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Encyclopedia Britannica - Main :: LUP-MAL |
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MACHINES . A more general form of the problem of harmonic analysis presents itself in astronomy, in the theory of the tides, and in various magnetic and meteorological investigations. It may happen, for instance, that a variable quantity f(t) is known theoretically to be of the form f(t)=Ao+A1cos nit+Bisin nit +A2cos n2t+B2sin nit+ . . . (2) 957 (3) In a " normal mode " (4) (5) where the periods 22r/n1, 2w/n2, . . . of the various simple-harmonic constituents are alzeady known with sufficient accuracy, although they may have no very simple relations to one another. The problem of determining the most probable values of the constants Ao, A1, B1, A2, B2, . . . by means of a series of recorded values of the function f(t) is then in principle a fairly simple one, although the actual numerical work may be laborious (see TIDE). A much more difficult and delicate question arises when, as in various questions of meteorology and terrestrial magnetism, the periods 2a/ni, 2g/n2, . . . are themselves unknown to begin with, or are at most conjectural. Thus, it may be desired to ascertain whether the magnetic declination contains a periodic element
axis
series of the type (2), can be effected in an infinite variety of ways. Plausible inferences can, however, be drawn
The question so far chiefly considered has been that of the representation of an arbitrary function of the time in terms of functions of a special
illustration
string
PayTax2 where T is the tension, and p the line-density. of vibration y will vary as ei' ", so that ax +key=o, k2 =n2p/T. where If p, and therefore k, is constant, the solution of (4) subject to the condition that y=o for x=o and x=l is y = B sin kx (6) provided kl =sir, [s =I, 2, 3, ...]. (7) This determines the various normal modes of free vibration, the corresponding periods (grin) being given by (5) and (7). By analogy with the theory of the free vibrations of a system of finite freedom it is inferred that the most general free motions of the string
f(x) = Bisin7+B2sin2 x+B3sin3ix+... (8) So far, this is merely a restatement, in mathematical language, of an argument given in the first part of this article. The series (8) may, moreover, be arrived at otherwise, as a particular case of Fourier
y= Cu(x)eini where u is a solution of the equation d2u n2p dx2+ TT u=o (to) The condition that u(x) is to vanish for'x=o and x=l leads to a transcendental equation in n (corresponding to sin kl=o in the previous case). If the forms of u(x) which correspond to the various roots of this be distinguished by suffixes, we infer, on physical grounds alone, the possibility of the expansion of an arbitrary initial form of the string in a series f(x) =C1u1(x)+C2U2(x)+C3u3(x)+ . . . (II) It may be shown further that if r and s are different we have the conjugate or orthogonal relation flpu,.(x)ue(x)dx=0. 0 (9) This enables us to determine the coefficients, thus C,= f 1pf(x)u,(x)dx4- f 1p{u,(x)}2dx. (13) The extension to spaces of two or three dimensions, or to cases where there is more than one dependent variable, must be passed over. The mathematical theories of acoustics, heat-conduction, elasticity, induction of electric currents, and so on, furnish an in-definite supply of examples, and have suggested in some cases methods which have a very wide application. Thus the transverse vibrations of a circular membrane lead to the theory of Bessel's Functions; the oscillations of a spherical sheet of air suggest the theory of expansions in spherical harmonics, and so forth. The physical, or intuitional, theory of such methods has naturally always been in advance of the mathematical. From the latter point of view only a few isolated questions of the kind had, until quite recently, been treated in a rigorous and satisfactory manner. A more general and comprehensive method, which seems to derive some of its inspiration from physical considerations, has, however, at length been inaugurated, and has been vigorously cultivated in recent
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