Simple Nature - Light and Matter

h/Dependence of the amplitude

and phase angle on the driving

frequency, for an undamped

oscillator. The amplitudes were

calculated withFm,m, andωo, all

set to 1.

the steady-state motion, we’re going to look for a solution of the
form
x=Asin(ωt+δ).
In contrast to the undriven case, here it’s not possible to sweepA
andδunder the rug. The amplitude of the steady-state motion,A,
is actually the most interesting thing to know about the steady-state
motion, and it’s not true that we still have a solution no matter how
we fiddle withA; if we have a solution for a certain value ofA, then
multiplyingAby some constant would break the equality between
the two sides of the equation of motion. It’s also no longer true that
we can get rid ofδsimply be redefining when we start the clock;
hereδrepresents adifferencein time between the start of one cycle
of the driving force and the start of the corresponding cycle of the
motion.
The velocity and acceleration arev=ωAcos(ωt+δ) anda=
−ω^2 Asin(ωt+δ), and if we plug these into the equation of motion,
[1], and simplify a little, we find

[2] (k−mω^2 ) sin(ωt+δ) +ωbcos(ωt+δ) =

Fm

A

sinωt.

The sum of any two sinusoidal functions with the same frequency
is also a sinusoidal, so the whole left side adds up to a sinusoidal.
By fiddling withAandδwe can make the amplitudes and phases
of the two sides of the equation match up.

Steady state, no damping

Aandδare easy to find in the case where there is no damping

at all. There are now no cosines in equation [2] above, only sines, so

if we wish we can setδto zero, and we findA=Fm/(k−mω^2 ) =

Fm/m(ω^2 o−ω^2 ). This, however, makesAnegative forω > ωo. The

variableδwas designed to represent this kind of phase relationship,

so we prefer to keepApositive and setδ= πforω > ωo. Our

results are then

A=

Fm

m|ω^2 −ω^2 o|

and

δ=

{

0, ω < ωo

π, ω > ωo

.

The most important feature of the result is that there is a reso-

nance: the amplitude becomes greater and greater, and approaches

infinity, asωapproaches the resonant frequencyωo. This is the phys-

ical behavior we anticipated on page 175 in the example of pushing

a child on a swing. If the driving frequency matches the frequency

of the free vibrations, then the driving force will always be in the

Section 3.3 Resonance 181