Theory of Sound
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. 1877 Excerpt: ...of equilibrium. If no frictional forces act, the motion is necessarily resolvable into normal vibrations. Assume y = ol sin mx + /3 cos mx) cos (mat--e) (1). The conditions at the ends are that when x = 0, My + /iy= Tl dv (2). when x = l, Mij + fiy =-Tj which give a _/S tanml--a_fi--Mam 0-atimrnl + (3--mT W' two equations, sufficient to determine m, and the ratio of /3 to a. Eliminating the latter ratio, we find tan ml = t (4), Equation (3) has an infinite number of roots, which may be found by writing tan 6 for v, so that tan ml = tan 26, and the result of adding together all the corresponding particular solutions, each with its two arbitrary constants a and e, is necessarily the most general solution of which the problem is capable, and is therefore adequate to represent the motion due to an arbitrary initial distribution of displacement and velocity. We infer that any function of x may be expanded between x = 0 and x = I in a series of terms (i/jSinm + cosro) + ff2 (v% sin rajc + cos m%x) + (5), mv mv &c. being the roots of (3) and vt, vv &c.the corresponding values of v. The quantities $v ft, &c. are the normal co-ordinates of the system. From the symmetry of the system it follows that in each normal vibration the value of y is numerically the same at points equally distant from the middle of the string, for example, at the two ends, where x = 0 and x = I. Hence vt sin mj, + cos mtl = + 1, as may be proved also from (4). The kinetic energy T of the whole motion is made up of the energy of the string, and that of the masses M. Thus T= p 2 j (v sin mx + cos mx)f dx Jo + iMf1 + f1+...Y + iM1(v1smm1l + cosmir)+...2. But by the characteristic property of normal co-ordinates, terms containing their products cannot be really present in the expressi...
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- PublisherPeter Smith Pub
- Publication date1976
- ISBN 10 0844630284
- ISBN 13 9780844630281
- BindingTextbook Binding
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