Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSdiscussed in the next chapter. Nevertheless, we now consider D’Alembert’s solutionu(x, t) of the wave equation subject to initial conditions (boundary conditions) inthe following general form:∂u(x, 0)initial displacement, u(x, 0) = φ(x); initial velocity, = ψ(x).∂tThe functions φ(x) andψ(x) are given and describe the displacement and velocityof each part of the string at the (arbitrary) time t =0.It is clear that what we need are the particular forms of the functions f and gin (20.26) that lead to the required values at t = 0. This means thatφ(x) =u(x, 0) = f(x − 0) + g(x +0), (20.27)∂u(x, 0)ψ(x) = = −cf ′ (x − 0) + cg ′ (x +0), (20.28)∂twhere it should be noted that f ′ (x − 0) stands for df(p)/dp evaluated, after thedifferentiation, at p = x − c × 0; likewise for g ′ (x +0).Looking on the above two left-hand sides as functions of p = x ± ct, buteverywhere evaluated at t = 0, we may integrate (20.28) between an arbitrary(and irrelevant) lower limit p 0 and an indefinite upper limit p to obtain∫1 pψ(q) dq + K = −f(p)+g(p),c p 0the constant of integration K depending on p 0 . Comparing this equation with(20.27), with x replaced by p, we can establish the forms of the functions f andg asf(p) = φ(p) − 1 ∫ pψ(q) dq − K 2 2c p 02 , (20.29)g(p) = φ(p) + 1 ∫ pψ(q) dq + K 2 2c p 02 . (20.30)Adding (20.29) with p = x − ct to (20.30) with p = x + ct gives as the solution tothe original problem∫ x+ctu(x, t) = 1 2 [φ(x − ct)+φ(x + ct)] + 1 ψ(q) dq, (20.31)2c x−ctin which we notice that all dependence on p 0 has disappeared.Each of the terms in (20.31) has a fairly straightforward physical interpretation.In each case the factor 1/2 represents the fact that only half a displacement profilethat starts at any particular point on the string travels towards any other positionx, the other half travelling away from it. The first term 1 2φ(x − ct) arises fromthe initial displacement at a distance ct to the left of x; this travels forwardarriving at x at time t. Similarly, the second contribution is due to the initialdisplacement at a distance ct to the right of x. The interpretation of the final694