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.  I  will  review  the  assumptions  later,  and  discuss  how  well they hold up in practice. The  problem  of  sound  propagation in horns  is  a  complicated  one,  and  has  not yet  been  rigorously  solved  analytically. Initially,  it  is  a  three-dimensional  problem, but solving the wave equation in 3D is  very  complicated  in  all  but  the  most elementary  cases. The  wave  equation  for three  dimensions. This equation describes how sound waves of  very  small  (infinitesimal)  amplitudes behave  in  a three-dimensional  medium. I  will  not  discuss  this  equation,  but  only note that it is not easily solved in the case of horns. In  1919,  Webster presented  a solution to the problem by simplifying equation 1from  a  three-dimensional  to  a one-dimensional  problem.  He  did  this by  assuming  that  the  sound  energy  was uniformly distributed over a plane wave-front perpendicular to the horn axis, and considering  only  motion  in  the  axial  direction. The  result  of  these  simplications is  the  so-called  “Webster’s  Horn  Equation,”  which  can  be  solved  for  a  large number of cases: the wave number or spatial frequency (radians per meter),φ is  the  velocity  potential,  and  S  is  the  cross-sectional area of the horn as a function of x. The  derivation  of  equation  2  is  given in the appendix. You can use this equation  to  predict  what  is  going  on  inside a  horn,  neglecting  higher  order  effects, but  it  can’t  say  anything  about  what  is going  on  outside  the  horn,  so  it  can’t predict directivity.

 

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