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Problem 8: In electromagnetic scattering by an infinite cylinder of radius a, under certain conditions, the time-harmonic electric field is given by E = u(p, ø) î, where u satisfies the Helmholtz wave equation %3D 1 ди 1 02u + k2u = 0, 0 < p< a, 0 < 0< 27 p dp u(0, ø) is finite, u(а, ф) = eika cos o u(p,0) = u(p, 27), Us(p,0) = us(p, 2m) where k is a constant. Solve it. Hint: Use the generating function of the Bessel function when obtaining the unknown coefficient.

Problem 8: In electromagnetic scattering by an infinite cylinder of radius a, under certain conditions, the time-harmonic electric field is given by E = u(p, ø) î, where u satisfies the Helmholtz wave equation %3D 1 ди 1 02u + k2u = 0, 0 < p< a, 0 < 0< 27 p dp u(0, ø) is finite, u(а, ф) = eika cos o u(p,0) = u(p, 27), Us(p,0) = us(p, 2m) where k is a constant. Solve it. Hint: Use the generating function of the Bessel function when obtaining the unknown coefficient.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 8: In electromagnetic scattering by an infinite cylinder of radius a, under certain conditions, the time-harmonic electric field is given by E = u(p, ø) î, where u satisfies the Helmholtz wave equation 1 ди 1 02u + k?u = 0, 0 < 0< 27 0 <p< a, p dp ika cos o u(0, ø) is finite, u (α, φ ) - = e u(p, 0) = u(p, 27T), us(p, 0) = us(p,2m) where k is a constant. Solve it. Hint: Use the generating function of the Bessel function when obtaining the unknown coefficient.
Transcribed Image Text:Problem 8: In electromagnetic scattering by an infinite cylinder of radius a, under certain conditions, the time-harmonic electric field is given by E = u(p, ø) î, where u satisfies the Helmholtz wave equation 1 ди 1 02u + k?u = 0, 0 < 0< 27 0 <p< a, p dp ika cos o u(0, ø) is finite, u (α, φ ) - = e u(p, 0) = u(p, 27T), us(p, 0) = us(p,2m) where k is a constant. Solve it. Hint: Use the generating function of the Bessel function when obtaining the unknown coefficient.
Problem 7: Prolate spheroidal coordinates {0 <n<∞0,0<0 < T, 0<¢ < 2n} are described by the metric coefficients h1 = h2 = a?(sinh? 7 + sin? 0) h3 = a? sinh? nsin? 0 where a is a constant. Derive Laplace equation and separate it into three ODESS. Don't solve the separated ODES.
Transcribed Image Text:Problem 7: Prolate spheroidal coordinates {0 <n<∞0,0<0 < T, 0<¢ < 2n} are described by the metric coefficients h1 = h2 = a?(sinh? 7 + sin? 0) h3 = a? sinh? nsin? 0 where a is a constant. Derive Laplace equation and separate it into three ODESS. Don't solve the separated ODES.
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