Multi-Page Printing

2.4. WAVE EQUATION 69 (ii) We see from (8) that if g ∈ Ck and h ∈ Ck−1, then u ∈ Ck but is not in general smoother. Thus the wave equation does not cause instantaneous smoothing of the initial data, as does the heat equation. A reflection method. To illustrate a further application of d’Alembert’s formula, let us next consider this initial/boundary-value problem on the half-line R+ = {x 0}: (9) ⎧ ⎨ ⎩ utt − uxx = 0 in R+ × (0, ∞) u = g, ut = h on R+ × {t = 0} u = 0 on {x = 0} × (0, ∞), where g, h are given, with g(0) = h(0) = 0. We convert (9) into the form (3) by extending u, g, h to all of R by odd reflection. That is, we set ˜(x, u t) := u(x, t) (x ≥ 0, t ≥ 0) −u(−x, t) (x ≤ 0, t ≥ 0), ˜(x) g := g(x) (x ≥ 0) −g(−x) (x ≤ 0), ˜(x) h := h(x) (x ≥ 0) −h(−x) (x ≤ 0). Then (9) becomes ˜tt u = ˜xx u in R × (0, ∞) ˜ u = ˜ g, ˜t u = ˜ h on R × {t = 0}. Hence d’Alembert’s formula (8) implies ˜(x, u t) = 1 2 [˜(x g + t) + ˜(x g − t)] + 1 2 x+t x−t ˜(y) h dy. Recalling the definitions of ˜ u, ˜ g, ˜ h above, we can transform this expression to read for x ≥ 0, t ≥ 0: (10) u(x, t) = 1 2 [g(x + t) + g(x − t)] + 1 2 x+t x−t h(y) dy if x ≥ t ≥ 0 1 2 [g(x + t) − g(t − x)] + 1 2 x+t −x+t h(y) dy if 0 ≤ x ≤ t. If h ≡ 0, we can understand formula (10) as saying that an initial dis- placement g splits into two parts, one moving to the right with speed one and the other to the left with speed one. The latter then reflects off the point x = 0, where the vibrating string is held fixed. Note that our solution does not belong to C2, unless g (0) = 0.