Notes-PDE pt4 - Second Order Linear Partial Differential...

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Unformatted text preview: Second Order Linear Partial Differential Equations Part IV One-dimensional undamped wave equation; DAlembert solution of the wave equation; damped wave equation and the general wave equation; two- dimensional Laplace equation The second type of second order linear partial differential equations in 2 independent variables is the one-dimensional wave equation. Together with the heat conduction equation, they are sometimes referred to as the evolution equations because their solutions evolve, or change, with passing time. The simplest instance of the one-dimensional wave equation problem can be illustrated by the equation that describes the standing wave exhibited by the motion of a piece of undamped vibrating elastic string. Undamped One-Dimensional Wave Equation: Vibrations of an Elastic String Consider a piece of thin flexible string of length L , of negligible weight. Suppose the two ends of the string are firmly secured (clamped) at some supports so they will not move. Assume the set-up has no damping. Then, the vertical displacement of the string, 0 < x < L , and at any time t > 0, is given by the displacement function u ( x , t ). It satisfies the homogeneous one- dimensional undamped wave equation : a 2 u xx = u tt Where the constant coefficient a 2 is given by the formula a 2 = T / , such that a = horizontal propagation velocity of the wave motion, T = force of tension exerted on the string, = mass density (mass per unit length). It is subjected to the homogeneous boundary conditions u (0, t ) = 0, and u ( L , t ) = 0, t > 0. The two boundary conditions reflect that the two ends of the string are clamped in fixed positions. Therefore, they are held motionless at all time. The equation comes with 2 initial conditions, due to the fact that it contains the second partial derivative term u tt . The two initial conditions are the initial (vertical) displacement u ( x , 0), and the initial (vertical) velocity u t ( x , 0), both are arbitrary functions of x alone. (Note that the string is merely the medium for the wave, it does not itself move horizontally, it only vibrates, vertically, in place. The resulting wave form, or the wave-like shape of the string, is what moves horizontally.) Hence, what we have is the following initial-boundary value problem: (Wave equation) a 2 u xx = u tt , 0 < x < L , t > 0 , (Boundary conditions) u (0, t ) = 0 , and u ( L , t ) = 0 , (Initial conditions) u ( x , 0) = f ( x ) , and u t ( x , 0) = g ( x ) . We first let u ( x , t ) = X ( x ) T ( t ) and separate the wave equation into two ordinary differential equations. Substituting u xx = X T and u tt = X T into the wave equation, it becomes a 2 X T = X T ....
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This note was uploaded on 07/16/2011 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.

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Notes-PDE pt4 - Second Order Linear Partial Differential...

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