A solid slab of thickness $H_{1}$ is initially at a uniform temperature $T_{0}$. At time $t=0$, the temperature of the top surface at $y=H_{1}$ is increased to $T_{1}$, while the bottom surface at $y=0$ is maintained at $T_{0}$ for $t \geq 0$. Assume heat transfer occurs only in the $y$-direction, and all thermal properties of the slab are constant. The time required for the temperature at $y=H_{1} / 2$ to reach $99 \%$ of its final steady value is $\tau_{1}$. If the thickness of the slab is doubled to $H_{2}=2 H_{1}$, and the time required for the temperature at $y=\mathrm{H}_{2} / 2$ to reach $99 \%$ of its final steady value is $\tau_{2}$, then $\tau_{2} / \tau_{1}$ is
- $2$
- $\frac{1}{4}$
- $4$
- $\frac{1}{2}$