Small transverse oscillations of a heavy uniform chain of length

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Ordinary differential equations

Conditions of Validity, Ch Sturm-Liouville theory: the spectral theorem for a differential operator

Problem

Small transverse oscillations

y(x,t) = Y(x)e^{-i\omega t}

of a heavy uniform chain of length

L

, hanging under gravity

g

with its upper end clamped at

x = L

and its lower end free at

x = 0

, obey the eigenvalue problem

-\big(xY'\big)' = \frac{\omega^2}{g}Y, \qquad 0 < x \le L,

with

Y(L) = 0

and

Y

required to stay bounded as

x \to 0^+

. ((a)) Identify

p

q

and

w

for this problem, and explain why no ordinary boundary condition is imposed at

x=0

and what the boundedness requirement does there instead. ((b)) Two modes

Y_m

Y_n

with distinct eigenvalues

\omega_m \ne \omega_n

both satisfy the stated conditions. Starting from

-(xY_m')' = \frac{\omega_m^2}{g}Y_m, \qquad -(xY_n')' = \frac{\omega_n^2}{g}Y_n,

derive that

\int_0^L Y_m(x)Y_n(x)\,\mathrm dx = 0