is also a solution of the driven equation. Here, as before,ωf is the
frequency of the free oscillations (ωf ≈ωofor smallQ),ωis the
frequency of the driving force,Aandδare related as usual to the
parameters of the driving force, andA′andδ′can have any values
at all. Given the initial position and velocity, we can always choose
A′andδ′to reproduce them, but this is not something one often
has to do in real life. What’s more important is to realize that the
second term dies out exponentially over time, decaying at the same
rate at which a free vibration would. For this reason, theA′term is
called a transient. A high-Qoscillator’s transients take a long time
to die out, while a low-Qoscillator always settles down to its steady
state very quickly.
Boomy bass example 51
In example 48 on page 187, I’ve already discussed one of the
drawbacks of a high-Q speaker, which is an uneven response
curve. Another problem is that in a high-Qspeaker, transients
take a long time to die out. The bleeding-eardrums crowd tend
to focus mostly on making their bass loud, so it’s usually their
woofers that have highQs. The result is that bass notes, “ring”
after the onset of the note, a phenomenon referred to as “boomy
bass.”Overdamped motion
The treatment of free, damped motion on page 178 skipped
over a subtle point: in the equation ωf =√
k/m−b^2 / 4 m^2 =
ωo√
1 − 1 / 4 Q^2 ,Q < 1 /2 results in an answer that is the square
root of a negative number. For example, suppose we hadk= 0,
which corresponds to a neutral equilibrium. A physical example
would be a mass sitting in a tub of syrup. If we set it in motion,
it won’t oscillate — it will simply slow to a stop. This system has
Q= 0. The equation of motion in this case isma+bv= 0, or, more
suggestively,
m
dv
dt
+bv= 0.One can easily verify that this has the solutionv= (constant)e−bt/m,
and integrating, we findx= (constant)e−bt/m+ (constant). In other
words, the reasonωf comes out to be mathematical nonsense^14 is
that we were incorrect in assuming a solution that oscillated at a
frequencyωf. The actual motion is not oscillatory at all.
In general, systems withQ < 1 /2, called overdamped systems,
do not display oscillatory motion. Most cars’ shock absorbers are
designed withQ≈ 1 /2, since it’s undesirable for the car to undulate
up and down for a while after you go over a bump. (Shocks with ex-
tremely low values ofQare not good either, because such a system
takes a very long time to come back to equilibrium.) It’s not par-
(^14) Actually, if you know about complex numbers and Euler’s theorem, it’s not
quite so nonsensical.
Section 3.3 Resonance 189