This Demonstration analyzes in which way the (high-

limit) Lorentzian lineshapes of a driven damped harmonic oscillator differ from the exact resonance lineshapes.

The equation of motion of a damped harmonic oscillator (with mass

, eigenfrequency

, and damping constant

) driven by a periodic force

is

.

The general solution can be written as

,

and

.

The solution can thus be parametrized either by the amplitude and phase (|

|,

) or by the in-phase and quadrature components (

. The explicit frequency dependence of those parameters is obtained by inserting the general solution into the equation of motion, yielding

;

;

.

The expressions can be rewritten using the dimensionless frequency parameter ξ and the quality factor

, defined by

, and

, to yield

;

.

The four resonance lineshapes are shown in the plots as black solid lines. In order to avoid rescaling during the manipulation of the quality factor

, all signals are normalized to their largest value.

The high-

limit:

for

, that is, for

, the expressions can be simplified, yielding

;

;

;

,

or, in dimensionless units,

;

;

;

.

The corresponding lineshapes are shown as dashed blue lines.

An alternative parametrization consists in introducing the dimensionless detuning

, for which the resonances are given by

;

;

;

,

which shows that the phase and quadrature signals in the high-

limit are dispersive and absorptive Lorentzians, respectively.