Page 28 - 6637

Chapter 4

Flows, Vibrations and Diffusions

4.1 Solved Problems

Exercise 88. Formulate a problem on logitudinal oscillation in the rod with the
length l, where one end is rigidly fixed, and the stretching force F is acting by the

free end. And at the moment of time t = 0 the force effects suddenly stop.
Solution: The equation describing longitudinal vibrations in the rod has the form:

2
2
∂ u ∂ u E
2
= a 2 , a = .
∂t 2 ∂x 2 ρ
If the force acts on the left end one has
∂u F
= .
∂x ES

Intagrating the equation by x we obtain u = Fx . Hence, the initial conditions are
ES
following
(
u(x, 0) = Fx ,
ES
∂u(x,0) = 0,
∂t
and boundary conditions
(
u(0, t) = 0,
∂u(l,t) = 0.
∂x

Exercise 89. Formulate a problem of mathematical physics concerning the equa-
tion:
2
2
∂ u 2 ∂ u 2 E
= a , a = .
∂t 2 ∂x 2 ρ
( (
u(x, 0) = hx , u(0, t) = 0,
l
2
∂u(x,0) = 0, ∂u(l,t) + M ∂ u(l,t) = 0.
∂t ∂x ES ∂t 2

24