Table 2.

Parameters for the linear model accounting for wave propagation in bubbly liquid systems.

Model Parameter	Mathematical Representation
Complex wave number,$k_m$	$k_m^2=\frac{{\omega }^2}{c_l^2}\left(1+\frac{4\pi c_l^2{n}_b{a}_o}{{\omega }_0^2-{\omega }^2+2ib\omega }\right)$(30)
Damping factor,$b$	$b=\frac{2{\mu }_l}{{\rho }_l{a}_0^2}+\frac{P_b}{2{\rho }_l\omega a_0^2}\mathfrak{I}\mathrm{\Phi }+\frac{{\omega }^2{a}_0}{2c_l}$(31)
Complex dimensionless parameter,$\mathrm{\Phi }$	$\mathrm{\Phi }=\frac{3\gamma }{1-3\left(\gamma -1\right)i\chi \left[{\left(i/\chi \right)}^{1/2}{\mathit{coth}\left(i/\chi \right)}^{1/2}-1\right]}$(32)
Dimensionless parameter,$\chi$	$\chi =D/\omega a_0^2$(33)
Bubble resonance frequency,${\omega }_0$	${\omega }_0^2=\frac{P_b}{2{\rho }_l{a}_0^2}\left(\mathfrak{R}\mathrm{\Phi }-\frac{2{\sigma }_l}{a_0{P}_b}\right)$(34)
Volume fraction of bubbles,$\beta$	$\beta =\frac{4\pi }{3}{n}_b{a}_0^3$(35)