120 DYNAMICAL THEOBY OF SOUND 



of the cross-section. But even in the other extreme, when all 

 the dimensions of the body are large compared with the wave- 

 length, the circumstances may be profoundly modified by the 

 existence of a free boundary. A new type of waves, called 

 after the discoverer the " Rayleigh waves " (1885), make their 

 appearance, and under some conditions may become, from the 

 observational point of view, predominant. These are surface 

 waves in which the agitation penetrates only to a relatively 

 small depth. Their velocity is somewhat less than that of 

 the distortional waves ; thus for an incompressible solid it 

 is *9554 b, whilst on Poisson's hypothesis (cr = J) it is '91946. 

 In modern observations of the tremors due to distant earth- 

 quakes three phases of the disturbance are often recognized. 

 The first is interpreted as due to the arrival of the dilatational 

 waves, propagated directly through the substance of the earth, 

 the second as due to that of the distortional waves, also 

 propagated directly, and the third to that of the Rayleigh 

 waves, which have travelled over the surface and are therefore 

 delayed more than in proportion to the difference of wave- 

 velocity*. The latter waves as they spread over the surface 

 are less attenuated than the former, which diverge in three 

 dimensions. It has even been attempted to deduce estimates 

 of the volume-elasticity and rigidity of the materials of the 

 earth from the various wave-velocities, as inferred from the 

 seismic records f. 



45. Plexural Vibrations of a Bar. 



We proceed to the transverse jv&F 



vibrations of a bar naturally straight. 

 To avoid unnecessary complications 

 we will suppose that the bar has a 

 longitudinal plane of symmetry, and 

 that the flexure takes place parallel 

 to this plane. We will also assume 

 for the present that the total longi- 

 tudinal stress on any section is zero. 

 The resultant stress at a section there- Fig. 42. 



* E. D. Oldham, Phil. Tram. A, 1900. 



t Prof. A. E. H. Love, Phil. Trans. A, vol. ccvu., p. 215 (1908). 



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