Question

Vibrating string equation physical interpretation of boundary condition.

(∂^2 u)/(∂t^2 )=a^2 (∂^2 u)/(∂x^2 ) x∈(0,l)
u(0,t)=0 ; ∂u/∂x(L,t)=0
u(x,o)=u1; ∂u/∂t(x,o)=x

Could someone please help me with 2. boundary condition ∂u/∂x(L,t)=0- what does it mean (physicaly) ?

Answer 1

∂u/∂x(L,t)=0 means that no perpendicular force is acting on right end of the string.

It means the string is free to move sideways and not constrained.
In order to achieve this, you need to tie its end to a small massless ring, able to slide frictionless on a rod.


You can see the force acted by the rope on the ring is parallel to x-axis and the tangent to the string is also horizontal.