[Reprinted from the PROCEEDINGS OF THE ROYAL Socrrty, A. Vol. 81] 
The Propagation of Groups of Waves in Dispersive Media, with 
Application to Waves on Water produced by a Travelling 
Disturbance. 
By T. H. Havetocx, M.A., D.Sc., Fellow of St. John’s College, Cambridge ; 
Lecturer in Applied Mathematics, Armstrong College, Newcastle-on-Tyne. 
(Communicated by Prof. J. Larmor, Sec. R.S. Received August 26,—Read 
November 19, 1908.) 
CONTENTS. 
SECTION PAGE 
Py ein troductiontd Steves cers acco tc: hke reer teee aan kk ae! 398 
2a DetinitionjofisimplelGroupyessseeeeee eee 399 
3. The Fourier Integral regarded as a Collection of Groups ...... 400 
(a) Damped Harmonic Wave-train .........scccccccceceseeseeeee 401 
(6) Interrupted Simple Wave-train ..........cesccscceeescceeeeee 402 
4. Features of the Integrals Involved ....c.....cccccccccsesessccceceeeece 403 
5. Initial Line Displacement on Deep Water.............ce0ecceeseeeee 404 
6. Initial Displacement of Finite Breadth .............cccceeeeeeeeeeeee 406 
7. Limited Train of Simple Oscillations .............ccsessccsceeceseeses 407 
8. Initial Impulse on Deep Water .........cccseccccecececeeesesscceceeee 410 
9. Moving Line Impulse on Deep Water ...........ccccseseccceeseeeee 41] 
TOS CapillanygS urtacemWiav.eslaseeneeces eae oe ener 413 
11. Water Waves due to Gravity and Capillarity ............es00000-. 415 
12. Surface Waves in Two Dimensions...........0.....cecccssseeseeeeeeee 415 
13. Point Impulse Travelling over Deep Water ............sss0e0c0ee- 417 
(a) The Transverse Wave System ..........s:sssssssesseeseseeeees 418 
(6) The Diverging Wave System ...............ccceceseeeeeeeesee 420 
(@) WNe TING GF CHEN csaccococooonsnoscescHsboSoNcovsocsbadbtdoooaDss 421 
14. Point Impulse for Different Media.................ccccseeseeeceseeees 424 
15. Point Impulse Moving on Water of Finite Depth ............... 426 
§ 1. Introduction. 
The object of this paper is to illustrate the main features of wave propa- 
gation in dispersive media. In the case of surface waves on deep water it 
has been remarked that the earlier investigators considered the more difficult 
problem of the propagation of an arbitrary initial disturbance as expressed 
by a Fourier integral, ignoring the simpler theory developed subsequently by 
considering the propagation of a single element of their integrals, namely, 
an unending train of simple harmonic waves. The point of view on which 
stress 1s laid here consists of a return to the Fourier integral, with the idea 
that the element of disturbance is not a simple harmonic wave-train, but a 
simple group, an aggregate of simple wave-trains clustering around a given 
central period. In many cases it is then possible to select from the integral 
