2 PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



things which we are obliged to assume as a basis of calculation, it is hoped that the 

 solution of the problems here considered may not be altogether irrelevant. 



It is found that the surface disturbance prodxiced by a single impulse of short 

 duration may be analysed roughly into two parts, which we may distinguish as the 

 "minor tremor" and the "main shock," respectively. The minor tremor sets in 

 at any place, with some abruptness, after an interval equal to the time which 

 a wave of longitudinal displacement would take to traverse the distance from the 

 source. Except for certain marked features at the inception, and again (to a lesser 

 extent) at an epoch corresponding to that of direct arrival of transversal waves, it 

 may be described, in general terms, as consisting of a long undulation leading up to 

 the main shock, and dying out gradually after this has passed. Its time-scale is 

 more and more protracted, and its amplitude is more and more diminished, the 

 greater the distance from the source. The main shock, on the other hand, is pro- 

 pagated as a solitary wave (with one maximum and one minimum, in both the 

 horizontal and vertical displacements) ; its time-scale is constant ; and its amplitude 

 diminishes only in accordance with the usual law of annular divergence, so that its 

 total energy is maintained undiminished. Its velocity is that of free Rayleigh waves, 

 and is accordingly somewhat less than that of waves of transversal displacement in 

 an unlimited medium.* 



The . paper includes a number of subsidiary results. It is convenient to begin by 

 attacking the problems in their two-dimensional form, calculating (for instance) the 

 effect of a pressure applied uniformly along a line of the surface. The interpretation 

 of the results is then comparatively simple, and it is found that a good deal of 

 the analysis can be utilized afterwards for the three-dimensional cases. Again, the 

 investigation of the effects of a simple-harmonic source of disturbance is a natural 

 preliminary to that of a source varying according to an arbitrary law. 



Incidentally, new solutions are given of the well-known problems where a periodic 

 force acts transversely to a line, or at a point, in an unlimited solid. These serve, to 

 some extent, as tests of the analytical method, which presents some features of 

 intricacy. 



2. A few preliminary formula? and conventions as to notation may be put in 

 evidence at the outset, for convenience of reference. 



The usual notation of BESSEL'S Functions " of the first kind " is naturally adhered to ; 

 thus we write : 



2 f^" 

 JG () = - cos ( cos &>) dot (1). 



77 JQ 



* Compare the concluding passage of Lord RAYLEIGH'S paper : 



" It is not improbable that the surface-waves here investigated play an important part in earthquakes, 

 and in the collision of elastic solids. Diverging in two dimensions only, they must acquire at a great 

 distance from the source a continually increasing preponderance." 



The calculations indicate that the preponderance is much greater than would be inferred from a mere 

 comparison of the ordinary laws of two-dimensional and three-dimensional divergence. 



