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Abstract.An integral equation method is used to obtain improvable lower bounds for the second eigenvalue of the second-order "reduced" problem obtained from the problem described in the title by singular perturbation methods. These lower bounds are compared with results obtained directly by invariant embedding. The computational aspects of the integral equation method are stressed. The method is shown to be quite general and can be applied to a variety of boundary-value problems including those in which the eigenvalue parameter appears in the boundary conditions as well as in the differential operator.

Abstract.Methods are developed to study the problem described in the title. Improvable lower bounds for the first eigenvalue are obtained for the low velocity-thin pipe wall case. It is shown that the eigenvalue changes from real to imaginary as the fluid velocity increases through a "critical" velocity. It is the methods which we wish to emphasize in that while we discuss them only for the present problem they are very general and especially powerful when applied to differential equations with constant coefficients.

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