Mathematic Methods HW Solutions 64

# Mathematic Methods HW Solutions 64 - .12 u = 400 π 64 o dd...

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Unformatted text preview: Chapter 13 10.12 u = 400 π 64 o dd n 1r na 2n sin 2nθ 10.14 Same as 9.12 10.15 u = 1r n 10 400 π 6n sin 6nθ = 2(10r)6 sin 6θ 200 arc tan π 1012 − r12 o dd n √ 10.16 v 5/(2π ) 10.17 νmn , n = 0; the lowest frequencies are: ν11 = 1.59ν10 , ν12 = 2.14ν10 , ν13 = 2.65ν10, ν21 = 2.92ν10 , ν14 = 3.16ν10 10.18 νmn , n = 3, 6, · · · ; the lowest frequencies are: ν13 = 2.65 ν10 , ν23 = 4.06 ν10 , ν16 = 4.13 ν10 , ν33 = 5.4 ν10 a2 10.19 u = E0 r − cos θ r vλl where λl = zeros of jl , a = radius of sphere, v = speed of sound 10.20 ν = 2πa 3 2 2 2 10.21 u = 3 P0 (cos θ) + 5 rP1 (cos θ) − 3 r2 P2 (cos θ) + 5 r3 P3 (cos θ) 7 11 10.22 u = 1 − 1 rP1 (cos θ) + 8 r3 P3 (cos θ) − 16 r5 P5 (cos θ) · · · 2 (al rl + bl r−l−1 )Pl (cos θ) where 10.23 u = 100 o dd l 2A(A + 1) 2A + 1 cl , b l = − cl , A = 2 l , al = 2A2 − 1 2A2 − 1 1 cl = (2l + 1) 0 Pl (x) dx (Chapter 12, Problem 9.1). The ﬁrst few terms are u = (107.1r − 257.1r−2 )P1 (cos θ) −(11.7r3 − 99.2r−4 )P3 (cos θ) + (2.2r5 − 70.9r−6 )P5 (cos θ) · · · 4(D − A) nπ nπx 10.24 T = A + sinh (b − y ) sin nπ sinh (nπb/a) a a o dd n 4(C − A) nπ nπy + sinh (a − x) sin nπ sinh (nπa/b) b b o dd n 4(B − A) nπy nπx + sinh sin nπ sinh (nπb/a) a a o dd n v 10.26 ν = (kmn /a)2 + λ2 where kmn is a zero of Jn 2π 2 n2 m2 λ + + a b π 200 ∞ sin k cos kx cosh ky dk 10.28 u(x, y ) = π 0 k cosh k 10.27 ν = v 2 . ...
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