I have a confusion regarding DIRICHLET and NEUMANN boundary conditions in vibrating string. Is it possible for a vibrating string to have inhomogeneous Dirichlet boundary conditions. As in "partial differential equations" by Walter A. Strauss, It is written that if one end of the string is free to move simply (not transversely) then, it would have nonhomogeneous Dirichlet BCs at that end. Now, what does it mean by "moving SIMPLY". As there are only two possible directions for a vibrating string to move; transversely or horizontally. If it moves horizontally then it would no more remain an END point as it has moved in the direction of x. Same is in the case of vibrating drum head(two dimensional wave equation).
The second question is that if we say that on one end u(x,t), the solution of wave equation, has inhomogeneous Dirichlet boundary condition i.e., u(x,t) is specified by a function f(t), then its derivative in normal direction (in direction of x, in this case) is also zero as it is depending purely on t. Thus we can say that at that end, ∂u/∂x is zero; which is Neumann boundary condition. So do both types of BCs coincide in this case???
thank you
