2_23 - .214384 1,0 ? 0.004641 2.13% 0.20974 0,1 ? 0.004641...

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PHY 5346 HW Set 5 Solutions – Kimel 2. 2.23 The system is pictured in the following figure: a) As suggested in the text and in class, we will superpose solutions of the form (2.56) for the two sides with V Ý x , y , z Þ = V . 1) First consider the side V Ý x , y , a Þ = z : f 1 Ý x , y , z Þ = > n , m = 1 K A nm sin Ý J n x Þ sin Ý K m y Þ sinh Ý L nm z Þ with J n = n ^ a , K m = m ^ a , L nm = ^ a n 2 + m 2 . Projecting out A nm using the orthogonality of the sine functions, A nm = 16 V sinh Ý L nm a Þ nm ^ 2 where both n , and m are odd. (Later we will use n = 2 p + 1, m = 2 q + 1 Þ 2) In order to express f 2 Ý x , y , z Þ in a form like the above, we make the coordinate transformation x v = y , y v = x , z v = ? z + a So f 2 Ý x , y , z Þ = f 1 Ý x v , y v , z v Þ = f 1 Ý y , x , = z + a Þ f Ý x , y , z Þ = f 1 Ý x , y , z Þ + f 2 Ý x , y , z Þ b) f Ý a 2 , a 2 , a 2 Þ = 16 6 V ^ 2 > p , q = 0 K Ý ? 1 Þ p + q Ý 2 p + 1 ÞÝ 2 q + 1 Þ cosh L nm a 2 where I have used the identity sinh Ý L nm a Þ = 2sinh Ý L nm a 2 Þ cosh Ý L nm a 2 Þ Let f Ý p , q Þ f > p , q = 0 K Ý ? 1 Þ p + q Ý 2 p + 1 ÞÝ 2 q + 1 Þ cosh Ý 2 p + 1 Þ 2 + Ý 2 q + 1 Þ 2 ^ 2
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(p,q) f Ý p , q Þ Error Sum 0,0 0.213484 4.4%
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Unformatted text preview: .214384 1,0 ? 0.004641 2.13% 0.20974 0,1 ? 0.004641 0.013% 0.20510 1,1 0.0002835 0.015% 0.20539 The first three terms give an accuracy of 3 significant figures. f a 2 , a 2 , a 2 = 16 6 0.20539 ^ 2 V = 0.33296 V f av a 2 , a 2 , a 2 = 2 6 V = 0.333. ... V c) a x , y , a = ? P / / z f | z = a a x , y , a = ? 16 P ^ 2 V = ? 16 P ^ 2 V > n , m odd K sin J n x sin K m y cosh L nm a ? 1 sinh L nm a a x , y , a = ? 16 P ^ 2 V > n , m odd K sin J n x sin K m y tanh L nm a 2...
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This note was uploaded on 01/10/2012 for the course PHYSIS 506 taught by Professor Smith during the Spring '08 term at University of Arkansas Community College at Morrilton.

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2_23 - .214384 1,0 ? 0.004641 2.13% 0.20974 0,1 ? 0.004641...

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