- Consider A Wave Equation 0 Y 20 T 0 Eq Q5 1 With The Given Boundary Conditions And Initial Conditions U 0 1 0 U 2 1 (67.33 KiB) Viewed 30 times
Consider a wave equation 0<y <20, t>0 Eq. (Q5-1) with the given boundary conditions and initial conditions u(0,1)=0, u(20,1)=0, t > 0; u(v,0) = $(y), u, (v,0) = 0, 0<< 20 where the initial deflection 6(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., u(y,t) = Y(y)T(t). Show that Y and T satisfy Y" - KY = 0, T" - kt = 0 where k is a constant. (a). It is known that k must be negative. Show that k=-(m.) 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 nny nga uly,t) = § 4, sin Eq.(Q5-2) 20 Obtain an expression for An and hence find An for y 0<y s 10 "(y)= 2' 20- - y 10<y s 20 2 (f). Show that the solution Eq.(Q5-2) can also be written as u(4,1) = 1:(+8) + (-1)] y where Pext is the half-range sine expansion of øy) on 0 <y<20. COS 20 2=1