Page 93 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SOLUTIONS USING FOURIER SERIES AND INTEGRALS 83
or

2
r(rR) ∓ λ R = 0

(5.47)
2
2

[(1 − x )X ] ± λ X = 0
The second of these is Legendre’s equation, and we have seen that it has bounded solutions at
2
r = 1 when ±λ = n(n + 1). The first equation is of the Cauchy–Euler type with solution
n
R = C 1 r + C 2 r −n−1 (5.48)
Noting that the constant C 2 must be zero to obtain a bounded solution at r = 0, and using
superposition,

∞
n
u = K n r P n (x) (5.49)
n=0
and using the condition at f (r = 1) and the orthogonality of the Legendre polynomial
π π

2 2K n
f (θ)P n (cos θ)dθ = K n P (cos θ)dθ = (5.50)
n
2n + 1
θ=0 θ=0
π
2n + 1
K n = f (θ)P n (cos θ)dθ (5.51)
2
θ=0

5.2 VIBRATIONS PROBLEMS
We now consider some vibrations problems. In Chapter 2 we found a solution for a vibrating
string initially displaced. We now consider the problem of a string forced by a sine function.
Example 5.6 (Resonance in a vibration problem). Equation (1.21) in Chapter 1 is

2
y tt = a y xx + A sin(ηt) (5.52)
Select a length scale as L, the length of the string, and a time scale L/a and defining
ξ = x/L and τ = ta/L,

y ττ = y ξξ + C sin(ωτ) (5.53)

2 2
where ω is a dimensionless frequency, ηL/a and C = AL a .
The boundary conditions and initial velocity and displacement are all zero, so the bound-
ary conditions are all homogeneous, while the differential equation is not. Back in Chapter 2 we