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The Helmholtz equation governs the dynamics of membranes and waveguides. Using the mode matching method, trapped modes are found for an infinite strip with a segment of inhomogeneity. The exact frequencies and mode shapes are determined as a function of density ratio and length of the segment. Nonuniqueness and nonexistence are demonstrated.
The vibration of a cantilever beam with constant thickness and linearly tapered sides is solved using a novel accurate, efficient initial value numerical method. The effects of tip mass, base fixity, and taper on the natural frequencies are determined. This geometrically anisotropic beam vibrates in a mixture of modes in two perpendicular directions.
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