Duality Principle for Horns
Robert W. Pyle · The Journal of the Acoustical Society of America · 1965
The concept of duality often encountered in electrical network theory may be extended to acoustic horn theory. For “plane” waves in two horns whose cross-sectional areas vary as S1(z) and S2(z), one can show that if S1(z)S2(z)=const, then the horns are acoustic duals. That is, the axial distribution of pressure in one horn is the same as the distribution of volume velocity in the other. As a result, input, output, and transfer acoustic impedances of one horn have the same form as the corresponding acoustic admittances of the other. A solution of the Webster horn equation for the pressure and volume velocity corresponding to a given horn contour thus yields the solution for the reciprocal contour as well. Whereas the exponential horn and its limiting case, the uniform tube, are their own duals, the dual of the conical horn is generated by revolving a rectangular hyperbola about its asymptote. Consideration of the latter pair of duals contributes to the understanding of brass wind instruments. [Work supported in part by the U. S. Office of Naval Research.]