Transient phenomena under the Crighton–Westervelt–Klein–Gordon equation: Controlling the onset of gradient catastrophe in bubbly liquids

Employing a combination of analytical and numerical methodologies, we investigate the propagation and evolution of acoustic acceleration waves in lossless bubbly liquids under the (1D) finite-amplitude model termed the Crighton–Westervelt–Klein–Gordon (CWKG) equation. Exact acceleration wave results for the acoustic pressure field are derived and analyzed, and special/limiting cases are identified. It is shown that by varying the value of the parameter that represents the volume concentration of bubbles, one can control the instant at which “gradient catastrophe” (i.e., shock formation) occurs. Results obtained are also compared with those for both the bubble-free limiting case of the CWKG equation and the classic Klein–Gordon equation, which is the linearized version of the CWKG equation. Lastly, a link between the present model and an extension of the Fermi–Pasta–Ulam–Tsingou-$\alpha$ case is noted.