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A vibrating object undergoes simple harmonic motion (SHM) if the
restoring force is proportional to the displacement,
F = - kx.
The maximum displacement is called the amplitude.
The period, T, is the time required for one complete cycle
(back and forth), and the frequency,f, is the number of cycles per
second; they are related by
f = 1/T
The period of vibration for a mass m on the end of a spring
is given by
T
= 2p(m/k)1/2
SHM is sinusoidal, which means that the displacement as a
function of time follows a sine or cosine curve.
During SHM, the total energy E = ½ mv2 + ½ kx2 is continually changing from potential to kinetic and back again.
A simple pendulum of length L approximates SHM if its
amplitude is small and friction can be ignored. Its period is then
given by (for small amplitudes)
T = 2p(L/g)1/2
where g is the acceleration of gravity.
When friction is present (for all real springs and pendulums), the
motion is said to be damped. The maximum displacement decreases in
time, and the energy is eventually all transformed to heat.
If an oscillating force is applied to a system capable of
vibrating, the system's amplitude of vibration can be very large if
the frequency of the applied force matches the natural (or resonant)
frequency of the oscillator. This is called resonance.
Vibrating objects act as sources of waves that travel outward from
the source. Waves on water and on a string are examples. The wave may
be a pulse (a single crest) or it may be continuous (many crests and
troughs).
The wavelength of a continuous sinusoidal wave is the distance between
two successive crests.
The frequency is the number of full wavelengths (or crests) that pass
a given point per unit time.
The wave velocity (how fast a crest moves) is equal to the product of
wavelength and frequency,
v = lf
The amplitude of a wave is the maximum. height of a crest, or depth
of a trough, relative to the normal (or equilibrium) level.
In a transverse wave, the oscillations are perpendicular to the
direction in which the wave travels. An example is a wave on a
string.
In a longitudinal wave, the oscillations are along (parallel to)
the line of travel; sound is an example.
Waves reflect off objects in their path. When the wave front (of a
two- or three-dimensional wave) strikes an object, the angle of
reflection is equal to the angle of incidence. When a wave strikes a
boundary between two materials in which it can travel, part of the
wave is reflected and part is transmitted.
When two waves pass through the same region of space at the same
time, they interfere. The resultant displacement at any point and
time is the sum of their separate displacements; this can result in
constructive interference, destructive interference, or something in
between, depending on the amplitudes and relative phases of the
waves.
Waves traveling on a string (or other medium) of fixed length
interfere with waves that have reflected off the end and are
traveling back in the opposite direction. At certain frequencies,
standing waves can be produced in which the waves seem to be standing
still rather than traveling. The string (or other medium) is vibrating
as a whole. This is a resonance phenomenon and the frequencies at
which standing waves occur are called resonant frequencies. The points
of destructive interference (no vibration) are called nodes. Points of
constructive interference (maximum amplitude of vibration) are
called antinodes.
Sound travels as a longitudinal wave in air and other materials. In
air, the speed of sound increases with temperature; at 20°C, it is
about 343 m/s.
The pitch of a sound is determined by the frequency; the higher
the frequency, the higher the pitch.
The audible range of frequencies for humans is roughly 20 to 20,000
Hz (1 Hz = 1 cycle per second).
The loudness or intensity of a sound is related to the amplitude of
the wave. Because the human ear can detect sound intensities from 10-12
W/m2 to over 1 W/m2, intensity levels are
specified on a logarithmic scale. The intensity level,
b, specified in decibels, is defined in terms of intensity I as
b = 10 log (I/Io), where the reference intensity 10 is usually taken to be 10-12
W/m2.
Musical instruments are simple sources of sound in which standing
waves are produced.
The strings of a stringed instrument may vibrate as a whole with
nodes only at the ends; the frequency at which this occurs is called
the fundamental. The string can also vibrate at higher frequencies,
called overtones or harmonics, in which there are one or more
additional nodes. The frequency of each harmonic is a whole-number
multiple of the fundamental.
In wind instruments, standing waves are set up in the column of air
within the tube.
The vibrating air in an open tube (open at both ends) has
displacement antinodes at both ends. The fundamental frequency
corresponds to a wavelength equal to twice the tube length. The
harmonics have frequencies that are 2, 3, 4, . .. times the fundamental frequency, just as for strings.
For a closed tube (closed at one end), the fundamental corresponds
to a wavelength four times the length of the tube. Only the odd
harmonics are present, equal to 1, 3, 5, 7, times the fundamental
frequency.
Sound waves from different sources can interfere with each other.
If two sounds are at slightly different frequencies, beats can be
heard at a frequency equal to the difference in frequency of the two
sources.
The Doppler effect refers to the change in pitch of a sound due to
the motion either of the source or of the listener. If they are
approaching each other, the pitch is higher; if they are moving apart,
the pitch is lower. |