Definitions, Theorems, Examples, Topics for Asmar 2nd edition Chapter 3: 3.7 and 3.8 only 3.7 The Two-Dimensional Wave and Heat Equations ==== The wave model: u_tt = c^2(u_xx + u_yy) u=0 on the edges of the plate for all t>=0 u=f(x,y) at t=0 (initial shape) u_t=g(x,y) at t=0 (initial velocity) SEPARATION of VARIABLES DEF. A PRODUCT SOLUTION is u=X(x)Y(y)T(t) THEOREM. A product solution satisfies the wave equation an u=0 on the edges of the plate for all time t>=0 provided X'' + mu^2 X = 0, X(0)=X(a)=0, Y'' + nu^2 Y = 0, Y(0)=Y(b)=0, T'' + c^2 k^2 T = 0, X,Y,T all nonzero k^2 = mu^2 + nu^2 THEOREM. mu[m] = m Pi/a, nu[n] = n Pi/b, k[m,n] = Pi^2((m/a)^2+(n/b)^2) X_m(x) = sin(mu[m] x) Y_n(y) = sin(nu[n] y) T[m,n](t) = B[m,n] cos(k[m,n] t) + B*[m,n] sin(k[m,n] t) DEF. A NORMAL MODE is a product solution u[m,n]=X_m(x)Y_n(y)T[m,n](t) THEOREM. Double Fourier Series coefficient formulas The superposition of the product solutions is a function u(x,y,t) that satisfies the wave equation and it is zero on the edges of the rectangle. It will have the prescribed shape f(x,y) and velocity g(x,y) at t=0 provided the series coefficients B[m,n] and B*[m,n] satisfy the following equations (Euler formulas): 4 B[m,n] = -- int( int(F,y=0..b), x=0..a) [double integral] ab where F=f(x,y)X[m](x)Y[n](y) 4 B*[m,n]= -------- int( int(G,y=0..b), x=0..a) [double integral] abk[m,n] where G=g(x,y)X[m](x)Y[n](y) ============================== How to remember these formulas ============================== No one is supposed to memorize such complicated expressions. What we do remember is the equation for the normal modes: DEF. A NORMAL MODE is a product solution u[m,n]X_m(x)Y_n(y)T[m,n](t) Then separation of variables will lead to problems for X, Y, T that are easy to solve, giving more detailed formulas X_m(x) = sin(mu[m] x) Y_n(y) = sin(nu[n] y) T[m,n](t) = B[m,n] cos(k[m,n] t) + B*[m,n] sin(k[m,n] t) Superposition of the normal modes gives the answer u(x,y,t). All that is left is determine formulas for B[m,n] and B*[m,n]. We do it from inner product abstraction by setting t=0 (which makes T[m,n]=1) in formulas for u and u_t: B[m,n] = / where h=X_m(x)Y_n(y) k[m,n] B*[m,n]= / where h=X_m(x)Y_n(y) and =double integral of f(x,y)h(x,y) over the set 0<=x<=a, 0<=y<=b. =================== THEOREM 1. Double sine series representation Let =double integral of f(x,y) over the rectangle 0<=x<=a, 0<=y<=b. Let h[m,n]=sin(m Pi x/a)sin(n Pi y/b). Then f(x,y) = double sum of B[m,n] h[m,n] for m=1..infinity, n=1..infinity where B[m,n] = /, h=h[m,n] EXAMPLE 1. A unit square membrane with wave speed c=1/Pi has edges held at position u=0. It is stretched to shape f(x,y)= x(x-1)y(y-1), then held and released (velocity g(x,y)=0).Find the position u(x,y,t) of the membrane. ANSWER. B[m,n]=64/(Pi^6 m^3 n^3) for m,n odd, else zero. DEF. A NODAL LINE is a solution of the equation Normal Mode = 0 Physically, these are lines along which sand gathers, after a vibrating membrane has been sprinkled with sand. EXAMPLE. See Figure 3, section 3.7. # 3.7, Example 1, rectangular membrane, dimensions a=b=1, # speed c=1/Pi, # u_{tt} = c^2(u_{xx}+u_{yy}), # u=0 on the edges of the membrane, # u(x,y,0)=f(x,y)=x(x-1)y(y-1), # u_t(x,y,0)=g(x,y)=0. A:=unapply(2*((-1)^m-1)/(Pi^3*m^3),m); B:=unapply(4*A(n)*A(m),(m,n)); C:=unapply(sqrt(m^2+n^2),(m,n)); V:=unapply(B(m,n)*sin(m*Pi*x)*sin(n*Pi*y) *cos(C(m,n)*t),(m,n,x,y,t)); S:=(x,y,t,N)-> sum(sum(V(2*k+1,2*j+1,x,y,t),k=0..N),j=0..N); plot3d(S(x,y,0,16),x=0..1,y=0..1); plot3d(S(x,y,2,16),x=0..1,y=0..1); plot3d(S(x,y,10,16),x=0..1,y=0..1); ============================================ Two-Dimensional Heat Equation on a Rectangle ============================================ u_t = c^2 (u_xx + u_yy) u=0 on the edges of the rectangle 0<=x<=a, 0<=y<=b, u=f(x,y) at t=0 THEOREM. The solution is the superposition of product solutions v[m,n] = sin(m Pi x/a)sin(n Pi y/b) exp(-lambda[m,n]^2 t) lambda[m,n]=c Pi sqrt((m/a)^2+(n/b)^2) THEOREM. The solution of the heat problem for the plate is u(x,y,t) = sum of constants A[m,n] times V[m,n] where A[m,n]=(4/ab)(double integral of f(x,y)V[m,n]) over the rectangle 0<=x<=a, 0<=y<=b EXAMPLE 2. A plate with unit sides is held to temperature zero on the edges with initial temperature f(x,y)=100 (at t=0). Solve the heat problem. ANSWER. A[m,n]=(1600/Pi^2)(1/mn) when m,n are both odd, zero otherwise. Problem 3.7-5, rectangular membrane problem, animation u_{tt} = c^2(laplacian u), u=0 on the 4 edges position and velocity given u=1, u_t=0 Separation of variables leads to the answer in 3.7, display before thm 1. we calculate on paper from 3.7 eq (8) that B_{mn}=0 for all m,n, and then from equation (9) that B_{mn}^* = (16/sqrt(m^2+n^2))(1/(mn))(1/pi^2) for m,n odd, B_{mn}^* = zero otherwise # Maple code for problem 3.7-5 Bstar:=(m,n)->(16/sqrt(m^2+n^2))*(1/(m*n))(1/Pi^2); phi:=(x,y,t,m,n)->sin(m*Pi*x)*sin(n*Pi*y)*sin(sqrt(m^2+n^2)*t); N:=20: # Number of partial sum terms is N^2=400. u:=(x,y,t)-> sum(sum(Bstar(2*K+1,2*L+2)*phi(x,y,t,2*K+1,2*L+1),L=0..N),K=0..N); plot3d(u(x,y,0.5),x=0..1,y=0..1); plots[animate](plot3d,[u(x,y,t),x=0..1,y=0..1],t=0..3) plots[display](%,axes=none); 3.8 Laplace's Equation in Rectangular Coordinates ==== Steady-state heat in 2 dimensions Consider a plate of sides a and b, initially heated. Formally we set u_t=0 in the heat equation u_t = c^2 (u_xx + u_yy) to get Laplace's equation for the steady-state temperature: u_xx + u_yy = 0 u(x,y) given on the boundary of the rectangle. u(x,0)=f1(x), u(x,b)=f2(x) for 0y and a<==>b. COEFFICIENTS A_n, B_n, C_n, D_n. We multiply the product solutions above by these constants, resp., then sum from n=1 to infinity. Individual sums are the solutions u1, u2, u3, u4 to the four subproblems. EVALUATING the COEFFICIENTS. To evaluate the coefficients, we set (x,y) to the values along and edge and then u(x,y) equals one of f1(x), f2(x), g1(y), g2(y). The product solutions along an edge are zero, except for just one product solution. For example, choose edge 0