I've just started a course in Fourier analysis, and have some problem understanding the initial condition of wave equation, and would appreciate if someone would like to explain to me in the easiest possible way, and give an easy example of what it might look like.
So we have the wave equation;
$u_{tt} = c^2 u_{xx}$
with boundary condition $u(0,t)=u(l,t)=0$
And the initial condition $u(x,0) = f(x)$ and $u_t(x,0) = g(x)$
What does the initial condition actually mean? From my understanding $u(x,0)$ is simply the height at point $x$ at time $t=0$, for which we can put in any x?
And what does $u_t(x,0) = g(x)$ mean? Since it's the time derivate I assume we are talking about some kind of speed?
Would really appreciate if someone could explain further, in a non mathematical way so I can get a intuitive feeling about it.
Thanks!
$u(x, 0)$ tells you how the wave function looks like at time $t=0$. It could be a sine wave, a triangular function, or anything else. $u_t(x, 0)$ tells you the "initial speed" of each point.
I find it simple to look at the example of an oscillating string. $u(x, 0)$ is the shape of the string initially. After $t =0$, each point on the string moves up and down. The initial vertical speed of each point $x$ on the string is given by $u_t(x, 0)$. So, these conditions basically tell you the vertical location and vertical speed for each point $x$ on the string.
From a physical perspective, the wave function is simply Newton's second law applied to each point $x$ on the string, which is why it tells you exactly how each point will behave at every time $t$.
I hope this helps.