Differential Equations and The - Braun, Martin 7579 · Problem 2.12.Ex8

Introduction to Applied Nonlinear Dynamical Systems and Chaos — Braun, Martin 7579

Mathematics — Calculus · dirac-delta-resonance

Problem

(a) Solve the initial-value problem d2ydt2+y=∑j=0∞δ(t−jπ),y(0)=y′(0)=0,\frac{d^2y}{dt^2}+y=\sum_{j=0}^{\infty}\delta(t-j\pi), \quad y(0)=y'(0)=0, and show that y(t)={sin⁡t,n even0,n oddy(t)=\begin{cases}\sin t, & n \text{ even}\\ 0, & n \text{ odd}\end{cases} in the interval nπ<t<(n+1)πn\pi<t<(n+1)\pi. (b) Solve the initial-value problem d2ydt2+y=∑j=0∞δ(t−2jπ),y(0)=y′(0)=0,\frac{d^2y}{dt^2}+y=\sum_{j=0}^{\infty}\delta(t-2j\pi), \quad y(0)=y'(0)=0, and show that y(t)=(n+1)sin⁡ty(t)=(n+1)\sin t in the interval 2nπ<t<2(n+1)π2n\pi<t<2(n+1)\pi. This example indicates why soldiers are instructed to break cadence when marching across a bridge. To wit, if the soldiers are in step with the natural frequency of the steel in the bridge, then a resonance situation of the type (b) may be set up.

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