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Horn  Theory,

Horn Theory, as  it  has  been  developed, is  based  on  a  series  of  assumptions  and simplifications,  but  the  resulting  equations  can  still  give  useful  information about  the  behavior of a horn.  The  problem  of  sound  propagation in horns  is  a  complicated  one,  and  has  not yet  been  rigorously  and analytically solved.   In essence,  it  is  a  three-dimensional  problem, but solving the wave equation in 3D is  very  complicated,  in  all  but  the  most elementary  examples.  In the  wave  equation  for three  dimensions, the equation describes how sound waves of  very  small  (infinitesimal)  amplitudes behave  in  a three-dimensional  medium (e.g. "air").  In  1919,  Webster presented  a solution to the problem which was made possible by simplifying equation 1 [See: the very technical Wikipedia article linked below] from  a  three-dimensional  to  a one-dimensional  problem.  He  did  this by  assuming  that  the  sound  energy  was uniformly distributed as a wave-front issuing perpendicular to the horn axis, and by considering  only  motion of sound  in  the  axial  direction [forward, towards the listener]. The  result  of  these  simplifications is  the  so-called  “Webster’s  Horn  Equation,”  which  can  be  solved  for  a  large number of cases.  You can use this equation  to  predict  what  is  going  on  inside a  horn,  neglecting  the higher  order  effects, but  it  can’t  say  anything  about  what  is going  on  outside  the  horn; so  it  can’t predict directivity, once sound exits the horn.

The Tractrix Theory attempts to  address the "shape" and resultant character of the sound, taking into account the nature and shape of the horn itself,  and the "tugging" effects of natural forces, such as  friction with the air.  It provides the seeds for a much more perfected horn behavior, and permits a design that lets you have your cake and eat it, too, as the old saying goes.  Much research, both scientific and empirical, has led Charney Audio to its cutting-edge use of modern horn technology.  We invite you to Just Hear Everything.  Gloriously!