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Sturm-Liouville theory - Application to normal modes | | Sturm-Liouville theory - Application to normal modes: Encyclopedia II - Sturm-Liouville theory - Application to normal modes | | Suppose we are interested in the modes of vibration of a thin membrane, held in a rectangular frame, 0 < x < L1, 0 < y < L2. We know the equation of motion for the vertical membrane's displacement, W(x, y, t) is given by the wave equation:
The equation is separable (substituting W = X(x) × Y(y) × T(t)), and the normal mode solutions that have harmonic time dependence and sati ...
See also:Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville form, Sturm-Liouville theory - Examples, Sturm-Liouville theory - Sturm-Liouville differential operators, Sturm-Liouville theory - Some highly technical details, Sturm-Liouville theory - Useful consequences of the preceding technicalities, Sturm-Liouville theory - Example, Sturm-Liouville theory - Application to normal modes | | | Sturm-Liouville theory, Sturm-Liouville theory - Application to normal modes, Sturm-Liouville theory - Example, Sturm-Liouville theory - Examples, Sturm-Liouville theory - Some highly technical details, Sturm-Liouville theory - Sturm-Liouville differential operators, Sturm-Liouville theory - Sturm-Liouville form, Sturm-Liouville theory - Sturm-Liouville theory, Sturm-Liouville theory - Useful consequences of the preceding technicalities, normal mode | | |
| | | Sturm-Liouville theory: Encyclopedia II - Sturm-Liouville theory - Application to normal modes
Sturm-Liouville theory - Application to normal modes
Suppose we are interested in the modes of vibration of a thin membrane, held in a rectangular frame, 0 < x < L1, 0 < y < L2. We know the equation of motion for the vertical membrane's displacement, W(x, y, t) is given by the wave equation:
The equation is separable (substituting W = X(x) × Y(y) × T(t)), and the normal mode solutions that have harmonic time dependence and satisfy the boundary conditions W = 0 at x = 0, L1 and y = 0, L2 are given by
where m and n are non-zero integers, Amn is an arbitrary constant and
Since the eigenfunctions Wmn form a basis, an arbitrary initial displacement can be decomposed into a sum of these modes, which each vibrate at their individual frequencies ωmn. Infinite sums are also valid, as long they converge.
Other related archivesFourier series, Hermitian, Jacques Charles François Sturm, Joseph Liouville, Kronecker delta, L2, Rayleigh quotient, Sobolev space, boundary conditions, compact, converge, convergence of Fourier series, differential equation, differential operator, eigenfunctions, eigenvalue, eigenvalues, function space, functional analysis, harmonic, hermitian, integers, integrating factor, linear operator, mathematics, normal mode, normal modes, ordinary differential equations, orthonormal basis, partial differential equations, self-adjoint, separable, sequence, spectral method, spectral theorem, wave equation, well ordered
 Adapted from the Wikipedia article "Application to normal modes", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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