D'Alembert solution of wave equation with initial velocity given

[OneMoreName]

Hi there,

This is a problem concerning hyperbolic type partial differential equations. Currently I am studying the book of S. J. Farlow "Partial differential equations for scientists and engineers". The attached pages show my problems. Fig. 18.4 from case two (which starts in the lower part of page 139). It shows several regions for integration concerning the problem of the wave equation with initial velocity given (1 in the interval of [-1,1]). I really have problems understanding how one obtains these six regions. If it's for the integration I would have assumed three regions and not six. Somebody knows why I am wrong?

 

[haruspex]

"Region" here means within the position-time space, not just position.
Initially, there are only three states on the position axis. Over time, these influence different parts of the position-time space. The rate at which influence spreads depends on the speed of the wave. For each point of the position-time space, you can ask which segments of the position line at time zero can have influenced it. Based on a yes/no answer for each, that gives 8 combinations, but two of those are not possible: none; and the two sides but not the centre. That leaves 6 possible combinations.

 

[OneMoreName]

OK, if it's analogous to the the light cone in relativity I have two origins here at 1,-1. Everything that lies in the cones is affected in spacetime. This explains the 5 upper regions, but why is region 6 considered to be affected (its integral is not zero). I would have thought the integral shoud be zero because this region is not influenced by the initial conditions at all?

 

[OneMoreName]

I will have a look through other posts. Not much activity here.