THE WAVE EQUATION 1. Longitudinal Vibrations We describe the ...

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$Math 664 Homework #2: Solutions 1. Let Î© = R, F = {A â R : either A ...$

6 R. E. SHOWALTERAs before, each S(t) is an integral operator of the form∞∑∫ll[S(t)v 0 ](x) =nπα sin(nπα lt) (v 0 (s) sin( nπ l s)) ds2 l sin(nπ l x)=n=1∫ l02( ∑∞ln=10lnπα sin(nπα lt) sin(nπ l s) sin(nπ l x)) v 0 (s) dsn=1=∫ l0H(x, s, t)v 0 (s) dsfor which the kernelH(x, s, t) = 2 ( ∑∞ ll nπα sin(nπα lt) sin(nπ l s) sin(nπ l x))is the Green’s function for the problem.Example 3. Suppose the left end of the elastic rod is free while the right end has anelastic constraint given by u(l, t) + u x (l, t) = 0. The initial-boundary-value problem forthis situation is(11a)(11b)(11c)u tt (x, t) = α 2 u xx (x, t), 0 < x < l, t > 0,u x (0, t) = 0, u(l, t) + u x (l, t) = 0, t > 0,u(x, 0) = u 0 (x), u t (x, 0) = v 0 (x), 0 < x < l.We seek a solution in the formu(x, t) = X(x)T (t),where the boundary conditions imply that X x (0) = 0 and X(l)+X x (l) = 0. The methodof separation of variables leads us to the boundary-value problemX ′′ (x) + λX(x) = 0, 0 < x < l,X ′ (0) = 0, X(l) + X ′ (l) = 0.We obtain a sequence of eigenvalues λ n and corresponding eigenfunctions given byX n (x) = ( 2) 12cos (λ 1 2(12)n x).lNote that for large λ n we haveλ n ≈ ( nπ l )2 ,so the eigenvalues are asymptotically close to those of the preceding example. Combiningthese results with the time-dependent solutions and using the orthogonality of theeigenfunctions, we find solutions of the initial-boundary-value problem (11) in the form∞∑ ( √u(x, t) = cos(α λn t)(u 0 (·), X n (·)) + sin(α √ λ n t) (v 0(·), X n (·))α √ )Xn (x),λ nn=1
THE WAVE EQUATION 7with the eigenfunctions given by (12). Once again, this can be represented in the form(5) for an appropriate family of operators {S(t) : t ≥ 0}.Exercise 1. Suppose the rod (0, l) is elastic and that we account for the inertia of lateralextension. Set α 2 = k and ρβ2 = r 2 P , where P is Poisson’s ratio and r is the averageradius of a cross section as above. Assume there are no internal forces, i.e., f(x, t) = 0,and that both ends of the rod are fixed. The initial displacement and velocity are givenby u 0 (x) and v 0 (x), respectively. The initial-boundary-value problem is(13a)(13b)(13c)u tt (x, t) = α 2 u xx (x, t) + β 2 u xxtt (x, t), 0 < x < l, t > 0,u(0, t) = 0, u(l, t) = 0, t > 0,u(x, 0) = u 0 (x), u t (x, 0) = v 0 (x), 0 < x < l.Find the solution by separation of variables. Find the family of operators {S(t) : t ≥ 0}for which this can be represented in the form (5).Department of Mathematics, Oregon State University, Corvallis, OR 97331 - 4605E-mail address: show@math.oregonstate.edu
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THE WAVE EQUATION 1. Longitudinal Vibrations We describe the ...

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