Abstract
The stability of the Stokes boundary layer generated by an acoustic disturbance in a two‐dimensional waveguide having a slowly varying height is examined. It is shown that the stability of the Stokes layer to three‐dimensional disturbances is governed by the value of three small parameters ε, 1/R, and 1/S, where ε is the wall slope, 1/R is the ratio of the oscillatory boundary layer thickness to the typical duct height, and 1/S is the ratio of the oscillatory particle displacement to the duct height. A stability analysis is presented for the amplitude range ε2R/S2 = O(R). It is shown that instability to infinitesimal disturbances occurs when h″ε2R/S2 is greater than 49.3. As the disturbance amplitude becomes finite in value the solution bifurcates from that obtained using linear stability theory. We will show that the first bifurcation satisfies the equation d2A/dx2 + (γ1x + T1γ2)A = γ3A3, where A is the disturbance amplitude and, γ1, γ2, and γ3 are parameters dependent on the local behavior of acoustic wave as well as the duct geometry. [Work supported by the National Science Foundation.]
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