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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?

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Advanced Vector Analysis with Application to Mathematical Physics by C.E. Weatherburn


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