Past exam of the physics B course of the University of Cambridge / 2012 / Paper 2 / Section B / B8 / b

Table of contents
- Solution b

Solution

A small displacement

u

in a homogeneous isotropic elastic solid obeys linear elasticity. With Lamé constants

λ, μ

and mass density

ρ

, momentum balance gives

ρ \partial_{t}^{2} u = (λ + μ) \nabla (\nabla \cdot u) + μ \nabla^{2} u .

A longitudinal plane wave has displacement parallel to its propagation direction and changes local volume. A transverse wave has perpendicular displacement and produces shear. Substitution gives

c_{L} = \frac{λ + 2 μ}{ρ}, c_{T} = \frac{μ}{ρ} .

An ideal fluid has no static shear modulus and cannot support this bulk transverse elastic wave. A thin rod with free lateral surfaces instead supports an extensional long-wavelength mode with speed

E / ρ

; its transverse contraction changes the effective stiffness, so it should not be identified with the unbounded-solid longitudinal mode.

Stress and displacement boundary conditions produce reflection, transmission, mode conversion and guided waves. Viscoelasticity and scattering can attenuate waves; finite geometry can make guided modes dispersive even when the bulk elastic constants are frequency independent. Wave speed depends on both material stiffness and the allowed deformation.

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