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190 4. The method of multiple scales

Also, from the equations that define the third term in the expansions, show
that the solution is uniformly valid (as only if
Q4.7 Forcing near resonance I. Consider an oscillation described by a Duffing equation
with (weak) forcing:

with is the given frequency of the forcing and the initial condi-
tions are Given that (where is
a constant independent of find the equation which describes completely the
first term of a uniformly valid asymptotic expansion. [Hint: write the forcing
term as where are the fast/slow scales, respectively.]
Q4.8 Forcing near resonance II. See Q4.7; repeat this for in the equa-
tion

(see Q4.4), where is a constant independent of (and note the appearance of
subharmonics).
Q4.9 Failure of the method of multiple scales. An oscillator is described by the equation

with Introduce and (with and
analyse as far as the term at which ensures the complete description
of the solution as far as Show that a uniformly valid solution cannot be
obtained using this approach. (You may wish to investigate why this happens
by examining the energy integral for the motion.)
Q4.10 Nonlinear oscillation I. A (fully) nonlinear oscillation is described by the equation

with Show that a solution with is x = a cn[4K(m)t; m] for a
suitable relation between a and m. Now use the method of multiple scales,
with the scales T (where and to find the first term of
an asymptotic expansion which is periodic in T. (The periodicity condition
should be written as an integral, but this does not need to be evaluated.)
Q4.11 Nonlinear oscillation II. See Q4.10; follow this same procedure for the equation

where a solution with can be written for
suitable a, b and m.