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 Incoming links: Fundamental frequency

The lowest positive normal mode frequency of an oscillating system is its fundamental frequency. For an ideal uniform string of length

L

with fixed ends and phase speed

c

f_{1} = c / (2 L)

. A spatial finite difference approximation has its own lowest positive normal mode frequency, approaching this continuous value as the grid is refined.

Each supplied eigenvector gives relative displacements at

x = 1, 2, 3

in a standing wave. Insert zero at

x = 0, 4

to describe the whole sampled string. A real normal mode is

u_{j} (t) = C v_{j} cos (ω t + ϕ)

: its shape stays fixed while its amplitude oscillates.

$(0, 1/2, - 1/ 2, 1/2, 0)$ is the highest-frequency resolved normal mode. Adjacent interior displacements alternate in sign: the middle point moves opposite to both outer interior points. Its interpolated shape has three lobes, corresponding to the third string normal mode.
$(0, 1/ 2, 0, - 1/ 2, 0)$ is the middle-frequency normal mode. The midpoint is a node of a standing wave, and the two sides move in opposite directions. It corresponds to the second string normal mode.
$(0, 1/2, 1/ 2, 1/2, 0)$ is the lowest-frequency normal mode, corresponding to the fundamental frequency. All interior points move in the same direction, with greatest displacement at the centre; the only nodes of a standing wave are the endpoints in the smooth shape through these samples.

sin (kπ x /4)

, with

k = 3, 2, 1

in the order listed. More precisely, their interior vectors are

v_{j}^{(k)} = 2^{- 1/2} sin (kπj /4)

, which have unit Euclidean norm and form an orthonormal basis of

R^{3}

λ_{k} = - 4 sin^{2} (\frac{kπ}{8}), ω_{k} = 2 sin (\frac{kπ}{8}), k = 1, 2, 3.

Thus larger absolute value of a negative eigenvalue means faster oscillation, rather than faster decay. The numerical angular frequencies approximate the continuous fixed-string values

kπ c / L

; the finite difference spatial approximation changes them. Only three independent interior normal modes exist on this grid, and any interior displacement or initial velocity vector can be expanded in this orthonormal basis.