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Unformatted text preview: igenvalue problem. 146 1 0.75 0.5 1 0.25 0.5 0 -1 -1 0 -0.5 -0.5 0 0.5 1 -1 Figure 5.2: Graph of the function f0,1 (r, θ). Each of these eigenfunctions corresponds to a mode of oscillation for the circular vibrating membrane. Suppose, for example, that a = 1. Then α0,1 = 2.40483 ⇒ λ0,1 = −5.7832, α0,2 = 5.52008 ⇒ λ0,2 = −30.471, α0,3 = 8.65373 ⇒ λ0,3 = −74.887, α0,4 = 11.7195 ⇒ λ0,3 = −139.039, and so forth. The mode of oscillation corresponding to the function f0,1 will vibrate with frequency −λ0,1 α0,1 = = .38274, 2π 2π while the mode of oscillation corresponding to the function f0,2 will vibrate with frequency −λ0,2 α0,2 = = .87855. 2π 2π Similarly, α1,1 = 3.83171 ⇒ λ1,1 = −14.682, α2,1 = 5.13562 ⇒ λ2,1 = −26.3746, and hence the mode of oscillation corresponding to the functions f1,1 and g1,1 will vibrate with frequency −λ1,1 α1,1 = = .60984, 2π 2π 147 1 0.5 0 -0.5 -1 1 0.5 0 -1 -1 -0.5 0 0.5 1 Figure 5.3: Graph of the function f0,2 (r, θ...
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