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# 2_23 - .214384 1,0 0.004641 2.13 0.20974 0,1 0.004641 0.013...

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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 x , y , z = V . 1) First consider the side V x , y , a = z : fi 1 x , y , z = > n , m = 1 K A nm sin J n x sin K m y sinh L nm z 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 L nm a nm ^ 2 where both n , and m are odd. (Later we will use n = 2 p + 1, m = 2 q + 1 2) In order to express fi 2 x , y , z in a form like the above, we make the coordinate transformation x v = y , y v = x , z v = ? z + a So fi 2 x , y , z = fi 1 x v , y v , z v = fi 1 y , x , = z + a fi x , y , z = fi 1 x , y , z + fi 2 x , y , z b) fi a 2 , a 2 , a 2 = 16 6 V ^ 2 > p , q = 0 K ? 1 p + q 2 p + 1 2 q + 1 cosh L nm a 2 where I have used the identity sinh L nm a = 2sinh L nm a 2 cosh L nm a 2 Let f p , q fl > p , q = 0 K ? 1 p + q 2 p + 1 2 q + 1 cosh 2 p + 1 2 + 2 q + 1 2 ^ 2

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(p,q) f p , q Error
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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 Ý a 2 , a 2 , a 2 Þ = 16 6 0.20539 ^ 2 V = 0.33296 V f av Ý a 2 , a 2 , a 2 Þ = 2 6 V = 0.333. ... V c) a Ý x , y , a Þ = ? P / / z f | z = a a Ý x , y , a Þ = ? 16 P ^ 2 V = ? 16 P ^ 2 V > n , m odd K sin Ý J n x Þ sin Ý K m y Þ Ý cosh Ý L nm a Þ ? 1 sinh Ý L nm a Þ a Ý x , y , a Þ = ? 16 P ^ 2 V > n , m odd K sin Ý J n x Þ sin Ý K m y Þ tanh L nm a 2...
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2_23 - .214384 1,0 0.004641 2.13 0.20974 0,1 0.004641 0.013...

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