Starting from the plate equation with inertia restored,

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Waves in two and three dimensions, dispersion and wave packets · Vibrating bodies, musical and architectural acoustics

Bending, Ringing, Breaking and Flowing, Ch Vibrations of rods, beams and plates

Problem

Starting from the plate equation with inertia restored, D∇4w+ρhw¨=0D\nabla^4w + \rho h\ddot w = 0, consider a flexural wave travelling in one direction, w=W0ei(ωt−kx)w = W_0e^{i(\omega t - kx)}. ((a)) Derive the dispersion relation ω(k)\omega(k), and from it find the phase velocity vpv_p as a function of ω\omega and the group velocity vgv_g, showing vg=2vpv_g = 2v_p. ((b)) An aluminium plate (E=70 GPaE = 70\,\mathrm{GPa}, ν=0.33\nu = 0.33, ρ=2700 kg m−3\rho = 2700\,\mathrm{kg\,m^{-3}}) is 3 mm3\,\mathrm{mm} thick. Evaluate vpv_p and vgv_g for a 2000 Hz2000\,\mathrm{Hz} bending wave on this plate.

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