lec16 - Math 124A – Viktor Grigoryan 16 Waves with a...

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Unformatted text preview: Math 124A – November 23, 2010 Viktor Grigoryan 16 Waves with a source Consider the inhomogeneous wave equation u tt- c 2 u xx = f ( x,t ) ,-∞ < x < ∞ ,t > , u ( x, 0) = φ ( x ) , u t ( x, 0) = ψ ( x ) , (1) where f ( x,t ), φ ( x ) and ψ ( x ) are arbitrary given functions. Similar to the inhomogeneous heat equation, the right hand side of the equation, f ( x,t ), is called the source term. In the case of the string vibrations this term measures the external force (per unit mass) applied on the string, and the equation again arises from Newton’s second law, in which one now also has a nonzero external force. As was done for the inhomogeneous heat equation, we can use the superposition principle to break problem (1) into two simpler ones: u h tt- c 2 u h xx = 0 , u h ( x, 0) = φ ( x ) , u h t ( x,t ) = ψ ( x ) , (2) and u p tt- c 2 u p xx = f ( x,t ) , u p ( x, 0) = 0 , u p t ( x,t ) = 0 . (3) Obviously, u = u h + u p will solve the original problem (1). u h solves the homogeneous equation, so it is given by d’Alambert’s formula. Thus, we only need to solve the inhomogeneous equation with zero data, i.e. problem (3). We will show that the solution to the original IVP (1) is u ( x,t ) = 1 2 [ φ ( x + ct ) + φ ( x- ct )] + 1 2 c Z x + ct x- ct ψ ( y ) dy + 1 2 c Z t Z x + c ( t- s ) x- c ( t- s ) f ( y,s ) dy ds. (4) The first two terms in the above formula come from d’Alambert’s formula for the homogeneous solution u h , so to prove formula (4), it suffices to show that the solution to the IVP (3) is u p ( x,t ) = 1 2 c Z t Z x + c ( t- s ) x- c ( t- s ) f ( y,s ) dy ds. (5) For simplicity, we will seize specifying the superscript and write u = u p (this corresponds to the as- sumption φ ( x ) ≡ ψ ( x ) ≡ 0, which is the only remaining case to solve). Recall that we have already solved inhomogeneous hyperbolic equations by the method of character- istics, which we will apply to the inhomogeneous wave equation as well. The change of variables into the characteristic variables and back are given by the following formulas ξ = x + ct, η = x- ct, t = ξ- η 2 c , x = ξ +...
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lec16 - Math 124A – Viktor Grigoryan 16 Waves with a...

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