Question

5. (2+2+3 Points) Consider the linear wave equation for-oo <エ<oo,-oo < t <oo for - oo<<oo for utt-c2uzz = f(x, t) tr(r,0)- (r) _oo < x < oo. a. Sketch the domain of dependence of the point (,t) (4, 1) in the (x, t)-plane in 1. case c = 2, Do the same in the case c b. State the general solution forrnula for this problem! in terms of the data f, φ, ψ. c. Now suppose ψ = 0 and f = 0. Use the formula from 5b to show lu(x, t)I max 101 for all r, t.

Answer 1

Here the given wave equation is y_tt - c^2 u_xx = f(x, t) for - infinity < x < infinity, - infinity < t < infinity u(x, 0) = phi(x) for - infinity < x < infinity u_t(x, 0) = Psi(x) for - infinity < x < infinity} II (a) If we have to find domain of dependent for the fined point (x_0, t_0) Then we consider the (x, t) plane and the two characteristics passing through (x_0, t_0) x - ct = x_0 - c t_0 x + ct = x_0 + ct_0 These straight lines interest the x axis at the points (x_0 - ct_0, 0) and (x_0 + ct_0, 0) respectively. the triangle formed by the these characteristics and the interval [x_0 - ct_0, x_0 + ct_0] is called a characteristics triangle By D. A lambert's formula we how u(t_0, x_0) = p(x_0 + ct_0) + phi(x_0 - ct_0)/2