orthogonal functions from normalized standing wave functions
- To: mathgroup at smc.vnet.net
- Subject: [mg67899] orthogonal functions from normalized standing wave functions
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Wed, 12 Jul 2006 05:05:41 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
For several years I have been trying to get orthogonal functions out of
circular cycloidal standing waves.
By converting them to roses (by subtraction the case of a circle) that
are Fourier function like
I was finally successful today. ( these also work for n->{-1,-8}).
Clear[p, ps, t, n, Inm, Enm]
(* normalized standing wave functions*)
(* circular standing waves with circle Exp[I*t] subtracted gives
normalized \
roses ( related to Fourier functions)*)
w[n_] = FullSimplify[Sqrt[1/Integrate[((2 - 1/n)*Exp[I*t]/2 + Exp[I*t*(n - \
1)]/(2*n) - Exp[I*t])*((2 - 1/n)*Exp[-I*t]/2 +
Exp[-I*t*(n - 1)]/(2*n) - Exp[-I*t]), {t, -Pi, Pi}]]]
p[t_, n_] = FullSimplify[
w[n]*((2 - 1/n)*Exp[I*t]/2 + Exp[I*t*(n - 1)]/(2*n) - Exp[I*t])]
ps[t_, n_] = FullSimplify[w[n]*((2 - 1/n)*
Exp[-I*t]/2 + Exp[-I*t*(n - 1)]/(2*n) - Exp[-I*t])]
Inm = Table[2*(N[Integrate[p[x, n]*ps[x, m], {x, -Pi, Pi}]] - 0.5), {
n, 3, 8}, {m, 3, 8}]
MatrixForm[Inm]
Enm = Table[2*(N[Integrate[p[x, n]*(D[
ps[x, m], {x, 2}]), {x, -Pi, Pi}]] + .5), {n, 3, 8}, {m, 3, 8}]
MatrixForm[Enm]
(* here are the standing waves themselves*)
Clear[w,ps,g,g1]
w[n_] = FullSimplify[Sqrt[1/Integrate[((2 - 1/n)*Exp[I*
t]/2 + Exp[I*t*(n -
1)]/(2*n) - Exp[I*t])*((2 - 1/n)*Exp[-I*t]/2 +
Exp[-I*t*(n - 1)]/(2*n)), {t, -Pi, Pi}]]]
p[t_, n_] = FullSimplify[w[n]*((2 - 1/n)*Exp[I*t]/2 + Exp[I*t*(n -
1)]/(2*n))]
g = Table[ParametricPlot[{Re[p[t, n]], Im[p[t, n]]}, {t, -Pi, Pi}], {n,
3, 8}]
Show[g]
g1 = Table[ParametricPlot[{Re[p[t, n]], Im[p[t,
n]]}, {t, -Pi, Pi}], {n, -8, -1}]
Show[g1]