$\beta$

$v_n\left(t,x\right)$

$\frac{\partial }{\partial t}{v}_n\left(t,x\right)$

$v_z\left(t,0,h\right)$

$-\infty \le \beta <-\frac{6}{5}$

$v_n\left(t,x\right)\to -\infty$

$t\to t_2$

$\frac{\partial v_n}{\partial t}<0$

$v_z\left(t,0,h\right)\to -\infty$

$t\to t_2$

$\beta =-\frac{6}{5}$

$v_n\left(t,x\right)=-\frac{4}{9}\frac{{\eta }^2}{1-{\eta }^2}$

$\frac{\partial v_n}{\partial t}=0$

$v_z\left(t,0,h\right)=-\frac{2}{9}$

$-\frac{6}{5}<\beta <-1$

$v_n\left(t,x\right)\to 0$

$t\to t_2$

$\frac{\partial v_n}{\partial t}>0$

$v_z\left(t,0,h\right)\to 0$

$t\to t_2$

$\beta =-1$

$v_n\left(t,x\right)\to 0$

$t\to \infty$

Exponentially

$\frac{\partial v_n}{\partial t}>0$

$v_z\left(t,0,h\right)\to 0$

$t\to \infty$

Exponentially

$-1<\beta <-\frac{4}{5}$

$v_n\left(t,x\right)\to 0$

$t\to \infty$

$\frac{\partial v_n}{\partial t}>0$

$v_z\left(t,0,h\right)\to 0$

$t\to \infty$

$-\frac{4}{5}\le \beta <0$

$v_n\left(t,x\right)\to 0$

$t\to \infty$

$\frac{\partial v_n}{\partial t}>0$

$v_z\left(t,0,h\right)\to 0$

$t\to \infty$

$\beta =0$

$v_n\left(t,x\right)=0$

$\frac{\partial v_n}{\partial t}=0$

$v_z\left(t,0,h\right)\to 0$

$t\to \infty$

$0<\beta \le \infty$

$v_n\left(t,x\right)\to 0$

$t\to \infty$

$\frac{\partial v_n}{\partial t}<0$

$v_z\left(t,0,h\right)\to 0$

$t\to \infty$