The Chemistry Maths Book, Second Edition

364 Chapter 12Second-order differential equations. Constant coefficients

Then

and

This is equal toA 1 cos 1 ωtwhend 1 = 10 andc 1 = 1 A 2 (ω

2

0

1 − 1 ω

2

). The particular integral is

therefore

(12.85)

and the general solution of equation (12.83) for the oscillator in the external field is

x(t) 1 = 1 x

h

(t) 1 + 1 x

p

(t)

(12.86)

The solution shows that the behaviour of the system depends strongly on the relative

values ofωandω

0

. In particular, when the frequencyωof the external force is close

to the natural frequency of the oscillator, the maximum amplitude of oscillation (the

maximum value ofx(t)), is large, and tends to infinity asω 1 → 1 ω

0

. This phenomenon

of large oscillations is called resonance, and is an important factor in the study of

vibrating systems in both classical and quantum mechanics.

In the case of resonance, whenω 1 = 1 ω

0

, the function (12.85) is not the required

particular integral because it is already a solution of the homogeneous equation. The

required function is (by prescription (a)in Section 12.8)

x

p

(t) 1 = 1 t(c 1 cos 1 ω

0

t 1 + 1 d 1 sin 1 ω

0

t) (12.87)

Substitution in (12.83) then givesc 1 = 10 andd 1 = 1 A 22 πω

0

, and the particular integral

in this case is

(12.88)

The graph of this function, in Figure 12.9, shows how in the case of resonance the

amplitude of the oscillation increases indefinitely with time. In mechanical systems

resonance may be avoided by the application of a suitable damping force.

xt

A

tt

p

()= sin

2

0

0

ω

ω

=++

−

dtdt

At

1020

0

22

cos sin

cos

ωω

ω

ωω

xt

At

p

()

cos

=

−

ω

ωω

0

22

dx

dt

xctdt

p

p

2

2

0

2

0

22

+=−ωωω ω ω()(cossin)+

dx

dt

ctd t

dx

dt

ctd

pp

=−ωωω ωsin + cos , =−ω( cosω+

2

2

2

ssin )ωt