ABSTRACT
When the rod is unstrained and at rest we define the displacement u of any transverse plane in the rod to be zero. This displacement u is actually the unknown in our system since if we know u for any station x in the bar and for any time t we have completely documented the rod's longitudinal motion. The displacement u of any plane away from its equilibrium position is thus rightly called u(x, t) since it is a function of both location in the bar and time t. (This basic fact will lead us inexorably to partial differential equations since when we write derivatives of quantities which are functions of more than one variable we must write them as partial derivatives.) We now choose a rod element of infinitesimal length dx and at an arbitrary location x in the rod. Since this problem involves motion of bodies under the action of forces, it is natural to apply Newton's Jaw to the element dx in hopes of getting a system equation. Since no external forces are allowed by our assumptions, the forces at the two ends of the element due to internal stresses are the only ones we need to find.
