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Find eigenvalues, eigenfunctions, and normal modes 3 points possible igraded! Considera cylinder with two open ends of length L. Longitudinal airwaves/ pressure waves along the midline of the cylinder satisfy the PDF Jy 8p 0<x<LI>0. 842 Ap 89 8 0cc, t> 0. 2² O₂?! whare p is the pressure, is the horizontal displacement of air molecules, and e is the speed of sound in the ambient air Let us find solutions for Separating variables and looking for solutions of the form a (x, t)-v(x) w (t) leads to solving the ODES do = Av, 0<x<L dw = Ae²w₁ t>0 df² de Soving leads to a family of solutions un(x, () = ™n (2) w₂ (1) for different values An, which all satisfy the boundary condicions. Find A and w, for n-1,2,3.... Let the unknown constant in front of the cosine term be a, and the unknown term in front of the sine term te b.) s(t)- FORMULA INPUTHUR Subenit Solve the initial value problem 405 points graced Suppose that the initial displacement takes the form 1 (37,0) - 1, 2x L Ocach Let the initial velocity of the displacement be zero (for convenience. The general solution takes the form (₁) = n/2 + (₂ cos (wsl) + Un sin (nl)) cos(x) (u 008 Determine what type of periodic extension is needed of this initial condition to solve for the coefficients in the Fourier series Then use the initial condition to find the coefficients Hint: The triangle wave 7 (2) at period 2x has Fourier series T(z) - 4 cos(z) T 2 54 72² and The sawtooth wave (2) of period 2m has Fourier series W()-2-1)¹¹ sin (na) TL 8-1 ₁/2 a ✓ 7 od 0- Try again! 21 even. D node, b 0 neven, b 0 لاواز