Second Order Linear Partial Differential Equations Part IV

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2008, 2012 Zachary S Tseng E 4 1 Second Order Linear Partial Differential Equations Part IV One dimensional undamped wave equation; D Alembert solution of the wave equation; damped wave equation and the general wave equation; two dimensional Laplace equation The Second type of Second Order Linear Partial Differential Equations in 2 independent variables is the one dimensional wave equation. Together with the heat conduction equation, they are sometimes referred to as the evolution Equations because their solutions evolve , or change, with passing time. The simplest instance of the one dimensional wave equation problem can be illustrated by the equation that describes the standing wave exhibited by the motion of a piece of undamped vibrating elastic string. 2008, 2012 Zachary S Tseng E 4 2 Undamped One-Dimensional Wave Equation: Vibrations of an Elastic String Consider a piece of thin flexible string of length L, of negligible weight. Suppose the two ends of the string are firmly secured ( clamped ) at some supports so they will not move.

The second type of second order linear partial differential equations in 2 independent variables is the one-dimensional wave equation. Together with the heat conduction equation, they are sometimes referred to as the “evolution equations” because their solutions “evolve”, or change, with passing time.

  Linear, Equations