Answer Happy

Question 5 (28 points) Consider a wave equation 4, = U, 0< y < 20, 1>0 Eq.(Q5-1) with the given boundary conditions and initial conditions (0,1)=0, u(20,t) = 0, >0; u(y,0) = $(y), 4(7,0) = 0, 0< y < 20 where the initial deflection f(y) is not identically equal to 0. (a). Classify the equation Eq.(Q5-1) as parabolic, hyperbolic or elliptic. (b). Is the equation Eq.(Q5-1) (1). homogeneous or non-homogeneous? (ii). linear or non-linear? (C). Assume the solution is in product form, i.e., uly, 1) = Y(y)7(1). Show that Y and 7 satisfy Y" - KY = 0, T" - KT = 0 where k is a constant (). It is known that k must be negative. Show that k=-) for some positive integer n so that the product solution also satisfies the boundary conditions. (e). It can be shown that the solution is in the form ny (y,1)-Σ4, sin = Eq.(Q5-2) Obtain an expression for A, and hence find A, for 0<ys 10 "(y) = 10 < y s 20 (1). Show that the solution Eq. (Q5-2) can also be written as M( y,)=(y+1) + (y-t)] ko : where der is the half-range sine expansion of dov) on 0<y<20. 1. os content 20 20 20-