By C.E. Weatherburn
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91) The general features of this function may be simply described. For small frequency (co d a ) it will be less than 1. 91) will, however, as x becomes progressively larger, grow in amplitude, so that f ( x ) will be less than 1 for intervals about the points x = mr, where n is an integer, which decrease in size as n becomes larger. Calling the inter vals of (o for which \ f(x) | < 1 “allowed regions” (representing the ranges of frequencies which can be propagated without attenuation), it is evident that sufficiently low frequencies are allowed, that “allowed” and “for bidden” ranges alternate, and that the allowed regions, which in any case become successively narrower as the frequency increases, do so more rapidly the larger the value of X; that is, the larger the attached masses in relation to the mass of a section of the string.
In this case, again, only a 12. Orthogonality of Eigenfunctions 35 suitable choice of k will permit the satisfaction of the condition at the other boundary. 12. Orthogonality of Eigenfunctions This section is concerned with a very important theorem, which says that, under conditions to be specified, the eigenfunctions of the Sturm-Liouville equation are orthogonal. 132) where ym and yn are the eigenfunctions belonging to different eigenvalues km and kn, and r(jc) is the weighting function. 133) the left-hand side being the difference o f p (y my i —ynym) at the points b and a.
116) If, as an example, the ends o f the string at x = 0 and x = L are fixed, y and therefore f must take the value zero at each o f these points. 116) would be positive and negative exponentials (or alternatively hyperbolic sines and cosines), no combina- 32 The Vibrating String tion of which can be made equal to zero at two points. Thus no negative value of A is permissible if the boundary conditions are to be satisfied. ] If, on the other hand, A. 117) where A and ф are constants. 118) X = ( i?
Advanced Vector Analysis with Application to Mathematical Physics by C.E. Weatherburn