Propagation of Sound in a Bend Joining Two Straight Waveguides of Rectangular Cross-Section

The theory of guided waves in straight and curved ducts of rectangular cross-section is developed. The propagation of waves in curved bends is considered in some detail with particular reference to source properties and their effects on the propagation of modes higher than the fundamental one. One of the problems encountered is the calculation of the angular wave numbers in the curved sections. Previous methods used for finding these are reviewed and the conclusion is reached that they are inadequate for bends of moderate curvature, particularly for the fundamental mode. Methods for calculating numerical values of Bessel functions and their derivatives are developed to cover a wide range of order and argument. Programs have been written for a small computer to give results of a fair degree of accuracy over most of the field of values. The angular wave numbers, which are the roots of the governing equations, are found by Newton's method.

To define the sound field in the straight sections of duct a single mode source was used (generally the first cross mode) in order to give a clear picture of the mechanics of propagation. It is suggested that the sound pressure field in the straight duct immediately prior to the bend may be regarded as defining adequately the source for the bend. The differences in the forms of sound pressure distribution between straight and curved sections causes generation of extra modes. The modal amplitude constants may be calculated from the use of orthogonal properties of the cylinder functions with respect to order. In the straight section following the bend the process is similar: the sound pressure field at the end of the bend acts as a source for the following straight section. The modal amplitude constants are calculated as before but due allowance must be made for the phase difference between the modes. The interaction between the modes in the bend and following straight sections is demonstrated by calculating the distributions of sound pressure level across successive sections in the direction of propagation.

Extensive experimental Results are reported for bends of varying angular length and for duct systems of two different materials. The agreement between the measured distributions of sound pressure level and predicted values is fair.