STANDING WAVES of a VIBRATING STRING for an ideal Fixed Fixed String

Opposite in Fig 8, a string is attached and stretched across a solid "U" shaped structure.  The string is stretched by an amount where it will naturally vibrate at some predetermined pitch; a musical note is produced when struck.  The first harmonic wave pattern produced when the string is struck and set to vibrate, is the Fundamental Frequency (f) as shown, in the top diagram.  Other harmonic wave patterns are also produced, as shown in Fig 8 numbered 1st, 2nd, 3rd, and 4th, as previously said are called overtones. The overtones occur naturally between each cycle of the fundamental frequency f and are multiples of f, i.e., 2xf, 3xf, 4xf... nxf.    Only the first four overtones are shown here of several.   These wave patterns called standing waves are made up of segments, the area as labelled in the top wave pattern for Fig 8. The first harmonic wave pattern of the top diagram has only one segment, where as the next harmonic in the series the 1st overtone has two segments.  A segment is contained between two adjacent stationary nodes, with its greatest motion of vibration occurring at the antinodes (crests & troughs). Of particular interest for the first harmonic, is shown the Elastic Energy (E) of the string, when set to vibrate.  The value of energy E placed into the string when struck has amplitude as indicated by the arrows, shown within the central area for the first harmonic segment.

As with all waves, the greater the magnitude of the crests & troughs or antinodes are of a vibrating string, the more energy they will have available to affect other objects.  I.e. how strongly the guitar soundboard will be made to vibrate.  Simply the amplitude of the antinodes all just depends on how strongly the string is struck or plucked, by the musician.  But unlike this instantaneous control of amplification by the musicians hand, tonal sustain (the duration of the vibrating string) and its related Elastic Energy levels of each harmonic, is only available to the musician by the integrity of the  bracing structure (of the soundboard), holding the string tension!  In the real world of vibrating strings for a guitar, both ends of the string length are not firmly fixed. The attached end of the string at the bridge, actually oscillates in three directions. The attached string end at the bridge, is considered to be oscillating in these three directions when excited, due to the static Mass-Loading of the string. Therefore, the analysis for a Fixed, Mass-Loaded String, is much more complex than for our above case of a Fixed, Fixed String.  The behaviour of an oscillating attached string end at the bridge saddle, and Tonal sustain will be discussed further within this text; but first I think the following is an important read!  

To Understand the Behavioural Motion of Standing Wave Patterns on a Vibrating String, may seem somewhat confusing, and is by no means an easy to understand process.  But a short specific description is as follows:

A standing wave is a stationary wave; it does not travel along the medium (string).   A standing wave occurs when two waves of the same wavelength, travel along a medium in opposite directions to each other. (The origin of these two opposite travelling waves on a vibrating string length is discussed in WAVE REFLECTION below).   As the two waves pass each other in opposite directions, crests from one wave and troughs from the other wave will cancel any standing wave pattern to zero vibration.  Then as crests and opposing troughs start to slip past each other, an upper segment part of a standing wave begins to grow from the crests of each travelling wave lining up.  As the two travelling waves continue in opposite directions, crest and troughs line up again forcing the standing wave string vibration back down to zero.   After which the troughs will line up, of both opposite travelling waves and cause the bottom segment of the standing wave to form. The discussion for Fig 5, how waves affect each other shows these effects.  A standing wave pattern pulsates on and off; it's either forming or collapsing to zero at any given moment.  Ok; so why do they look like the diagrams in Fig 8 and Fig 8.5?  The upper and bottom parts of the standing wave take shape so quickly; that we can't even see them dropping back to zero, instead what we do see is a fuzzy segment.


WAVE REFLECTION

What's missing from the above discussion is why and how come there is, two separate waves travelling in opposite directions on a vibrating string.  Simply when the string is struck (say near the guitar bridge), a wave pulse is sent along the string.  This wave pulse is inversely reflected (turned upside down) at the other fixed end of the string, and then travels back to the bridge saddle, fixed end.  The wave pulse keeps going round and inverting its wave shape each time it reflects from a fixed end.  Now meanwhile the bridge is actually been set to oscillate (vibrate) so there are more wave pulses being generated, consequently a sinusoidal wave has now taken shape and is transmitting along the string one way and inversely reflecting back the other way, and so now the required condition for standing waves to form is set up!  Only the Fundamentals natural harmonic series can sync together to form standing wave patterns.

The above standing wave patterns of Fig 8 occur very quickly, after an initial settling in period from striking the string.  Standing wave patterns can vary interestingly enough; either odd or even numbered harmonic standing wave patterns are removed, if you should pluck a string rather than strike it, on any of the natural nodal positions.  For instance if you were to pull a string up and let it go (as you would do by plucking it) at say a string position of 1/5th its length, the 5th harmonic (4th overtone) standing wave would not form.  In fact any other multiple of the 5th harmonic, I.e. the 10th, 15th… etc, would not form or be heard.  If the string were to be plucked at ½ way along its length, then only the odd numbered harmonics would form, i.e. the 3rd , 5th, 7th …., etc.   When a string is initially plucked two travelling wave pulses are transmitted along the string in opposite directions and so the string behaves differently to being struck.  String vibration is an extremely varied and complex subject. The natural harmonic modes for a vibrating string as shown in Fig 8 are for an ideal case, where the string ends are attached to a ridged longitudinal structure and is referred to as a fixed fixed string length.  In the real world of vibrating strings, the vibration modes as mentioned also vary with the methods used to fix the string ends to a boundary.  I.e. a string end fixed to an oscillating (movable) boundary end, will behave differently to a string fixed to a ridged boundary.  More is said about boundary conditions later.

search the words in quotation marks above with Google. Or just type in your browser address bar: http://www.open.edu/openlearn/  -- in their search bar type in "Creating Musical Sounds" under the heading of 5. Primary Vibrators click on 5.1 Standing waves). Alternatively use the link provided below: on this page left in the "contents" click on 5 Primary Vibrators.

http://www.open.edu/openlearn/science-maths-technology/engineering-and-technology/technology/creating-musical-sounds/content-section-0