设方程 $\frac{\partial^2 z}{\partial x^2}-y \frac{\partial^2 z}{\partial y^2}-\frac{1}{2} \frac{\partial z}{\partial y}=0$ 在变换 $\left\{\begin{array}{l}u=x+a \sqrt{y}, \\ v=x+2 \sqrt{y}\end{array}\right.$ 下化为 $\frac{\partial^2 z}{\partial u \partial v}=0$, 求常数 $a$.