On Resonance in Infinite Gratings of Cylinders
On Resonance in Infinite Gratings of Cylinders
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This is in contrast to the situation for non-critical wavelengths • Artmann has shown also that the maximum amplitude occurs at slightly larger wavelengths than critical, and that the shift is independent of the order of the spectrum. To see this from our result we return to the formula for f(S ) at a non-critical wave- length. Consulting (7) and (9), and neglecting terms of order A_, we find, for a case when A^ is large but not infinite, f (o)f (e ) (30) f(9^) . F(- e^) - fj(©^) . Fj(- 9^) * (...i)a^) t f^(o) • From (12), A^ '^ 2(kd 5 )" ^, where, if 5 is negative, the positive imaginar7 root is taken. Since -Re ^f^(0)t is positive by the forward amplitude theorem, there can be no cancellation between l/A^ and fT(0) if 5 is positive. On the other hand, if 6 is negative, then l/Ay is positive imaginary. In (2?) we see that Im \f^(0)i is also positive. Thus the imaginary part of the denominator will vanish at a certain negative value of 5, given by the formula 2 2 °° (31) y^;;^ . _ kV r i^(t)rH^(-t)r at .
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To quickly assess the difficulty of the text, read a short excerpt:
This is in contrast to the situation for non-critical wavelengths • Artmann has shown also that the maximum amplitude occurs at slightly larger wavelengths than critical, and that the shift is independent of the order of the spectrum. To see this from our result we return to the formula for f(S ) at a non-critical wave- length. Consulting (7) and (9), and neglecting terms of order A_, we find, for a case when A^ is large but not infinite, f (o)f (e ) (30) f(9^) . F(- e^) - fj(©^) . Fj(- 9^) * (...i)a^) t f^(o) • From (12), A^ '^ 2(kd 5 )" ^, where, if 5 is negative, the positive imaginar7 root is taken. Since -Re ^f^(0)t is positive by the forward amplitude theorem, there can be no cancellation between l/A^ and fT(0) if 5 is positive. On the other hand, if 6 is negative, then l/Ay is positive imaginary. In (2?) we see that Im \f^(0)i is also positive. Thus the imaginary part of the denominator will vanish at a certain negative value of 5, given by the formula 2 2 °° (31) y^;;^ . _ kV r i^(t)rH^(-t)r at .
What to read after On Resonance in Infinite Gratings of Cylinders?
You can find similar books in the "Read Also" column, or choose other free books by Samuel Karp to read onlineMoreLess
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