances in a medium for 
which the velocity of propagation of homogeneous simple harmonic wave- 
trains is a definite function of the wave-length. The kinematically simplest 
group of waves is composed of only two simple trains, of wave-lengths A, 0’, 
differing by an infinitesimal amount dA; then with the usual approximation 
we have for the combined effect 
9 
y = Acos s @—Vi)+A cos 7 (e—V't) 
dn 2a 
= 2A cos a («— Ut) cos =e (w— Vb), (1) 
ner Gey". (2) 
dn ° 
The expression (1) may be regarded as representing at any instant a train 
of wave-length A, whose amplitude varies slowly with « according to the first 
cosine factor. Thus it does not represent a form which moves forward 
unchanged ; but it has a certain periodic quality, for the form at any given 
instant is repeated after equal intervals of time 4/(V—U), being displaced 
forward through equal distances X\U/(V—U). The ratio of these quantities, 
namely U, is called the group-velocity. It has also the following significance : 
in the neighbourhood of an observer travelling with velocity U the disturb- 
ance continues to be approximately a train of simple harmonic waves of 
length 2. 
The most general simple, or elementary, group may be defined in the 
following manner. Let the central form bea simple harmonic wave of length 
2qr/xo, and let the other members be similar waves whose amplitude, wave- 
length, and velocity differ but slightly from the central type; then, with 
similar approximation, we have 
y = XA cos {« (z—Vt)+ a} 
= ZA, cos {xo (#—Vot)-+ (w@—Uot) Senta}. (3) 
