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Chapter 2:Linear Second-Order Equations. Sec 2.5:Some Equations of Mathematical Physics. 1. imagine an elastic string stretched between two pegs as on a guitar. We want to describe the motion if the string is displaced and released to vibrate in a plane.
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Chapter 2:Linear Second-Order Equations Sec 2.5:Some Equations of Mathematical Physics 1 imagine an elastic string stretched between two pegs as on a guitar. We want to describe the motion if the string is displaced and released to vibrate in a plane
Chapter 2:Linear Second-Order Equations Sec 2.5:Some Equations of Mathematical Physics • Place the string along x-axi from 0 to L • Assume that the plane of motion is the x-y plane • We seek a function u(x,y) s.t. At any time t the graph y=u(x,y)is the shape of the string at that time.
Neglet air resistance , weight of the string • Assume that the tension T(x,t) acts tangentially to the string • Assume the mass per unit length is constant • Apply Newton’s second law of motion to the segement of string between and :
Chapter 2:Linear Second-Order Equations Sec 2.5:Some Equations of Mathematical Physics horizontal component vertical component
Chapter 2:Linear Second-Order Equations Sec 2.5:Some Equations of Mathematical Physics Since there is no acceleration in the horizontal direction However the vertical components must satisfy where is the coordinate to the center of mass Let: rearranging the equation becomes
Letting , the equation becomes To express this in terms of only terms of u. we note that The resulting equation in terms of u is and since h(t) is independent on x the resulting equation is
Chapter 2:Linear Second-Order Equations Sec 2.5:Some Equations of Mathematical Physics For small motions of the string, it is approximated that using the substitution that the wave equation takes its form of
