# New PDF release: Mathematics in Physics and Engineering

By J. Irving and N. Mullineux (Eds.)

ISBN-10: 1483230600

ISBN-13: 9781483230603

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Extra info for Mathematics in Physics and Engineering

Example text

E. 29) On OB: x = y. 31) 58 I. INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS On OA : x = — y. 33) l ]/2\ % ) ^ b. 34). 30) we use the relations θφ_3φ3Χ Jx~ Jxlhc' n . , , and similarly a<£ + Βφ_<ΓΥ^_ l 3φ W J x " " " 2Acos^ ä Z l θφ 1 dj>_ 2Asin^ ä Y . + θφ ι -^- = ^ ^ 7 + ^„ . 4. 24) to d<# 1 A^cos// d<& -—=-——1 CL-— dv y2 u du d& 1 A^sin// d& — = _ - ^ ^ — w dv 1/2 "U where now « = { ( * - f)(y - η)}112 = λ{(Υ and on OB on AO %) a sin a /*«r- ( - ï - f 1 ) a c o s V } 1 / 2 9 = J0{u). n J ί I' .

Now the current density iy is given by iy = oEy = OEQ exp {— ]/(μσω0/2) x) sin {œQt — ]j (μσω^β) χ} The current thus falls off exponentially with x. The current in a rectangular strip parallel to the y-axis and of unit width is 00 \iydx = {i0l ]/μσω0} sin (ω0ί — π/4) with i0 = σΕ0. The mathematical analysis of the skin-effect in the case of a cylindrical wire is treated in Chapter 3 (Example 5). Example 12. 10) is said to be inhomogeneous because of the presence of the term g(x). 10). We try to eliminate the term g(x).

The bending of an elastic plate—the biharmonic equation A more complicated fourth-order equation than that considered in § 11 arises in problems involving the loading of plates. This equation can, however, be solved by the method of separation of the variables in simple cases, though the analysis is a little more tedious than in the problems so far considered. 1) where u is the deflection of the plate at the point (x,y), q is the loading per unit area, and & is the flexural rigidity. In general q will be a given function of x and y.