An aluminium block of unknown thickness is tested by sending a longitudinal…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Elastic waves in a solid and at its surface

Problem

An aluminium block of unknown thickness is tested by sending a longitudinal ultrasonic pulse into one face and timing the echo from the back wall: the pulse returns after

t = 16.13\,\mathrm{\mu s}

. Recalling the thin-bar speed

c_{\rm bar} = \sqrt{E/\rho}

from a first waves course, a technician computes the thickness as

d = c_{\rm bar}t/2

. Take

E = 70\,\mathrm{GPa}

\nu = 0.33

\rho = 2700\,\mathrm{kg\,m^{-3}}

for the aluminium. ((a)) What thickness does the technician report? ((b)) Find the correct bulk longitudinal speed

c_L = \sqrt{(\lambda+2G)/\rho}

and the true thickness of the block, and state briefly why the technician's assumption is wrong for this specimen.

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