An acceleration reduction factor can be defined as the ratio of peak acceleration at depth z$,$ to the peak

ground surface acceleration, i$.$e$.$

rd$=\frac{U_{\text{m}\text{a}\text{x}}\text{'}\text{'}(z)}{U_{\text{m}\text{a}\text{x}}\text{'}\text{'}(z}=0)$

For the case of undamped soil overlying rigid bedrock, develop an expression for the reduction factor as a function of the thickness and the shear wave velocity of the soil layer and the frequency of the input motion. Plot this reduction factor for a $15$m thick soil deposit with a shear wave velocity of $300$ m$/$sec at input motion frequencies of $0\mathrm{.}5$ Hz and $50$ Hz$.$Hint: u$($z$,$ t$)=2$A cos $($kz$)$ e$^$iwt

where w is the circular frequency of ground shaking, k isthe wave number

k$=$ w$/$Vs$,$

A is the amplitude of the wave