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Supplementary Problems

Solution: It is a problem on longitudinal oscillations of rod, where one end is rigidly
fixed, and the second one contains load by weight M. The rod was displaced by

the size h (some force that was removed). The rod was at rest.

Exercise 90. Formulate a problem on transverse vibrations of infinite string, if in

the initial moment of time it was stationary and it has parabolic form:
Solution:
2
2
∂ u ∂ u E
2
= a 2 , a = .
∂t 2 ∂x 2 ρ


0, x < 0,




u(x, 0) = (l − x), −l < x < l,
 4hx

l 2

0, x > l,





u(0, t) = 0.


4.2 Supplementary Problems
Exercise 91. Carefully derive the equation of a string in a medium in which the
resist-ance is proportional to the velocity.

Exercise 92. A flexible chain of length l is hanging from one end x = 0 but
oscillates horizontally. Let the x axis point downward and the u axis point to the
right. Assume that the force of gravity at each point of the chain equals the weight
of the part of the chain below the point and is directed tangentially along the chain.
Assume that the oscillation are small. Find the PDE satisfied by the chain.

Exercise 93. On the sides of a thin rod, heat exchange takes place (obeying New-
ton’slaw of cooling — flux proportional to temperature difference) with a medium of

constant temperature T . What is the equation satisfied by the temperature u(x, t),
0
neglecting its variation across the rod?

Exercise 94. Suppose that some particles which are suspended in a liquid medium
would be pulled down at the constant velocity V > 0 by gravity in the absence of
diffusion. Taking account of the diffusion, find the equation for the concentration
of particles. Assume homogeneity in the horizontal directions x and y. Let the z

axis point upwards.

Exercise 95. Derive the equation of one-dimensional diffusion in a medium that

is moving along the x-axis to the right at constant speed V.

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