SOUNDWAVE PROPAGATION

For the air contained inside the hollow body of a guitar that is set to vibrate from the central bridge area of the soundboard in motion, compression of air particles may first take place leaving behind it an area of rarefaction, a void of air particles.  Due to this process of compression and by the fact that a void of air particles has taken place (rarefaction), air particles from outside the sound hole rush in to fill the void.   The void generally occurs in the primary top layer of air just under the soundboard and undertakes air to draw in, through the sound hole. The air in its self has its own elastic force and so helps to push out the excess air back through the sound hole along with the action of the vibrating soundboard, as shown in Fig 11 by the orange coloured curved lines of a, b, c and d.  As the bridge continues to vibrate the soundboard, the process reoccurs and the compressions of air inside the hollow body are said to reflect from the back boundary inside the guitar body and reverberate back and forwards.  
In Fig 11 a target like drawing is seen above the bridge area and represents an ideal orderly vibration pattern, for the plane area of a stress free soundboard.    A soundboard placed initially in a stress free state that is also able to vibrate with an up and down motion (y direction), would radiate vibrations from the central bridge area through the plane area of the soundboard with an orderly pattern.  The resulting uniform motion of the soundboard allows for a more singular directional projection of soundwave propagation to take place, as seen coming out of the sound hole and by the diagram "a." of Fig 11.  If the soundboard has areas in its plane that are under a pre stressed state, then vibrations travelling along the plane of the soundboard would encounter resistance from these areas.  The resulting waving motion of the soundboard would indeed be distorted and would not look anything like that represented by the target like drawing.

The soundboard of an x-braced guitar is placed under a great amount of pretension, by directly placing the string load tension onto the bridge, with no direct support.  The areas of the soundboard forward and backward of the bridge under tension, largely influences the way the soundboard vibrates.  The bridge central to these two areas of stress takes on a rocking motion, towards the other end of the string length. While the forward and backward areas of the soundboard central to the bridge are also set to rock in a waving motion with the bridge.  Consequently the air that's drawn in and pushed out of the sound hole also enters and leaves at an angle.  Propagation of airwaves through the sound hole is not congruent or perpendicular to the soundboard, due to this longitudinal waving motion of the soundboard; as indicated by b, c and d in Fig 11.  Of particular interest is diagram d, where wave interference patterns are shown forming and cause a scattering of wave propagation.  This scattering of soundwaves creates a confused sound field spectrum.  That can hover around the musician playing the guitar, which helps us to understand why some musicians use terms like "cloudy".


A mathematical proof for the occurrence of increased frequency modes due to nodal shortening

The following takes a mathematical look at changing and or set conditions for a vibrating string.                  Tips:  Always read the meaning into the symbols' of the formula -- once they are defined -- unless you are consciously aware of the meaning.  AB means A x B also 2L means 2 x L, but numerical subscripts f "1"   only identify which harmonic is used in an calculation, etc.    
The Tuned Natural Fundamental Frequency f 1 of a guitar string is related to its standing wave pattern as follows,     f 1 =  v / λ =  nv/2L , where n  is  the standing wave harmonic mode number in question, i.e. f 1 , f 2 , f 3 , …..f n ,  L is the length of the string fixed to a ridged support, v is the travelling pulse wave speed; while the standing wavelength λ is = to 2L for the fundamental   f 1 .

For the ideal case of a Fixed, Fixed String length:

A λ  occurring on any one of the guitar strings is physically constraint to the same constant length L, for the fundamental tuned-up frequency of each string.  For that matter the λ length(s) of all corresponding harmonics in the series for each string have the same set λ distances, for an ideal fixed, fixed string length.

If we rearrange the equation where  f = v / λ into the form of   λ = v / f we see that the only two other variables to consider next are frequency f and the pulse wave speed v travelling along the string.  Investigating the variable v for v = where t is the string load tension, m is the mass of the string and ul is the unit length of the string; shows these variables are also preset values for v and are constant.  Which only leaves the frequency f as the depended variable as to all other set conditions.   If however, the  λ n  in question  should experience nodal migration where the   λ n   is shortened  (compressed), due to either: (1) a  forceful change in tension t;  or (2) by the fixed end bridge position not held firmly fixed, but instead allowed to oscillate in the direction of the string length back and forth, causing a compression of the standing wave patterns;  then from the harmonic frequency formula      f n = v / λ n , the resulting harmonic frequencies f n are  increased.  In other words:
If λ n -- standing wave harmonics - become smaller in their numerical value of wavelength distance due to (2), and in some lesser instances should  v also becomes greater in numerical value due to (1) also.  Then the resulting calculated values by division, of  v / λ n  for the resulting  f n frequencies of the overtones especially but also for the fundamental frequencies, increase, and the instrument will exhibit discord.

However, if we only allow the bridge fixed end to oscillate up and down perpendicular to the soundboard, or even side to side transversely, string energy is maintained. This system would then be described as a Fixed, Resistance-Loaded String.  But otherwise if the bridge is allowed to oscillate in the direction of the string line longitudinally, string tension will loosen off quickly and consequently, nodal shorting producing discord with a combined loss in tonal sustain will take place.


  SPECIFICATIONS: IBS System vs X-brace: dB graphs compare both systems over a period of time, to show Sound-Intensity-Levels.

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