December 8, 192 1] 



NATURE 



46; 



Stanford's New Map of the Pacific Ocean. Size 

 30x22^ in. (London: E. Stanford, Ltd., n.d.) 

 Coloured sheet, 45. ; mounted to fold in case, 

 65. 6d. 

 A MAP of the Pacific Ocean showing the distribu- 

 tion of political interests should prove useful at 

 a time when international problems centre largely 

 on that ocean. Messrs. Stanford have produced 

 an excellent map which has the merit of being 

 on Mollweide's equal area projection, and showing 

 the main features of relief by layer colouring. 

 Spheres of interest are shown by distinctive bands 

 of colour and the principal submarine cables and 

 wireless stations are clearly marked. A few cor- 

 rections might be made in a later edition. The 

 Banks and Torres Islands are within the joint 

 Anglo-French administration of the New Hebrides 

 and not under the High Commissioner of the 

 Western Pacific. The Chesterfield Islets, although 

 of very slight importance, form part of the French 

 colony of New Caledonia. The Portuguese foot- 

 hold in Eastern Timor should be marked. The 

 small group of the Tasman Islands, north of the 

 Solomons, used to be German territon.- and should 

 presumably now be included within the area of the 

 Australian mandate. But these are all minor 

 points which do not affect the general usefulness 

 of this well-printed map. It is accompanied by 

 a sixteen-page pamphlet of statistical matter. 



Elementary Principles of Continuous-Current 

 Armature Winding. By F. M. Denton. (Pit- 

 man's Technical Primers.) Pp. x-fio2. 

 (London: Sir Isaac Pitman and Sons, Ltd., 

 IQ2I.) 25. 6d. net. 

 The greater part of this compact little treatise 

 has appeared in the columns of a contemporary. 

 The principles governing the arrangement and 

 proportioning of armature windings are worked 

 out by a simple progressive treatment with very 

 little mathematics, and some useful rules and com- 

 parative data are given. 



Letters to the Editor. 



[The Editor does not hold himself responsible for 

 opinions expressed by his correspondents. Neither 

 can he undertake to return, or to correspond with 

 the writers of, rejected manuscripts intended for 

 this or any other part of Nature. No notice is 

 taken of anonymous communications.] 



Propagation of Waves in an isotropic Solid. 



The velocity of waves in an isotropic solid is pro- 

 portional to the square root of the coefficient of that 

 kind of elasticity called into action by the displace- 

 ments which constitute the wave-motion. These dis- 

 placements may involve either the simple rigidity (n), 

 the volume elasticity (k), or more generally a com- 

 bination of both. The effective elasticity will depend 

 partly on the nature of the initial disturbance, and 

 partly on the boundary conditions 



If torsional vibrations are propagated along a rod, 

 the wave-velocity varies as s/n; or if the pressure 

 over the whole surface of a solid sphere varies simul- 

 taneously, a wave travels inwards with a velocity pro- 

 portional to \/k. If the wave-length is great com- 

 XO. 2719, VOL. 108] 



pared to the radius of the sphere, the latter is merely 

 compressed and dilated as a whole ; on the other hand', 

 when wave-length/radius is small, the amplitude of 

 the vibration increases as the wave travels inwards 

 on account of the same energy being embodied in a 

 shell of smaller mass. 



At the centre of the sphere, if the limits of elas- 

 ticity did not operate, the amplitude would be infinite, 

 but in any real case disruption of some sort would 

 occur. 



Analogies may be found in other cases where a 

 constant energy content is confined in a continually 

 decreasing mass, as, for instance, in the "cracking " 

 of a whip, or in the drop thrown up in the centre of a 

 circular basin of fluid after a small wave has been 

 initiated round the circumference. 



When longitudinal waves are propagated along a 

 rod of which the transverse dimension (A) is small 

 compared t o the wave-length (A), the velocity is propor- 

 tional to -/Young's modulus (E), which contains both 

 n and k. 



Such longitudinal waves always occasion lateral 

 motions of the particles proportional to Poisson's ratio 

 (/x) for the substance, but these are of little importance 

 as long as A/A is great. If, however, A/A is great, 

 the lateral motion at the surface of the rod might, 

 if the elasticity were still represented by Young's 

 modulus, become greater than the longitudinal ampli- 

 tude of the original wave ; in fact, the two amplitudes 

 would be equal when A/A = ,u. 



In reality, however, the surface deformation which 

 must accompany the longitudinal wave exerts a 

 normal force on the interior parts, and thus reduces 

 the lateral motion to a quantity which decreases ex- 

 ponentially from the surface inwards. 



The conditions for the elasticity defined by Young's 

 modulus are that the deformations shall produce no 

 normal force at the free surface, while for the elas- 

 ticity which governs the same class of displacements 

 in the far interior the conditions are that there shall 

 be no normal motion. 



If a force P parallel to Z acts on a unit cube and 

 causes a contraction, a, then if no forces act parallel 

 to X and y there is a lateral extension ua in both direc- 

 tions. If now keeping the stress P constant, a force 

 E//0 is caused to act parallel to x and y, the lateral 

 dimensions are restored to their constrained magni- 

 tude, and the longitudinal strain is decreased by 

 2/i'a. Thus if the coefficient of the interior elasticity 

 is denoted by B, B(i — 2u*) = E, or B=^E/i — 2/1*. 



The expression for E in terms of n and k is 

 gnKJ^K + n and for fi, 3K— 2n/2(3K-l-n). In Fig. i 

 curves are given showing the values of E and B as 

 multiples of n in terms of ,u. At the surface of a solid 

 the wave-velocity is always proportional to v'E, but 

 gradually increases to a/B in the interior. Assuming 



_y 

 that the la.ieral displacements vary as e e (y normal 

 to the free surface), it will be found that when A /A 



. . ^ E 



IS great c= . 



2jr n ' 



Thus at the depth of one wave-length the coefficient 

 of elasticity nearly approaches B. If a plane wave 

 surface starts from OY (Fig. 2) in the direction of Z, 

 its surface will, as it progresses, assume the form 

 sketched at Z,Y,. Earthquake waves must be affected 

 by the change from B in the interior to E at the sur- 

 face, and if a plane wave of compression were vertical 

 at its source it would afterwards cut the surface at a 

 more or less acute angle. 



In iron or glass \/ B> V E by something like 10 per 

 cent., and it would be Interesting to examine the 

 velocity of very short waves in rods of such materials. 



