Uniformly valid solution for acoustic propagation in weakly tapered circular waveguides: liquid jet example.

The propagation of ultrasound down laminar liquid jets has potential applications to the stimulation of liquid drop production [J. B. Lonzaga, C. F. Osterhoudt, D. B. Thiessen, and P. L. Marston, J. Acoust. Soc. Am. 121, 3323-3330 (2007)] as well as to the coupling of ultrasound to objects through contact with a jet. In normal gravity, a jet issuing from a nozzle becomes tapered as the jet accelerates downward. A uniformly valid solution for the acoustic propagation in a weakly tapered, liquid jet waveguide in air with a turning point is derived using Langer's transformation and the method of multiple scales. The loss of energy from transmission into the air and from thermal viscous absorption is neglected. A solvability condition is used to obtain the leading-order correction due to the taper of the waveguide. This asymptotic solution is validated using finite-element numerical calculations. The ultrasonic wave amplitude is enhanced in the region of the jet close to the cutoff of the excited mode.