The transmission of sound in a non-uniform two dimensional duct without flow is investigated by a method of weighted residuals which leads to a set of coupled "generalized telegraphists * equations". Results for several duct configurations are compared with those from, respectively, a variational method, a stepped duct approximation, and an eigenfunction expansion method based on linearly tapered duct segments. 106 w. EVERSMAN, E. L. COOK AND R. J. BECKEMEYER approach requires a set ofeigenfunctions and eigenvalues in each stepped section and in the case oflined ducts this can be a problem in terms of computational requirements (the method introduced in the present paper must have eigenvalues and eigenfunctions at stations along the duct, but, as will be described, a rapid numerical scheme has been devised for their computation). The variational approach suffers a dimensionality problem in that complicated acoustic fields (both axial and transverse) require a large number of basis functions in the trial solution.
Other recent approaches to the problem are more approximate in nature by virtue of restricting geometry or frequency range allowed. Nayfeh and his co-workers have published several studies of propagation in non-uniform ducts with and without flow. The paper by Nayfeh, Telionis and Lekoudis [10] is representative. They restrict themselves to ducts with slowly varying cross-section, lining properties and flow properties and employ a perturbation scheme. To within the level of accuracy which they retain, they do not predict the generation, reflection, and transmission of modes other than the one incident on the non-uniformity. It appears that a higher level of approximation is required to predict this.
Karamcheti and his co-workers have also made contributions in this area. King and Karamcheti [11] studied plane waves in ducts with one dimensional flow by a method of characteristics and Huerre and Karamcheti [12] used a short wave approximation for the same type of problem. Similar problems were studied earlier by Powell [13] and Eisenberg and Kao [14].
Tam [I 5] seems to have published the first paper dealing with a multi-modal approach to the problem of non-uniform ducts with flow. His technique is a perturbation scheme based on the assumption of slowly varying cross-sectional area. The first order approximate solution is obtained by Fourier transformation.
A recent paper by Cummings [16] studies the novel problem of the acoustics of a wine bottle. The wine bottle is a non-uniform duct without flow and the acoustic field is approximated by a Runge-Kutta integration scheme based on the Webster Horn Equation. This method allows only a plane wave mode of propagation and hence is limited to the lower frequency ranges. There have been many studies of the horn equation since Rayleigh first introduced it and it is a favorite topic in texts on acoustics.
In the present research program we are interested in the multi-modal propagation of sound in nonruniform ducts of fairly general shape. The final goal of the program is the study of propagation in non-uniform ducts with flow, so that we will be interested in methods for the no flow case which appear to be extendable to the case with flow. Of the two generally applicable techniques mentioned previously, the variational method does not seem to be readily extendable to the case with flow. The stepped duct approximation might have some application, although it is certainly questionable whether the non-uniform flow field can be represented in sutticient detail in a series of stepped uniform segments. We are thus led to look for another method with promise for the flow case. In this paper we will introduce the method and assess its utility in the case without flow, as there are equivalent results available against which a direct comparison can be made. In the course of the development and application of the method it has become apparent that the method of weighted residuals is an important alternate method for the duct with no flow, and indeed, may well be superior to the other two methods of general utility.
The method of weighted residuals (MWR) employed here actually was first employed in connection with electromagnetic waveguide problems by Schelkunoff [17,18]. The field equations for these problems are identical to the classical acoustic equations in certain cases. Schelkunoff's work followed work by Stevenson [19,20] which used MWR but led to a somewhat different formulation which does not seem to have been widely used. Stevenson appears to be the first to suggest the application of methods of this type to acoustic horn problems.