Lesson 4: Standing
Waves
Traveling Waves vs. Standing
Waves
A mechanical wave is a disturbance which is created by a
vibrating object and subsequently travels through a medium
from one location to another, transporting energy as it
moves. The mechanism by which a mechanical wave propagates
itself through a medium involves particle interaction; one
particle applies a push or pull on its adjacent neighbor,
causing a displacement of that neighbor from the equilibrium
or rest position.
As a wave is observed traveling through a medium, a crest is
seen moving along from particle to particle. This crest is
followed by a trough which is in turn followed by the next
crest. In fact, one would observe a distinct wave pattern
(in the form of a sine wave) traveling through the medium.
This sine wave pattern continues to move in uninterrupted
fashion until it encounters another
wave along the medium or until it encounters a
boundary with another medium. This type of wave pattern
which is seen traveling through a medium is sometimes
referred to as a traveling
wave.
Traveling waves are observed when a wave
is not confined to a given space along the medium. The most
commonly observed traveling wave is an ocean wave. If a wave
is introduced into an elastic chord with its ends held 3
meters apart, a wave introduced on one end becomes confined
in a small region. This wave will quickly reach the end of
the chord, reflect and travel back in the opposite
direction. Any reflected portion of the wave will then
interfere with the portion of the
wave incident towards the fixed end. This interference
produces a new shape in the medium which seldom resembles
the shape of a sine wave. Subsequently, a traveling wave (a
repeating pattern which is observed to move through a medium
in uninterrupted fashion) is not observed in the chord.
Indeed there are traveling waves in the chord, it is just
that they are not easily detectable. In such instances,
rather than observing the pure shape of a sine wave pattern,
a rather irregular and non-repeating pattern is produced in
the chord which tends to change appearance over time. This
irregular looking shape is the result of the interference of
an incident sine wave pattern with a reflected sine wave
pattern in a rather non-sequenced and untimely manner. Both
the incident and reflected wave patterns continue their
motion through the medium, meeting up with one another at
different locations in different ways. For example, the
middle of the chord might experience a crest meeting a
half crest; then moments later, a crest meeting a
quarter trough; then moments later, a
three-quarters crest meeting a one-fifth
trough, etc. This interference leads to a very irregular
and non-repeating motion of the medium. The appearance of an
actual wave pattern is difficult to detect amidst the
irregular motions of the individual particles.
It is possible however to have a wave
confined to a given space in a medium and still produce a
regular wave pattern which is readily discernible amidst the
motion of the medium. For instance, if an elastic rope is
held end to end and vibrated at just the right
frequency, a wave pattern would be produced which
assumes the shape of a sine wave and is seen to change over
time. The wave pattern is only produced when one end of the
rope is vibrated at just the right frequency. When the
proper frequency is used, the interference of the incident
wave and the reflected wave occur in such a manner that
there are specific points along the medium which appear to
be standing still. Because the observed wave pattern is
characterized by points which appear to be standing still,
the pattern is often called a
standing wave pattern.
There are other points along the medium whose displacement
changes over time, but in a regular manner. These points
vibrate back and forth from a positive displacement to a
negative displacement; the vibrations occur at regular time
intervals such that the motion of the medium is regular and
repeating - a pattern is readily observable.
The
diagram at the right depicts a standing wave pattern in a
medium. A snapshot of the medium over time is depicted using
various colors. Note that point A on the medium moves from a
positive to a negative displacement over time. The diagram
only shows one-half cycle of the motion of the standing wave
pattern. The motion would continue and persist, with point A
returning to the same positive displacement and then
continuing its back-and-forth vibration between the up to
the down position. Note that point B on the medium is a
point which never moves. Point B is a point of no
displacement; such points are known as
nodes and will be
discussed in more detail later in this
lesson. The standing wave pattern which is shown at the
right is just one of many different patterns which could be
produced within the rope. Other patterns will be discussed
later in the lesson.
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