$1.$ Using the following equation for Q factor for a wave energy device with mass m$,$ damping

coefficient R$,$ spring stiffness S$,$ natural frequency $\backslash$omega $0$ and damping ratio $\backslash$delta $=$R$/2$m

$(\backslash$delta small$)$

Q $=(1$ e$^(2\backslash$delta $/\backslash$omega d $)^)1$

derive the equations

Q $=\backslash$omega $0/2\backslash$delta $(1+\backslash$delta $/\backslash$omega $0-2\backslash$delta $^2/3\backslash$omega $0^2+$O$\{\backslash$delta $^3/$w$0^3\})=\backslash$omega $0/2\backslash$delta $=\backslash$omega $0$m$/$R$=$S$/\backslash$omega $0$R$=($Sm$)^1/2/$R

$($Hint: use Taylor series for an exponential function and for a binomial and$/$or use the

method of successive approximations.$)$