Many thanks and very interesting. An elegant derivation of Eq. (3) of Note 365(2). The only comment I have is that the derivation is based on the assumption in Eq. (33) of UFT363 that partial R (t, r, theta) / partial theta = 0. This was made to reduce complexity and to give Eq. (3) of Note 365(3), in which:
partial R(t, r, theta) / partial r = f(r, t)
within the approximation in Eq. (33) UFT363.
To: Emyrone@aol.com
Sent: 24/12/2016 11:33:12 GMT Standard Time
Subj: Notes 365(1-3)I tried a direct solution of eq.(44) of note 365(1). Using the substitution
u = 1/r
I obtained directly eq.(3) of note 365(3) when inserting suitable
integration constants, see section 2 of the protocol.
A direct solution with variable r gives a similar result but with phase
factors, see eq. o8 (section 1). Trying a theta-dependent function
Omega(theta) gives no analytical solution (section 3).I am not sure if Omega can be assumed not to depend on theta. According
to eq.(33) of the first note, bold R depends on theta, thereforepartial R / partial r
maintains this dependence. If Omega is not constant, then
f(theta) = 1 / sqrt(1+Omega)
reflects this dependence even by the indirect dependence
Omega(r(theta)), see eqs. 12-14 in note 2.I will try an example for f(theta) based on the numerical results of
paper 328.Horst