Transcription of The two dimensional wave equation - Trinity University
{{id}} {{{paragraph}}}
Example: marketing
The two dimensional wave equation - Trinity University
The 2D wave equationSeparation of variablesSuperpositionExamplesThe two dimensional wave equationRyan C. DailedaTrinity UniversityPartial Differential EquationsMarch 1, 2012 DailedaThe 2D wave equationThe 2D wave equationSeparation of variablesSuperpositionExamplesPhysical motivationConsider a thin elastic membrane stretched tightly over arectangular frame. Suppose the dimensions of the frame area band that we keep the edges of the membrane fixed to the the membrane from equilibrium results in somesort of vibration of the goal is to mathematically model the vibrations of themembrane 2D wave equationThe 2D wave equationSeparation of variablesSuperpositionExamplesWe letu(x,y,t) =deflection of membrane from equilibrium atposition (x,y) and a fixedt, the surfacez=u(x,y,t) gives the shape of themembrane at ideal assumptions ( uniform membrane density, uniformtension, no resistance to motion, small deflection, etc.) one canshow thatusatisfies thetwo dimensional wave equationutt=c2 2u=c2(uxx+uyy)(1)for 0<x<a, 0<y< 2D wave equationThe 2D wave equationSeparation of variablesSuperpositionExamplesRemarks:Fo r the derivation of the wave equation from Newton s secondlaw, see exercise in the one dimensional situation, the constantchas theunits of velocity.
The 2D wave equation Separation of variables Superposition Examples Representability The question of whether or not a given function is equal to a double Fourier series is partially answered by the following result. Theorem If f(x,y) is a C2 function on the rectangle [0,a] ×[0,b], then
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: