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Index 893
unit, 150 forcing term, 567, 575–577
zero, 149 Fourier transforms for solution of,
Velocity v, 30–31, 37–38, 349–354, 582–584, 586–587
568–570, 572–573, 579–581 infinite medium, motion in a, 579–584
acceleration a(t), 350 initial conditions, 566, 573–575
curvature κ(s) and, 349–354 initial-boundary value problem for,
defined, 349 566
first-order differential equation intervals, motion in, 567–577
applications for, 30–31, 37–38 Laplace transform techniques for,
nonzero initial, 572–573 587–593
speed v(t), 349 one-dimensional equation, 566
terminal, 30–31 position function y(x,t), 565–567
unwinding chain, 37–38 semi-infinite medium, motion in a,
vector analysis for, 349–354 585–587
wave motion and, 568–570, 572–573, vibrations in a membrane,
579–581 applications of, 602–610
zero initial, 568–570, 579–581
wave motion, 567–587, 596–610
Verhulst’s logistic equation, 15
Weight function p, 182–183, 511, 515
Vibrations, 602–610
Weighted dot product, 515
circular membranes, 602–608
Window function w(t), 483–485
frequencies of normal modes of, 604
Windowed Fourier transform, 483–485
normal modes of, 604–605
Wronskian (W), 46–47
periodicity conditions, 605–606
Wronskian test, 46
rectangular membranes, 608–610
wave equations for, 602–610
Voltage law, Kirchhoff’s, 33 Z
Vortex, 780 Zero function, 181
Zero initial displacement, 570–572,
W 581–582
Walk (path), 195 Zero initial velocity, 568–570, 579–581
Wave equation, 565–610 Zero matrix, 192
boundary conditions, 566–567 Zero temperature, heat equation for,
Cauchy problem for, 594–596 612–614
characteristics of, 594–601 Zero vector, 149
d’Alembert’s solution for, 594–601 Zeros of Bessel functions, 550–552
derivation of, 565–567 Zeros of Legendre polynomials,
displacement function z(x,y, t), 567 528–569
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