Acoustic Dipole in a Duct
An acoustic dipole oriented along the axis is placed in a duct. The dipole can be moved along the axis. The resulting sonic radiation can be approximated by an -term Fourier series, given in the Details section, with representing the Helmholtz number (the cut-off frequency of a duct, product of the wavenumber and the duct radius) in multiples of .
The solution for the pressure is
 M. Gupta, "Simulation and Visualization of Various Acoustic Sources in MATLAB." (2006) wenku.baidu.com/view/9ca5d912f18583d049645990.