Jl-t 



OIJVKK AM) DUK.MAN 



[chap. 8 



pipe or a banjo string. There are a fimdamental mode and an infinity of har- 

 monies or higher modes. The great difference, however, is that, because of the 

 pro])agating nature of the waves in the surface wave guide of the earth, a normal 

 mode is not represented by a single ])eriod as in the case of the banjo string, but 

 rather by a continuous set of allowed values of velocity as a function of ])eriod. 

 In practice, only the fundamental and first few higher modes are important, 

 largely because of greater attenuation with distance of the shorter period 

 waves which, in general, are associated with the higher modes. 



For a simple model, consisting, for example, of a single solid layer overlying a 

 sub-stratum, it is easy to describe and predict the general configuration of the 

 normal-mode dispersion curves. As a guide to discussion of more complex 

 models, let us consider this simple case first in some detail. In the case of Love 

 waves, as illustrated in Fig. 2, the phase-velocity curves for all modes approach 

 the shear velocity in the upper layer as the wave period approaches zero, and 

 the shear velocity in the sub-stratum as the periods become long. The funda- 

 mental mode includes the entire spectrum from zero to infinity, but the higher 

 modes have long-jjeriod cut-offs at successively shorter periods as the mode 

 number increases. The corresponding group-velocity curves have similar limits, 



10 



20 



30 40 



T (sec) 



50 



60 



70 



Fig. 3. Theoretical group velocity versus period for Rayleigh waves and higher modes on 

 a model of the continental crvist and mantle. Phase -velocity curves, not shown, have 

 the same general relations to their respective group-velocity curves as in the case of 

 Love waves. The curves are all based on roots of the same period equation. However, 

 as a point of difference with the Love-wave curves, the asymptotic velocity of the 

 fundamental or Rayleigh mode as the period increases is Fi}.^ = 0.9 194^32, whereas 

 higher or shear modes reach ^2 at their cut-off period. The higher-mode curves are not 

 complete at their short- and long-period ends because of computing difficulties. (See 

 Case 8008 of Oliver, Kovach and Dorman (1961) for further data on this theoretical 

 model.) 



