Question

12.4. The Bessel function (of the first kind of order ser is defined by Jo(+) 2 (nl) This function is of considerable importance in applied mathematics, with merous applications to problems involving cylindrical containers, such as temperature distribution in a se pipe (6) Write out the partial num for the Bessel function of order sero up to the terms. If you have access to a graphing calculator or computer, use the graph of this partial sum to approximate, to one decimal place, the smallest positive real number wy such that Jo(wy) = 0. (Remark. This partial sum is a decent approximation for Jola) when 0 Siswi but becomes a very poor approximation for larger values of (b) For which values of does Jo(1) converge? (c) The vibrations of drum can be described by a partial differential equation with boundary value and initial conditions. The position of the circular drumhead at a fixed moment in time satisfies the differential equation prp + p ) -0, where the distance from the center of the drumhead, and representa deplacement (perpendicular to the plane of the rear boundary of the drumhead). Show that pe r la solution to the differential equation (1). (Hint. Just substitute the Bessel function into the differential equation. You may assume the theorem stating that any power series with radius of convergence R can be differentiated term-by-term within the radius of convergence, i.e. (34-) - ] whenever -R<I<R.) Remark. The simple modes of vibration of a string have frequencies proportional to 1,2,3,4... (the resulting musical notes are called the fundamental of the string and its various overtones). By contrast, the simple modes of vibration of s drum hawwe frequencies proportional to wiwe. W where the i Arranged in increasing sequence, Are the positive solutions to the equation Jo(z) 0 (you approximated w in part (A)). Unlike 1, 2, 3, 4, the sequence wi . . . Increases in an erratic fashion, which explains why a drum sounds so much different from a violin.

Answer 1

(a) Jy()=30*** 11.(x)S 1,(x)=S, =1_**_* 4 64 S;=1-**+ 0.5 (1.466,0) The smallest Positive root is w, =1.466 - 1.5 [ Correct upto 1 decimal place] • (-1) (b) 1,(=30 (b) =30 =ga, [say] = ya say 22. (n!222n Then Using the ration test an+1 an = lim n+ him .22n+2 ((n + 1)!222n+2 + .22n = lim- n-04(n + 12 Hence J.(x) converges VXER (c) p(r)=J.()= -)= 3 2 ... ....(1) PW p'r= 1"(2n), 21 (2)4" (2n (2n-1 (-1)"(2n (2n-1) 2.1 22 bm=ge brengen 1. at sa pin= getrent en ben rp"(t)= IM. ...(111) t1 From (i),(ii), (iii) We get rp"(t)+ p'()+rp(r) =;: (+1)'(2m/2n-1) -- +£ C_1"(21)-+(-)"> (1274" * (d}4* H(1)+(2n+2H2n+1), 2nd+51 )(2n+2),20 + 2 “G (n+1)/4" ^ a (n+1)/4*” “xanh” 2 • (-1)", 2-11_(2n+2H2n+1) (2n+2)] (74) 4' [ 4(n+1) 4(n+1)? 2n+2 (2n+1)+(2n+2) 4(n+1) (-1)",21,_(2n+2)] = (-1)"4(n+1)?] 16(14- 4(+1)] = (a) - 1 A(n+1)) B [1 2n 2 m ) 2n + 2) rii Hence rp'()+pr+rpír) = 0