A thin straight elastic rod of density , Young's modulus and constant cross-sectional…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Mathematical Methods for Physics, Ch The classical partial differential equations of physics and their boundary conditions

Problem

A thin straight elastic rod of density

\rho

, Young's modulus

E

and constant cross-sectional area

A

lies along the

x

axis. A longitudinal disturbance displaces the cross-section originally at

x

by a small amount

u(x,t)

along the axis. Consider the slice of rod originally between

x

and

x+\Delta x

, of mass

\rho A\,\Delta x

. The internal force the rod exerts across a cross-section is

F(x,t) = A\sigma(x,t)

, where the longitudinal stress

\sigma

obeys Hooke's law,

\sigma = E\,\frac{\partial u}{\partial x},

an idealisation valid while the strain

\partial u/\partial x

stays small. ((a)) Apply Newton's second law to the slice and let

\Delta x\to0

to derive the partial differential equation obeyed by

u(x,t)

. ((b)) Identify the wave speed

c

in terms of

E

and

\rho

, and check that it has units of speed. ((c)) For a steel rod,

E = 2.0\times10^{11}\,\mathrm{Pa}

and

\rho = 7800\,\mathrm{kg\,m^{-3}}

. Compute

c

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