The 1-D Wave Equation 18.303 Linear Partial Differential Equations Matthew J. Hancock Fall 2006 1 1-D Wave Equation : Physical derivation Reference: Guenther & Lee §1.2, Myint-U & Debnath §2.1-2.4 [Oct. 3, 2006] We consider a string of length l with ends fixed, and rest state coinciding with x-axis. The string is plucked into oscillation. Let u (x,t) be the position of the string at time t. Assumptions: 1. Small oscillations, i.e. the displacement u (x,t) is small compared to the length l. (a) Points move vertically. In general, we don’t know that points on the string move vertically. By assuming the oscillations are small, we assume the points move vertically. (b) Slope of tangent to the string is small everywhere, i.e. |u x (x,t)|≪ 1, so stretching of the string is negligible  l  (c) arc length α (t)= 1+ u 2 dx ≃ l. 0 x 2. String is perfectly flexible (it bends). This implies the tension is in the tan- gent direction and the horizontal tension is constant, or else there would be a preferred direction of motion for the string. Consider an element of the string between x and x +Δx. Let T (x,t) be tension and θ (x,t) be the angle wrt the horizontal x-axis. Note that ∂u tan θ (x,t) = slope of tangent at (x,t) in ux-plane = (x,t) . (1) ∂x 1