On the propagation of sound waves in cylindrical tubes

It is shown that the two main parameters governing the propagation of sound waves in gases contained in rigid cylindrical tubes, are the shear wave number, s=R√{ρω/μ}, and the reduced frequency, k= ωR/a ₀. It appears possible to rewrite the most significant analytical solutions for the propagation constant, Γ, as given in the literature, as simple expressions in terms of these two parameters. With the aid of these expressions the various solutions are put in perspective and their ranges of applicability are indicated. It is demonstrated that most of the analytical solutions are dependent only on the shear wave number, s, and that they are covered completely by the solution obtained for the first time by Zwikker and Kosten (1949). The full solution of the problem has been obtained by Kirchhoff (1868) in the form of a complicated, complex transcendental equation. In the present paper this equation is rewritten in terms of the mentioned basic parameters and brought in the attractive form F<, s, k>=0, which is solved numerically by using the Newton-Raphson procedure. As first estimate in this procedure the value ofaccording to the solution of Zwikker and Kosten is taken. Results are presented for a wide range of s and k values.