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13 Nov 2019
1. Partial differential equations can be used to describe various physical phenomena. For example, a one-dimensional string, if given a small perturbation at time t 0, will start to vibrate as time evolves. The displacement of any point z on the string at a gievn time t can be described by a two-variable function u(x,t). It can be derived using some basic approximation techiniques that u(x, t) satisfies the wave equation Please verify that functions of the form F(z+ at) and G(x - at), where F(v) and G(u) are arbitrary twice differentiable single variable functions, are solutions of the wave equation. Hence we conclude that the vibration propagates along the lines satisfying x ± at = k, k E R, and a is the magnitude of the wave velocity.
1. Partial differential equations can be used to describe various physical phenomena. For example, a one-dimensional string, if given a small perturbation at time t 0, will start to vibrate as time evolves. The displacement of any point z on the string at a gievn time t can be described by a two-variable function u(x,t). It can be derived using some basic approximation techiniques that u(x, t) satisfies the wave equation Please verify that functions of the form F(z+ at) and G(x - at), where F(v) and G(u) are arbitrary twice differentiable single variable functions, are solutions of the wave equation. Hence we conclude that the vibration propagates along the lines satisfying x ± at = k, k E R, and a is the magnitude of the wave velocity.
Deanna HettingerLv2
13 Nov 2019