mpalenik (He-Jutsu)
Jan28-09, 06:00 AM
I'm trying to model a violin string, but I'm getting stuck on
something very simple. Although I'm sure this has already been solved,
I'm more interested in what I'm doing wrong at the moment.
[[Mod. note -- If you can get access to a copy, you might find the
following references of interest:
Gabriel Weinreich
"What science knows about violins-and what it does not know"
American Journal of Physics 61(12) [Dec 1993], 1067-1077
http://adsabs.harvard.edu/abs/1993AmJPh..61.1067W
Chris Waltham
"A balsa violin"
American Journal of Physics 77(1) [Jan 2009], 30-35
http://adsabs.harvard.edu/abs/2009AmJPh..77...30W
-- jt]]
I'm starting with the one dimensional wave equation with dispersion
and damping:
Ty'' - ny''' = my^^ + By^ (' indicates spacial derivitive, ^ indicates
time derivative)
The violin bow applies a driving force, so I now write:
F(x,t) + Ty'' - ny''' = my^^ + By^
First, I want to solve it for F(x,t) as a delta function
delta(x-x_0)*delta(t-t_0)*F_0 = -TG'' + nG''' + mG^^ + BG^
Now, transform into frequency space (forget about the factors of pi
for a moment)
exp[i*k(x-x_0)]*exp[-i*w(t-t_0)]*F_0 = (Tk^2 + nk^4 - mw^2 - iBw)G
so:
G = F_0*exp[i*k(x-x_0)]*exp[-i*w(t-t_0)]/(Tk^2 + nk^4 - mw^2 - iBw)
Now, I have to integrate this over w and k. To do so, I performed and
contour integral, since the denominator goes to infinity as |k| goes
to infinity.
The first contour integral over w seems correct. It gives the
relationship:
w = iB/2m +- sqrt(-B^2+4m(Tk^2-nk^4))/2m
and (ignoring factors of pi)
G = (F_0/m)*exp[B(t-t_0)/2m]Integral[dk exp[ik(x-x_0)]sin[w_0(t-t_0)]/
sqrt(-B^2+4m(Tk^2+nk^4))]
where w_0 = sqrt(-B^2+4m(Tk^2+nk^4))/2m
The problem is, this integral over k also goes to zero as |k| goes to
infinity, so we can perform a contour integration again. However,
since k is of degree 4, there are only 4 poles, which wipe out most
values of k from the solution.
Even worse, if the damping and dispersion constants go to zero, the
only pole is at k=0 and the value of the integral is equal to zero.
This can't be right.
something very simple. Although I'm sure this has already been solved,
I'm more interested in what I'm doing wrong at the moment.
[[Mod. note -- If you can get access to a copy, you might find the
following references of interest:
Gabriel Weinreich
"What science knows about violins-and what it does not know"
American Journal of Physics 61(12) [Dec 1993], 1067-1077
http://adsabs.harvard.edu/abs/1993AmJPh..61.1067W
Chris Waltham
"A balsa violin"
American Journal of Physics 77(1) [Jan 2009], 30-35
http://adsabs.harvard.edu/abs/2009AmJPh..77...30W
-- jt]]
I'm starting with the one dimensional wave equation with dispersion
and damping:
Ty'' - ny''' = my^^ + By^ (' indicates spacial derivitive, ^ indicates
time derivative)
The violin bow applies a driving force, so I now write:
F(x,t) + Ty'' - ny''' = my^^ + By^
First, I want to solve it for F(x,t) as a delta function
delta(x-x_0)*delta(t-t_0)*F_0 = -TG'' + nG''' + mG^^ + BG^
Now, transform into frequency space (forget about the factors of pi
for a moment)
exp[i*k(x-x_0)]*exp[-i*w(t-t_0)]*F_0 = (Tk^2 + nk^4 - mw^2 - iBw)G
so:
G = F_0*exp[i*k(x-x_0)]*exp[-i*w(t-t_0)]/(Tk^2 + nk^4 - mw^2 - iBw)
Now, I have to integrate this over w and k. To do so, I performed and
contour integral, since the denominator goes to infinity as |k| goes
to infinity.
The first contour integral over w seems correct. It gives the
relationship:
w = iB/2m +- sqrt(-B^2+4m(Tk^2-nk^4))/2m
and (ignoring factors of pi)
G = (F_0/m)*exp[B(t-t_0)/2m]Integral[dk exp[ik(x-x_0)]sin[w_0(t-t_0)]/
sqrt(-B^2+4m(Tk^2+nk^4))]
where w_0 = sqrt(-B^2+4m(Tk^2+nk^4))/2m
The problem is, this integral over k also goes to zero as |k| goes to
infinity, so we can perform a contour integration again. However,
since k is of degree 4, there are only 4 poles, which wipe out most
values of k from the solution.
Even worse, if the damping and dispersion constants go to zero, the
only pole is at k=0 and the value of the integral is equal to zero.
This can't be right.