Derive the equation of longitudinal vibrations of a…
Derive the equation of longitudinal vibrations of a homogeneous cylindrical shaft.
Explanation of Method
Apply Hooke's law to relate axial strain to internal normal force in an infinitesimal element, then apply Newton's second law.
Step 1 · Strain, stress, and internal force
With axial displacement $u(x,t)$: strain $\varepsilon=u_x$, stress $\sigma=Eu_x$, internal force $N=EAu_x$.
Step 2 · Newton's second law on an element
Net force $dN=(\partial N/\partial x)\,dx=EAu_{xx}\,dx$ equals mass times acceleration, $\rho A\,dx\,u_{tt}$: \[ EAu_{xx} = \rho Au_{tt}. \]
Step 3 · Derive the PDE
Dividing by $\rho A$: \[ u_{tt} = \frac{E}{\rho}u_{xx}. \]
Remark
The general solution $u=F(x+ct)+G(x-ct)$ describes compression/rarefaction waves with speed $c=\sqrt{E/\rho}$ (e.g. $\approx5060$ m/s for steel).
Result
The longitudinal wave equation is: