Fourier Series and Partial Differential Equations Lecture Notes

Chapter 4. The wave equation 35
where l, and τ are a typical lengthscale and timescale, respectively, for the problem under
consideration. Then
and substituting into the wave equation we have
∂ dt ∂ 1 ∂
= = ,
∂t dˆt ∂ˆt τ ∂ˆt
∂
(4.15)
2
dt ∂ 1 ∂
= =
∂t2 dˆt ∂ˆt τ ∂ˆt
1
τ2 ∂2 ∂ˆt 2,
(4.16)
∂ dx ∂ 1 ∂
= = ,
∂x dˆx ∂ˆx l ∂ˆx
∂
(4.17)
2
dx ∂ 1 ∂
= =
∂x2 dˆx ∂ˆx l ∂ˆx
1
l2 ∂2 ∂ˆx 2, (4.18)
l
τ2 ∂2ˆy ∂ˆt 2 = c2l l2 ∂2ˆy ∂ˆx 2.
Rearranging gives
∂ ˆ T
∂ˆt = c2τ2 l2 ∂2ˆ T
∂ˆx 2.
Considering the problem on a timescale where τ = l/c gives
Notice that now
since
∂ 2 ˆy
∂ˆt 2 = ∂2 ˆy
∂ˆx 2.
(4.19)
(4.20)
(4.21)
[ˆx] = 1, [ˆt] = 1, [y] = 1, (4.22)
[l] = m, [τ] =
l
= s. (4.23)
c
This gives a relationship between problems with different lengthscales and wave speeds.
4.3 Normal modes of vibration for a finite string
A string is stretched between x = 0 and x = L and the ends are held fixed. If the string
is plucked, what notes do we hear? The question suggests that we want a solution which
is periodic in time, with a period to be determined.
The displacement y(x,t) satisfies the wave equation
with boundary conditions
∂2y ∂t2 = c2∂2 y
∂x2, (4.24)
y(0,t) = 0 and y(L,t) = 0 for t > 0. (4.25)
Weusetheseparationofvariablestechniquei.e. weattempttofindsome(notall)solutions
of equations (4.24) and (4.25) in the separable form
y(x,t) = F(x)G(t). (4.26)