Definitions, Theorems, Examples, Topics for Asmar 2nd edition Chapter 4: 4.2 only 4.2 Radially Symmetric Drumhead ==== SUMMARY of RESULTS u_tt = c^2 ( u_rr + (1/r)u_r + (1/r^2)u_{theta theta} ) Radially symmetric case is u_theta = 0 u_tt = c^2(u_rr + (1/r)u_r) u(a,t)=0 [clamped edge] u(r,0)=f(r), [shape] u_t(r,0)=g(r) [velocity] SEPARATION OF VARIABLES. PRODUCT SOLUTION u = R(r)T(t) rR'' + R' + lambda^2 rR = 0, R(a)=0, T'' + c^2 lambda^2 T = 0, T nonzero The answers for the product solutions are T_n(t) = A_n cos(c alpha_n t/a) + B_n sin(c alpha_n t/a), R_n(t) = J_0(alpha_n r/a), where J_0 is the Bessel function of order zero, and alpha_n is the sequence of positive zeros of J_0(x). DETAILS. See the textbook, sections 4.2, 4.7, 4.8. BESSEL FUNCTIONS J0(x) = a solution of xy'' + y' + xy = 0 [4.7, bounded at x=0] Y0(x) = a second independent solution [unbounded at x=0] LEMMA. The change of variables x=br, R(r)=y(x) changes xy'' + y' + xy = 0 into r R'' + R' + b^2 r R = 0 INNER PRODUCT For Bessel functions, the appropriate inner product is = integral of f(x)*g(x)*x dx, for x=0..a This definition implies the 4 properties of an inner product. ORTHOGONALITY RELATIONS For Bessel functions R_n = J_0(alpha_n x/a), the following is true: THEOREM [Page 252, Section 4.8, Theorem 1] The sequence R_1, R_2, R_3, ... is an othogonal set relative to the inner product = int(f1(r)f2(r)r, r=0..a). This means: (1) = 0 for two different functions f1,f2 selected from the sequence R_1, R_2, R_3, ... (2) is positive for any f1 selected from the sequence R_1, R_2, R_3, ... SOLVING FOR A_n and B_n The solution u(r,t) is given by the series of product solutions R_n(r)T_n(r). We can solve for A_n and B_n in the relations f(r) = u(r,0) = sum A_n R_n(r) over n=1 to infinity g(r) = u_t(r,0) = sum (c*alpha_n/a)*B_n R_n(r) over n=1 to infinity Both equations are Bessel series, to be solved from results in section 4.8. According to the results of 4.8, the coefficients are uniquely determined by inner products. We can recover the formulas without reference to section 4.8 by taking the inner product of each equation against a Bessel series term for h(r)=R_m(r)=J_0(alpha_m r/a). Then for shape f and velocity g, = A_m , = (c*alpha_m/a)B_m These equations uniquely determine the coefficients A_m, B_m in the product solution series expansion of u(r,t). The relations are A_m = /, B_m = (a/(c*alpha_m)) / EVALUATION OF INNER PRODUCT THEOREM. = (a^2/2)(J_1(alpha_m))^2 PROOF. Use (12) in section 4.8, part of Theorem 1, with p=0. It is impossible to compute this value from scratch. It must be memorized or referenced. EXAMPLE 1. Radially symmetric drumhead, f(r)=0, g(r)=-100 meters per second. Find u(r,t) and animate the solution on time t. SOLUTION: u(r,t) = sum of B_n sin(100 alpha_n t) J_0(alpha_n r) B_n = -2/(alpha_n*alpha_n*J_1(alpha_n)) = -0.6662, 0.1929, -0.0984, 0.0619, -0.0434, ... or u(r,t) = -0.6662 J_0(2.4 r)sin(240 t)+0.1929 J_0(5.52 r)sin552 t)+... PLOTS. The animation of u(r,t) was done in class by Ethan Bennett. Here is a maple implementation of the same plots, using four terms instead of three used by the book's plots. Ethan used four terms. B:=[-0.6662,0.1929,-0.0984,0.0619,-0.0434]; alpha:=[2.4048,5.5201,8.6537,11.7915,14.9309]; u:=unapply(sum(B[n]*sin(100*alpha[n]*t)*J0(alpha[n]*r),n=1..4),(r,t)); J0:=x->BesselJ(0,x); plots[animate](plot3d,[u(sqrt(x^2+y^2),t), x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),thickness=2, orientation=[63,16,64]], t=0..0.4,frames=20); # More detail: replace frames=20 by frames=200 # Be patient for frames=200, it takes a long time. EXAMPLE 2. Radially symmetric drumhead, shape=1-r^2, velocity=0. Find the solution u(r,t) of the drumhead problem, given a=c=1. SOLUTION. u(r,t)=sum of terms A_n cos(alpha_n t) J0(alpha_n r) for n=1 to infinity A_n = 8/((alpha_n)^3*J1(alpha_n)) alpha_n = nth positive zero of J0(x) =====================================================================