Geography Reference
In-Depth Information
property of the harmonic oscillator is that the period, or time required to execute
a single oscillation, is independent of the amplitude of the oscillation. For most
natural vibratory systems, this condition holds only for oscillations of sufficiently
small amplitude. The classical example of such a system is the simple pendulum
(Fig. 7.1) consisting of a mass M suspended by a massless string of length l, free to
perform small oscillations about the equilibrium position θ
=
0. The component
of the gravity force parallel to the direction of motion is
−
Mg sin θ . Thus, the
equation of motion for the mass M is
Ml
d
2
θ
dt
2
=−
Mg sin θ
≈
Now for small displacements, sin θ
θ so that the governing equation becomes
d
2
θ
dt
2
ν
2
θ
+
=
0
(7.1)
where ν
2
≡
g/l. The harmonic oscillator equation (7.1) has the general solution
=
+
=
−
θ
θ
1
cos νt
θ
2
sin νt
θ
0
cos (νt
α)
where θ
1
,θ
2
,θ
0
, and α are constants determined by the initial conditions (see
Problem 7.1) and ν is the frequency of oscillation. The complete solution can thus
be expressed in terms of an amplitude θ
0
and a phase φ(t)
=
νt
−
α. The phase
varies linearly in time by a factor of 2π radians per wave period.
Propagating waves can also be characterized by their amplitudes and phases.
In a propagating wave, however, phase depends not only on time, but on one or
more space variables as well. Thus, for a one-dimensional wave propagating in
the x direction, φ(x,t)
α. Here the
wave number
, k, is defined as
2π divided by the wavelength. For propagating waves the phase is constant for an
=
kx
−
νt
−
Fig. 7.1
A simple pendulum.
