Plane Wave Backscattering by a Liquid Sphere

→ See also Johnson  (1977)

Consider a fluid sphere with radius a and $\rho _{{1}}$, $c_{{1}}$, $k_{{1}}$ for density, sound speed, free field wave number, respectively, of the sphere fluid in an outer medium with $\rho _{{0}}$, c${}_{{0}}$, k${}_{{0}}$, respectively. Ratios of densities: $g=\rho _{{1}}/\rho _{{0}}$; of sound velocities $\gamma=c_{{1}}/c_{{0}}$.

The backscattering cross section $\sigma$ for an incident plane wave is:

$$\begin{array}[]{@{}l@{\ }c@{\ }l}\displaystyle\frac{\sigma}{\pi a^{2}}&=&\displaystyle\frac{2}{k_{0}a}\left|{\sum\limits _{{m\geq 0}}{\displaystyle\frac{(-1)^{m}(2m+1)}{1+j\cdot C_{m}}}}\right|\;,\\ C_{m}&=&\displaystyle\frac{\displaystyle\frac{{\alpha}^{{\prime}}_{m}}{\alpha _{m}}\,\displaystyle\frac{y_{m}(k_{0}a)}{j_{m}(k_{1}a)}-\displaystyle\frac{{\beta}_{m}}{\alpha _{m}}\, g\gamma}{\displaystyle\frac{{\alpha}^{{\prime}}_{m}}{\alpha...