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.

 

[blue_raver22]

Hello,

Thank you for reaching out and sharing your question. I understand your confusion and I would be happy to provide some clarification on the D'Alembert solution of the wave equation with initial velocity given.

The D'Alembert solution is a method for solving the wave equation, which is a partial differential equation that describes the propagation of waves. This solution involves breaking the problem into two parts – the homogeneous solution and the particular solution. The homogeneous solution is the solution to the wave equation without any initial conditions, while the particular solution takes into account the given initial velocity.

In the attached pages from Farlow's book, Fig. 18.4 shows the regions for integration for the particular solution. These regions are determined by the initial velocity given in the problem. Since the initial velocity is given in the interval of [-1,1], the integration must be done in six regions – three for positive values of the initial velocity and three for negative values. This is because the particular solution involves both positive and negative values of the initial velocity.

I hope this helps to clarify the six regions for integration in the D'Alembert solution. If you have any further questions or concerns, please don't hesitate to reach out. Best of luck with your studies!

Sincerely,