A string, clamped at x = 0 and at x = L, is vibrating freely. Its...

  

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A string, clamped at x = 0 and at x = L, is vibrating freely. Its motion is described by the wave equation ə²u(x, t) _ „²d²u(x, t) = a1² əx² Assume a Fourier expansion of the form ηπί u(x, t) = [bn(t) sin L n=1 and determine the coefficients bn (t). The initial conditions are u(x,0) = f(x) and ə at -u(x, 0) = g(x). ( Note. This is only half the conventional Fourier orthogonality integral interval. How- ever, as long as only the sines are included here, the Sturm-Liouville boundary condi- tions are still satisfied and the functions are orthogonal. ANS. bn(t) = A₁, cos j An = 2 L ηπυt L - + Bn sin - f(x) sin ηπχ ηπυt L L - dx, Bn= = 2018 ηπυ g(x) sin nπX L dx. A string, clamped at x = 0 and at x = L, is vibrating freely. Its motion is described by the wave equation ə²u(x, t) _ „²d²u(x, t) = a1² əx² Assume a Fourier expansion of the form ηπί u(x, t) = [bn(t) sin L n=1 and determine the coefficients bn (t). The initial conditions are u(x,0) = f(x) and ə at -u(x, 0) = g(x). ( Note. This is only half the conventional Fourier orthogonality integral interval. How- ever, as long as only the sines are included here, the Sturm-Liouville boundary condi- tions are still satisfied and the functions are orthogonal. ANS. bn(t) = A₁, cos j An = 2 L ηπυt L - + Bn sin - f(x) sin ηπχ ηπυt L L - dx, Bn= = 2018 ηπυ g(x) sin nπX L dx.