Proceedings - London Mathematical Society Volume 20 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1889 Excerpt: ...(fc2-l2/2) sinh rzx sinh sz...(48), y =--eipteif" 2ifrs sinh szx cosh rz + if(k2--2f) sinh rzx coshsz... (49). If we now introduce the assumption thut the plate is thin, we find, by expanding the hyperbolic functions in (47), 4f (/--#) l+iz (Ir-h1) = (/ '-2f)-. The first approximation gives k2 = 0, signifying that the notes are infinitely grave. The second approximation is = / (-') (50), or, in terms of p, m, n, p, = _mn_ YX r m+n 'if) K' Again, if we drop out a common factor (k2rzx), (48), (49) take the forms a-fz e' eif y = ife'"' et" (52). Hence a =--zdy/dx, signifying that to this order of approximation every line originally perpendicular to the middle surface retains its straightness and perpendicularity during the vibrations. The third approximation to the value of A;2 from (47) gives, mn 4/V ( r 4, 71), KO, so that, when the thickness is increased beyond a certain point, the rise of pitch begins to be less rapid than according to the second approximation (51). When zx is infinitely great, we get, from (38) or (47), 4/Vs=(fc2-2/2)2 (54), the equation considered in the paper already referred to upon surfacewaves. From (43), (53) we learn that p2 is positive, or the equilibrium is stable, so long as m is positive. On the other hnnd, it was proved by Green many years ago that a solid body would be unstable if m were less than n, m--n being in fact the dilatation modulus. The reconciliation of these apparently contradictory results depends upon This is upon the supposition that r and s aru real, equation would havo no definite limit. In the contrary case the principles similar to those recently applied by Sir W. Thomson, to show that a solid, every part of the boundary of which is held fixed, is st;ibk', so long ...