How Does D'alembert Solution Work in Wave Equations?

[Niall101]

Hi I have been trying to figure out how this method works.

Searched the forum but got no results. Have attached an example I am trying to do. If someone could show me it would be great.

Or even show me a similar example worked out.

Thanks in advance

 

[LCKurtz]

Do you have this equation to work with:

$$u(x,t)=\frac{f(x-ct)+f(x+ct)}{2}+\frac 1 {2c}\int_{x-ct}^{x+ct}v_0(s)\,ds$$

to work with or are you expected to derive it?

 

[Niall101]

Yes I have this equation. Does the integral go to a constant. ITs after this i don't know what to do

 

[LCKurtz]

Yes I have this equation. Does the integral go to a constant. ITs after this i don't know what to do

Think about this:

$$\int_a^b 0\ dt = c|_a^b = c - c = 0$$