Question

Problem . Consider the function a) Find the full Fourier Series of F(x) a(0, y, t) = u(a, y, t) 0 u(z, 0, t ) = u(z, b, l) = 0 u(z,y,0) = f(z,y), u(x, y,0)-g(x,y), 0<y< b,t0 a) b) Solve the initial-boundary value problem for 2D wave equation. What is the physical interpretation of these boundary conditions

This is PDE problem. Please show all steps in detail with neat handwriting.

Answer 1

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U_tt = c^2 (u_xx + u_yy) Let u (x y_1 t) = X (x) u (y) T (t) So, T''/c^2 T = x''/x + y''/y Take x''/x = -p^2 y''/y = -2^2 x'' + p^2 x = 0 u'' + 2^2 y = 0 x (x) = c_1 cos (px) + c_2 sin (px) , u (y) = c_3 cos (2y) + c_4 sin (2y) x (0) = 0 implies c_1 = 0 u (0) = 0 implies c_3 = 0 x)a) = 0 p = n pi/a u (b) = 0 implies q = m pi/b T''/c^2 T = p^2 - q^2 T''/c^2 T = -(n^2 pi^2/a^2 + m^2 pi^2/b^2) T'' = -c^2 pi^2 (x^2/a^2 + m^2/b^2) T \r\n so, T (t) = c_5 cos (squareroot n^2/a^2 + m^2/b^2 pi ct) + c_5 sin (squareroot n^2/a^2 + m^2/b^2 pi ct) So, u (x_1 y_1 + 1 = sigma^infinity_m = 1

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