Two wires A and B , possessing identical geometrical construction,...

The speed of a transverse wave on a stretched wire is given by $v = \sqrt{\dfrac{T}{\mu}}$

The tension $T$ in the wire is $T = \dfrac{YA \Delta L}{L}$

The linear mass density $\mu$ is $\mu = \rho A$

Substituting $T$ and $\mu$, we get $v = \sqrt{\dfrac{Y \Delta L}{\rho L}}$

The time taken by the transverse signal to travel the length $L$ is $t = \dfrac{L}{v} = \sqrt{\dfrac{\rho L^3}{Y \Delta L}}$

Since $L$ and $\Delta L$ are identical for both wires, $t \propto \sqrt{\dfrac{\rho}{Y}}$

For wire $A$, $t_1 \propto \sqrt{\dfrac{\rho_A}{Y_A}}$ and for wire $B$, $t_2 \propto \sqrt{\dfrac{\rho_B}{Y_B}}$

Given that $Y_A > Y_B$ and $\rho_A > \rho_B$, both the numerator and the denominator for wire $A$ are greater than those for wire $B$. The ratio $\dfrac{\rho_A}{Y_A}$ can be greater than, equal to, or less than $\dfrac{\rho_B}{Y_B}$.

Therefore, the given information is insufficient to determine the relation between $t_1$ and $t_2$.