A standing wave pattern is a vibrational
pattern created within a medium when the vibrational frequency of the source
causes reflected waves from one end of the medium to interfere with incident waves
from the source. This interference occurs in such a manner that specific points
along the medium appear to be standing still. Because the observed wave pattern
is characterized by points that appear to be standing still, the pattern is
often called a standing wave pattern. Such patterns are
only created within the medium at specific frequencies of vibration. These
frequencies are known as harmonic frequencies, or merely harmonics. At any frequency other than a harmonic frequency, the interference of
reflected and incident waves leads to a resulting disturbance of the medium
that is irregular and non-repeating.
But how are standing wave formations formed? And why are they
only formed when the medium is vibrated at specific frequencies? And what makes
these so-called harmonic frequencies so special and magical? To answer these questions, let's consider asnakey stretched
across the room, approximately 4-meters from end to end. (A "snakey" is a slinky-like device that consists of a
large concentration of small-diameter metal coils.) If an upward displaced
pulse is introduced at the left end of the snakey,
it will travel rightward across the snakey until
it reaches the fixed end on the right side of the snakey.
Upon reaching the fixed end, the single pulse will reflect and undergo
inversion. That is, the upward displaced pulse will become a downward displaced
pulse. Now suppose that a second upward displaced pulse is introduced into
the snakey at the precise moment that the
first crest undergoes its fixed end reflection. If this is done with
perfect timing, a rightward moving, upward displaced pulse will meet up with a
leftward moving, downward displaced pulse in the exact middle of the snakey. As the two pulses pass through each other, they
will undergo destructive interference. Thus, a point of no displacement in the exact
middle of the snakey will be produced. The
animation below shows several snapshots of the meeting of the two pulses at
various stages in their interference. The individual pulses are drawn in blue
and red; the resulting shape of the medium (as found by the principle of
superposition) is shown in green. Note that there is a point on the diagram in
the exact middle of the medium that never experiences any displacement from the
equilibrium position.
An upward displaced pulse introduced at one end will
destructively interfere in the exact middle of the snakey with
a second upward displaced pulse introduced from the same end if the
introduction of the second pulse is performed with perfect timing. The same
rationale could be applied to two downward displaced pulses introduced from the
same end. If the second pulse is introduced at precisely the moment that the
first pulse is reflecting from the fixed end, then destructive interference
will occur in the exact middle of the snakey.
The above discussion only explains why two pulses might interfere
destructively to produce a point of no displacement in the middle of the snakey. A wave is certainly different than a pulse. What if
there are two waves traveling in the medium? Understanding why two waves
introduced into a medium with perfect timing might produce a point of
displacement in the middle of the medium is a mere extension of the above
discussion. While a pulse is a single disturbance that moves through a medium,
a wave is a repeating pattern of crests and troughs. Thus, a wave can be
thought of as an upward displaced pulse (crest) followed by a downward
displaced pulse (trough) followed by an upward displaced pulse (crest) followed
by a downward displaced pulse (trough) followed by... . Since
the introduction of a crest is followed by the introduction of a trough, every
crest and trough will destructively interfere in such a way that the middle of
the medium is a point of no displacement.
Of course, this all demands that the timing is perfect. In
the above discussion, perfect timing was achieved if every wave crest was
introduced into the snakey at the precise
time that the previous wave crest began its reflection at the fixed end. In
this situation, there will be one complete wavelength within the snakey moving to the right at every instant in time;
this incident wave will meet up with one complete wavelength moving to the left
at every instant in time. Under these conditions, destructive interference
always occurs in the middle of thesnakey. Either a
full crest meets a full trough or a half-crest meets a half-trough or a quarter-crest meets a quarter-trough at this point. The
animation below represents several snapshots of two waves traveling in opposite
directions along the same medium. The waves are interfering in such a manner
that there are points of no displacement produced at the same positions along the
medium. These points along the medium are known as nodes and are labeled with an N. There are also points along the medium that vibrate back and forth
between points of large positive displacement and points of large negative
displacement. These points are known as antinodes and are labeled with an AN. The two individual waves are drawn in blue and
green and the resulting shape of the medium is drawn in black.
There are other ways to achieve this perfect timing. The main idea behind the
timing is to introduce a crest at the instant that another crest is either at
the halfway point across the medium or at the end of the medium. Regardless of
the number of crests and troughs that are in between, if a crest is introduced
at the instant another crest is undergoing its fixed end reflection, a node
(point of no displacement) will be formed in the middle of the medium. The
number of other nodes that will be present along the medium is dependent upon
the number of crests that were present in between the two timed crests. If a crest is introduced at the instant another crest is at the
halfway point across the medium, then an antinode (point of maximum
displacement) will be formed in the middle of the medium by means of
constructive interference. In such an instance, there might also be nodes and
antinodes located elsewhere along the medium.
A standing wave pattern is an interference phenomenon. It is
formed as the result of the perfectly timed interference of two waves passing
through the same medium. A standing wave pattern is not actually a wave; rather
it is the pattern resulting from the presence of two waves (sometimes more) of
the same frequency with different directions of travel within the same medium.
The physics of musical instruments has a basis in the conceptual and
mathematical aspects of standing waves. For this reason, the topic will be
revisited in the Sound and Music unit at The Physics
Classroom Tutorial.