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

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Second Order Linear Partial Differential Equations Part IV One-dimensional undamped wave equation; D’Alembert 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.
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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 / ρ , such that a = horizontal propagation velocity of the wave motion, T = force of tension exerted on the string, ρ = 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.)
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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 ″. Dividing both sides by a 2 X T : T a T X X 2 = As for the heat conduction equation, it is customary to consider the constant a 2 as a function of t and group it with the rest of t -terms. Insert the constant of separation and break apart the equation: λ - = = T a T X X 2 λ - = X X X ″ = - λX X ″ + λX = 0, λ - = T a T 2 T ″ = - a 2 λ T T ″ + a 2 λ T = 0.
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The boundary conditions also separate: u (0, t ) = 0 X (0) T ( t ) = 0 X
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