Background for Lab 4
This laboratory concerns, in part, the phenomenon of beats. Recall that a mass m hanging from a
spring and subject to a periodic force obeys the differential equation
my 00 + γy 0 + ky = F0 cos(ωt)
where
•
•
•
•
•
We have
y is the position of the mass measured with respect to its equilibrium position.
my 00 is the mass times acceleration term in Newton’s law.
γy 0 is (minus) the frictional force acting on the mass.
k is the spring constant and ky is (minus) the force exerted by the spring on the mass.
F0 cos(ωt) is the periodic external force.
seen in class that, if γ > 0, this equation has general solution of the form

−γt/2m
cos(µt − δ)
underdamped

 Ae
−γt/2m
−γt/2m
y(t) = R cos(ωt − δ) + C1 e
+ C2 te
critically damped


C1 e−r1 t + C2 e−r2 t
overdamped
and that the Ae−γt/2m cos(µt−δ) or C1 e−γt/2m +C2 te−γt/2m or C1 e−r1 t +C2 e−r2 t part of the solution
tends to zero exponentially as t → ∞. This part of the solution is called the transient part of the
solution. It damps to zero because of energy losses arising from the presence of the damping term γy 0
in the differential equation.
Beats arise when there is no damping (or at least the damping is small enough to be ignored),
so that γ = 0. Then the differential equation simplifies to
my 00 + ky = F0 cos(ωt)
As ert obeys the homogeneous equation my 00 + ky = 0 if and only if r = ±iω0 with ω0 =
00
iω0 t
−iω0 t
q
k
,
m
the
general solution of my + ky = 0 is D1 e
+ D2 e
, or equivalently C1 cos(ω0 t) + C2 sin(ω0 t)
(with C1 = D1 + D2 and C2 = iD1 − iD2 ). Subbing y = R cos(ωt) into my 00 + ky gives F0 cos(ωt) if
F0
F0
and only if R = k−mω
2 = m(ω 2 −ω 2 ) . (We shall assume that ω0 6= ω.) Hence the general solution of
0
my 00 + ky = F0 cos(ωt) is
y(t) = C1 cos(ω0 t) + C2 sin(ω0 t) +
F0
m(ω02 −ω 2 )
cos(ωt)
If the mass starts out at rest in its equilibrium position then y(0) = y 0 (0) = 0. These initial conditions
0
give C1 = − m(ωF2 −ω
2 ) and C2 = 0 so that
0
y(t) =
F0
m(ω02 −ω 2 )
0
cos(ωt) − cos(ω0 t) = 2 m(ωF2 −ω
2 ) sin
0
ω0 −ω
2 t
sin
ω+ω0
2 t
a+b
b−a
by the trig identity cos a−cos b = 2 sin
2 sin 2 . If the forcing frequency ω and the natural
frequency
ω0 −ω
ω+ω0
t
oscillates
at
frequency
roughly
ω,
while
sin
t
oscillates
ω0 are reasonably close, sin
2
2
0
quite slowly. Hence we can think of y(t) as a vibration with frequency ω+ω
≈
ω
and
amplitude
2
ω0 −ω
ω0 −ω
0
2 m(ωF2 −ω
t
.
The
amplitude
oscillates
slowly
in
time,
with
frequency
. Guitar players
2 ) sin
2
2
0
exploit this phenomenon, called beats, to tune their guitars. They play two neighbouring strings at
the same time, while placing a finger of their left hand on one string at the fret that should make the
two strings have the same pitch. If the guitar is not quite in tune, the pitches of the two strings are
not quite the same and the resulting sound gets alternately louder and softer with a frequency that
is half the difference of the frequencies produced by the two vibrating strings.