Energy flux streamlines versus acoustic rays for modeling an acoustic lens: Energy flow inside and in the focal region for a carbon dioxide filled spherical balloon in air

As an extension of recently published experimental work [Dean and Parker, “A ray model of sound focusing with a balloon lens: An experiment for high school students,” J. Acoust. Soc. Am. 131, 2459–2462 (2012)], a comparison of ray acoustics and wave analysis via energy flux streamlines [Chapman, “Using streamlines to visualize acoustic energy flow across boundaries,” J. Acoust. Soc. Am. 124, 48–56 (2008)] as a means of visualizing the sound field for a positive acoustic lens in the form of a carbon dioxide filled spherical balloon in air is made. The sound field is expanded in the usual Legendre polynomials and spherical Bessel functions [Anderson, “Sound scattering from a fluid sphere,” J. Acoust. Soc. Am. 22, 426–431 (1950)], and the energy flux vectors at points throughout the regions of interest are calculated [Adin Mann III, et al., “Instantaneous and time-averaged energy transfer in acoustic fields,” J. Acoust. Soc. Am. 82, 17–30 (1987)]. Then, energy flux streamlines are plotted using Mathematica r...