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[Physltest] [Phys-L] Re: pulse on a vertical rope



That's what we've been talking about all along. Each component varies
sinusoidally in time and Besseloidally in space, in accordance with
the equation
f(kx)*cos(wt-d)

Thanks John, I've learned something but I'm still confused. Let me
review my present state of mind, and perhaps you can help me figure
out my next step.

The modified wave equation MWE (is there a better name?) for my case is:

d^2y/dt^2 = gx*d^2y/dx^2 + g*dy/dx

I am seeking the group velocity v of a small traveling wave pulse.

But isn't f(kx)*cos(wt-d) a standing wave, and hence not what I want?

Instead, I am thinking I should use exp[i(kx-wt)], where generality
would require k to be a complex function of x. The imaginary part is
needed because of the first derivative in the MWE and gives rise to
dissipation (damping).

I know that a zeroth approximation to k is to drop this last term and
ignore derivatives of k to get:

w^2 = gx*k^2

and thus v = dw/dk = sqrt(gx), which is the "naive" solution that
gives good agreement with experiment.

However, if I retain both the derivatives of k and the last term in
the MWE, I get a complicated pair of nonlinear second-order
differential equations for the real and imaginary parts of k.

So I'm thinking that I want to make some kind of iterative
approximations, starting from the zeroth solution above. I'm stuck.

ps: I'm assuming that what I'd do in next order of approximation is evaluate
dw/dK at K0 where K is the real part of k and K0 is the peak
wavenumber of my wave pulse. Say I construct a Gaussian wavepacket
with peak K0 and some spread sigma_K, chosen to simultaneously give
me not too large a spread for the Fourier transform in real space x
(compared to the length L of the rope) and yet not involve too many
Fourier components so that K0 is a good approximation to all K in the
pulse.

Am I on the right track? Carl
--
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5040
mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/
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