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8.3 Neumann conditions

We now apply the method of separation of variables to solve the problem of a vibrating
string without external forces and with two clamped but free ends. Let u(x, t) be the
amplitude of the string at the point x and time t, and let f and g be the amplitude and
the velocity of the string at time t = 0. We need to solve the problem

2
u tt − c u xx = 0, 0 < x < L, t > 0, (8.16)
u x (0, t) = u x (L, t) = 0, t ≥ 0, (8.17)
u(x, 0) = f(x), 0 ≤ x ≤ L, (8.18)
u t (x, 0) = g(x), 0 ≤ x ≤ L, (8.19)
(8.20)

where f, g are given functions and c is a positive constant. The compatibility conditions
are given by
0
0
0
0
f (0) = f (L) = g (0) = g (L) = 0.
The problem (8.17)-(8.19) is a linear homogeneous initial boundary value problem. As
mentioned above, the conditions (8.17) are called Neumann boundary conditions.
Recall that at the first stage of the method, we compute nontrivial separated solutions
of the PDE (8.16), i.e. solutions of the form

u(x, t) = X(x)T(t), (8.21)

that also satisfy the boundary conditions (8.17). Here, as usual, X, T are functions of
the variables x and t respectively. At this stage, we do not take into account the initial
conditions (8.18)-(8.19).
Differentiating the separated solution (8.21) twice in x and twice in t, and then sub-
stituting these derivatives into the wave equation, we infer

2
XT tt = c X xx T.
By separating the variables, we see that

T tt X xx
= .
2
c T X
It follows that there exists a constant λ such that

T tt X xx
= = −λ. (8.22)
2
c T X
Equation (8.22) implies

2
d X
= −λX, 0 < x < L, (8.23)
dx 2
2
d T
2
= −λc T, t > 0. (8.24)
dt 2

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