Notes 10-7 - SECTION 10.7 THE WAVE EQUATION Suppose we have an elastic string(violin string guitar string guy wire electric power line stretched

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SECTION 10.7 THE WAVE EQUATION Suppose we have an elastic string (violin string, guitar string, guy wire, electric power line, . . . ) stretched and fastened at its ends x = 0 and x = L . Set this string in motion, by plucking for example, so that it vibrates in a vertical plane. Let u ( x,t ) denote the vertical displacement of the very very very small piece of string at location x at time t . Now we use Newton along with a whole bunch of simplifying assumptions, among which are damping eﬀects such as air resistance are negligible, each point along the string moves only in a vertical line, the weight of the string is negligible, T cos θ = T when θ is small a 2 = tension linear density is constant. We get (pp. 653-654) the one-dimensional wave equation a 2 u xx = u tt . This applies not just to guitar strings, but to nearly any wave problem, including ocean waves, sound waves, electromagnetic waves, or waves in a solid body, at least if we generalize slightly. For example, the motion of a drumhead is governed by

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This note was uploaded on 11/28/2010 for the course M 56840 taught by Professor Schurle during the Spring '10 term at University of Texas at Austin.

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Notes 10-7 - SECTION 10.7 THE WAVE EQUATION Suppose we have an elastic string(violin string guitar string guy wire electric power line stretched

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