Consider a string fixed at and , as in Fig. 3.2
It satisfies the wave equation
(3.14) |
with boundary conditions
(3.15) |
and initial conditions,
(3.16) |
Consider a string with ends fastened to air bearings that are fixed to a rod orthogonal to the -axis. Since the bearings float freely there should be no force along the rods, which means that the string is horizontal at the bearings, see Fig. 3.3 for a sketch.
It satisfies the wave equation with the same initial conditions as above, but the boundary conditions now are
(3.17) |
These are clearly of von Neumann type.
To illustrate mixed boundary conditions we make an even more complicated contraption where we fix the endpoints of the string to springs, with equilibrium at , see Fig. 3.4 for a sketch.
Hook’s law states that the force exerted by the spring (along the axis) is , where is the spring constant. This must be balanced by the force the string on the spring, which is equal to the tension in the string. The component parallel to the axis is , where is the angle with the horizontal, see Fig. 3.5 .
For small we have . Since both forces should cancel we find
These are mixed boundary conditions.