Semi-infinite string with free end

Question

I want to understand, one dimensional semi infinite string with free end geometrically..

Consider the semi infinite string with free end

$u_{tt}=c^2u_{xx}$

$u(x,0)=f(x), 0\leq x<\infty$

$u_t(x,0)=g(x), 0\leq x<\infty$

$u_x(0,t)=0, t\geq 0$

My understanding

Suppose a rod is fixed at $x=0$. Assume that the knot of the string with rod is lose. If we pull the string from any point $x$, then, at $t=0$, we have an initial shape and initial stress in string, which implies,

$u(x,0)=f(x), 0\leq x<\infty$

$u_t(x,0)=g(x), 0\leq x<\infty.$

Now, if we leave the string, then a wave travels towards left hit the boundary. If knot is tight, then there is no displacement, and hence we have $u(0,t)=0$, which is fixed end condition.

If knot is lose, then we have displacement along the rod but this rod will not shift backward (that is, rod will not shift from x=0 to x=-0.001,say,) and hence we have

$u_x(0,t)=0, t\geq 0.$ that is no change with $x$ axis, which is free end condition.

Please tell whether I am correct or not?

Also, can anybody elaborate, what will happen if $u_x(0,t)\neq 0$?

Answer 1

Your interpretation of the whole problem is correct.

About your question, if $u_x(0,t)\neq 0$, it means $u_x(0,t)= h(t)$.

$h(t)$ may vary by time. This means you have a source of energy which may vary by time.

This source is responsible for additional waves on -Let's say like disturbances- starting from the fixed edge of the rod and spreading through the entire rod.

Another analogy is a rod which has heat transfer phenomena instead of displacement. When $u_x(0,t)\neq 0$ this means you have a heat flux in $x=0$ that makes the rod warmer/cooler.