4. WAVE EQUATION IN HALF-SPACE The upper half-space in R3 is given by R* = {x = (21, 22, 23) R|13 >0} with boundary R} = {x = (21, 22, 23) R3|13 =0}. We consider for an unknown function u: : [0,00) * R} + R the initial value problem for the wave equation in the half-space (with Dirichlet boundary conditions) given by ( au :- Au = 0) in (0, 0) R}, u(0,-) = g on {t = 0} 1). Consider four times ti = 0, ti = 3, t2 = 5, and t3 = 8. Please visualize the support (ie. the region which the solution does not vanish) of u(ti,) : R} R for the different times and explain your visualization. Hint: Compare with exercise 8.1 to gain some intuition. 4. WAVE EQUATION IN HALF-SPACE The upper half-space in R3 is given by R* = {x = (21, 22, 23) R|13 >0} with boundary R} = {x = (21, 22, 23) R3|13 =0}. We consider for an unknown function u: : [0,00) * R} + R the initial value problem for the wave equation in the half-space (with Dirichlet boundary conditions) given by ( au :- Au = 0) in (0, 0) R}, u(0,-) = g on {t = 0} 1). Consider four times ti = 0, ti = 3, t2 = 5, and t3 = 8. Please visualize the support (ie. the region which the solution does not vanish) of u(ti,) : R} R for the different times and explain your visualization. Hint: Compare with exercise 8.1 to gain some intuition