Vibrations and the wave equation

[Tony11235]

An infinite string vibrates according to the homogenenous wave equation $$u_{tt}-u_{xx} = 0$$ with initial data given by $$u(x, 0) =f(x)$$ and $$u_{t}(x, 0) = g(x)$$ for -infinity<x<infinity where both f and g are smooth functions that are positive on the intervals -4<x<-3 and 2<x<3 and both zero everywhere else along the x-axis. A person stands at location x=0.

The question is during what intervals of time will the person notice the string vibrating? I know the interval is t=x to t=infinity but what is the process of getting x? Sorry if this is a stupid question.

 

[member 553083]

The process of getting x involves solving the wave equation. This can be done by using Fourier series or separation of variables. You would need to find the solution to the wave equation with initial data given by f and g. Once you have the solution to the wave equation, you can then determine when the person at x=0 will notice the string vibrating. This can be done by evaluating the solution at x=0 and finding the intervals of time for which the solution is not zero.