e.g. the string of a guitar (both ends fixed and tensioned, undamped)
As I understood the analytical solution of the fundamental frequency (first eigenvalue) is :
f_1 = \sqrt{\frac{T}{\mu}}*\frac{1}{2*L}
while :
mu - mass per length (Kg/m)
T- Force in N
L - lenght of a string
by describing the string using wave equation
mu y_tt = T y_xx
while:
y_tt is second derivative with respect to time
y_xx is second derivative with respect to x
x = [0, L];
and discretisation using central differences. The first eigenvalue converge to the analytical solution, if the grid in space coordinates get smaller and smaller.
Is there way to represent to string with n nodes; while the n eigenvalues of the discrete system match the first n eigenvalues of the continuous string.
Your question is not clear. Break it into parts. Do not use Maple notation. Attach the text with a detailed description. All the best.
To the best of my knowledge, the answer is negative - this simply happens because you are looking for the eigenpairs of two different operators (exact and discretized).
The only way I see is to expand the solution in Fourier series of eigenfunctions and apply the true evolution operator to the truncated series.
An idea that comes to my mind is to try to formulate your problem as a variational one and to add the conditions on the eigenvalues as constraints to the variational problem.
But to do this is probably quite an intricate problem.
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