PHYS 302
: Unit 1 Reading Notes –
Athabasca University
1
PHYS 302: Vibrations and Waves – Unit 1 Reading Notes
French, A. P.
Vibrations and Waves
. New York: W.W. Norton & Company, 1971.
Chapter 1: Periodic Motions (pp. 316).
Chapter 3: The Free Vibrations of Physical Systems (pp. 4162, up to “The Decay of Free Vibrations”).
Chapter 1: Periodic Motions
(p. 3) The introduction gives several examples of oscillatory systems, stressing the idea of
periodicity
: the repetition
of a pattern regularly in time. The period of vibration is given the symbol
T
.
Sinusoidal Vibrations
(p. 4)
Page 5 gives a general expression for a restoring force on an object near equilibrium:
2
2
dx
dt
mk
x
. The linear term
in this expression is often the most important one. Consider Hooke’s law for extension of a spring. In this law, the
restoring force is F=–kx, which is linear. When we apply Newton’s second law (in the form ma=F) to such a
situation, we get
2
2
dt
x
. The general solution,
)
cos(
t
A
x
, differs from Eq. 11 only in notation
and could be converted to the form in that equation by the substitution
δ
→
φ
0

π
/2 (recall, or show with a small
sketch, that sin(
θ
+
π
/2)=cos
θ
.
δ
or
φ
0
are initial phase angles, determined by the conditions at time
t
=0). For this to
be the solution, we require
k
m
, which is the
angular
frequency of the motion, since
ω
t
is an angle (in
radians).
Sinusoidal vibrations arise naturally for small amplitudes. All periodic phenomena can be built up from suitable
combinations of sinusoidal variations of certain frequencies, amplitudes, and phases.
The Description of Simple Harmonic Motion
(p. 5)
Eq. 11 can be written
0
sin( (
))
xA
t
, so that it is at time
t
=
φ
0
/
ω
that the displacement is first 0 (see Fig.
12). If the motion actually began (physically) at time
t
=0, as suggested by the dashed lines before this time, then
φ
0
is the angular argument of the sin function at
t
=0; since the argument is called the
phase
, this is the
initial phase
.
Since the amplitude of the sin function itself is 1, it must be multiplied by
A
to describe motion of amplitude
A
.
When the phase increases by 2
π
, the sin function repeats. To make the phase increase by 2
π
,
t
must increase to
t+2
π
/
ω
. This corresponds to one period of the sin function, so the period
T
=2
π
/
ω
. It is stated without proof that
motion is specified if the initial position and velocity are given, and the general form for velocity is the time
derivative of Eq. 11,
00
(s
i
n
(
)
c
o
s
(
)
d
dt
At
A
t