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and initial conditions,
∂u
u(x, 0) = f(x), (x, 0) = g(x).
∂x

Example 5.2 A string with freely floating endpoints Consider a string with ends
fastened to air bearings that are fixed to a rod orthogonal to the x-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. 5.5 for a sketch.

x = 0

x = a

Figure 5.5: A string with floating endpoints

It satisfies the wave equation with the same initial conditions as above, but the
boundary conditions now are

∂u ∂u
(0, t) = (a, t) = 0, t > 0.
∂x ∂x
These are clearly of von Neumann type.

Example 5.3 A string with endpoints fixed to strings 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 y = 0, see Fig. 5.6 for a sketch.

x = a

x = 0

Figure 5.6: A string with endpoints fixed to springs

Hook’s law states that the force exerted by the spring (along the y axis) is F =
−ku(0, t), where k is the spring constant. This must be balanced by the force the
string on the spring, which is equal to the tension T in the string. The component
parallel to the y axis is T sin α, where α is the angle with the horizontal, see Fig. 5.7

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