Re: orthogonal functions from normalized standing wave functions
- To: mathgroup at smc.vnet.net
- Subject: [mg67934] Re: orthogonal functions from normalized standing wave functions
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Mon, 17 Jul 2006 06:51:50 -0400 (EDT)
- References: <e92eja$lte$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
If you do a search on standing waves you still find only the ancient
plate/ thin sheet sound wave
Chiladni standing waves ( they don't even seem to know that Legendre
polynomials are a kind of standing wave).
Here is a normalized version of the Chladni plate sound waves on a
circles base:
x = 2*Pi*Cos[t];
y = 2*Pi*Sin[t];
clad[t_, n_, m_] = Cos[n*x] + Cos[m*y]
norm[n_] = Sqrt[1/Integrate[clad[t, n, n], {t, -Pi, Pi}]]
Table[N[norm[n]], {n, 0, 10}]
nclad[t_, n_, m_] = Re[norm[n]*norm[m]*clad[t, n, m]]
g = Table[ParametricPlot[{nclad[t,
n, m]*Cos[t], nclad[t, n, m]*Sin[t]}, {t, -Pi,
Pi}], {n, 0, 5}, {m, 0, 5}]
Show[g]
These standing waves resemble things like p and d orbital s
and are closely related to what are called spherical harmonics.
If you don't normalize them,
they are all inside a radius one circle.
This formulation can be expanded to 3 dimensions
and two angles with only a little trouble in Mathematica.
Roger Bagula wrote:
>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]
>
>
>