Transcription of The two dimensional wave equation - Trinity University
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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.)
As in the one dimensional situation, the constant c has the units of velocity. It is given by c2 = τ ρ, where τ is the tension per unit length, and ρ is mass density. The operator ∇2 = ∂2 ∂x2 + ∂2 ∂y2 is called the Laplacian. It will appear in many of our subsequent investigations. Daileda The 2D wave equation
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