Vibrations and Waves
Simple Harmonic Motion
What is it?
The motion of an object under the influence of either
a force characterized by Hook's law:
F = kx
or equivalently
a potential given by
PE_{sho} = (1/2)kx^{2}
where sho stands for simple harmonic oscillator.
Force and potential (energy) are two ways to describe a conservative interaction.
Force is equal to the negative of the slope of the corresponding potential.
Examples to be studied: The motion of an object attached to a spring and
the motion of a pendulum.
Why is it important?
Almost all types of motion that have an equilibrium position can be
approximated by an simple harmonic oscillator if their amplitudes are sufficiently
small.
The understanding of simple harmonic motion is also the basis for understanding
the wave motion.
What happens
in a simple harmonic motion?
Understanding based on Newton's second law:
a = F/m = (k/m)x

The motion is back and forth around the equilibrium position.

Acceleration is zero at the equilibrium position (where the velocity
is greatest), and it is greatest at the maxima distances away from
the equilibrium (where the velocity is zero).
Understanding based on the conservation of energy:
a) No friction (no damping)
E = KE + PE = const
(1/2)mv^{2} + (1/2)kx^{2} = E = const
from which it is clear that energy can be written in terms of the amplitude
as
E = (1/2)kA^{2}
It is also clear that the maximum velocity is achieved at the equilibrium
position (x=0) and
E = (1/2)mv_{max}^{2}
(1/2)mv_{max}^{2} = E = (1/2)kA^{2}
In a simple harmonic motion, kinetic energy and potential energy are converting
back and forth into each other. The energy is purely kinetic at the equilibrium
position, and it is pure potential energy at the maxima positions.
b) With friction
DE = W_{friction} = fs
Example 13.4 Motion with and without friction
Parameters
that characterize a simple harmonic motion
Amplitude A and frequency f.
Amplitude: the maximum distance traveled by an object away from its equilibrium
position.
Amplitude depends on the initial condition, i.e. how you get the motion
started; how much energy you put into the system.
Frequency: the number of cycles per unit time.
Frequency is an intrinsic property of the oscillator and is independent
of the initial conditions (as long as you do not stretch it too
hard).
f = (1/2p)(k/m)^{1/2}
This equation is derived by solving the Newton's equation of motion. The
solution gives us position as a function of time:
x = A cos w t
where
w = 2pf = (k/m)^{1/2}
is called the angular frequency (rad/s).
There is another parameter that is related to frequency. It is the period
T.
Period T: time it takes to complete one cycle.
T = 1/f = 2p(m/k)^{1/2}
(f = 1/T may be even easier to see).
Example 13.7 The vibrating massspring system
***
There is in fact a third parameter that characterizes a simple harmonic
motion. It is called phase and is determined by initial conditions.
The concept of phase is very important in understanding the interference
between two waves. Briefly, the more general description of displacement
as a function of time is
x = A cos(w t+f
)
in which f is called the phase.
***
Pendulum
A pendulum is a good example of the motion near an equilibrium being well
approximated by a simple harmonic oscillator. For small angles, the restoring
force for a pendulum is
F = mg sin q » mg q
= (mg/L)s
which corresponds to a simple harmonic oscillator with an effective spring
constant
k = mg/L
The period of a pendulum is therefore given by
T = 2p(L/g)^{1/2}
Example 13.8 What is the height of that tower?
Wave
What is it?
Propagation of energy or disturbance.
A spatially correlated oscillations.
Parameters that characterize
a (monochromatic) wave
Amplitude and two out of the following three parameters:

frequency f

velocity v

wavelength l
which are related by
v = f l
Using light (an electromagnetic wave) as an example, the amplitude is the
parameter that determines the intensity of the light and frequency (or
wavelength) is the parameter that determines the color of the light.
Example 13.9 A traveling wave
Transverse and longitudinal
waves
Transverse wave: a wave in which the displacement is perpendicular to the
direction of propagation.
Longitudinal wave: a wave in which the displacement is parallel to the direction
of propagation.
Figure 13.19 is a illustration of two types of waves.
Examples
a) Light (electromagnetic) wave
Light waves of all frequency travel at the same speed in vacuum. The speed
of light in vacuum:
c = 2.99792458 ´ 10^{8}
m/s
The speed of light in a medium is generally frequencydependent. This phenomenon
is called (or gives rise to what is called) dispersion.
Light wave is transverse in nature.
b) Transverse wave on a string
The speed of a transverse wave on a string is determined by
v = (F/m)^{1/2}
in which F is the tension in the string and m is
mass per unit length.
Example 13.12 A pulse traveling on a string
c) Sound wave
Sound waves can be either transverse or longitudinal.
Doppler effect
Superposition and interference of waves
The displacement at a point which is under the influence of multiple waves
is equal to the sum of displacements induced by each individual wave at
that point.
Constructive interference: The influences of two waves add to each other.
Destructive interference: The influences of two waves cancel each other.
Figure 13.25 illustrates constructive interference and Figure 13.26 illustrates
destructive interference.
The two waves in Figure 13.25 are said to be in phase (oscillations induced
by both waves at a specific point reach their maxima at the same time).
The two waves in Figure 13.26 are said to be 180 degrees out of phase (oscillation
induced by one of waves is at its minimum when the oscillation induced
by the other is at its maximum).
***
Waves are described mathematically by so called wave equations. Waves satisfying
the superposition principle are described by equations which are linear.
Maxwell's equation for an electromagnetic wave in vacuum and the Schrodinger
equation in quantum physics are examples of wave equations, both of which
are linear.
A medium may become nonlinear if the wave travelling through it is too
intense (amplitude too big).
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Copyright © 1997 by Bo Gao. All rights reserved.