The boundary layer attenuation caused by viscous and thermal losses of all higher order modes of propagation in rigid walled rectangular and circular tubes has been calculated using the concept of boundary layer admittance introduced by Cremer. In the case of the rectangular tube Cremer's admittance formula for obliquely incident waves is applied directly in conjunction with Morse's duct formulas to obtain the attenuation of (ny, 0) and (ny, n2) modes. The boundary layer admittance of higher order modes in circular tubes is obtained from an extension of Cremer's method for plane waves. An electromagnetic wave-guide technique for calculating “skin” losses is applied to determine the attenuation of the asymmetric (m, n) circular tube modes. It is shown that the symmetric (0, n) circular tube modes are the least attenuated of all higher order modes in both rectangular and circular tubes; above a certain frequency the attenuation is less than the plane wave attenuation.
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Ralph E. Beatty (1950) studied this question.