$($continuing from Problem $2$ of Homework No$.3)$

$($a$)$ Provide general expressions for the convective and the unsteady terms of the horizontal

acceleration of the fluid particles, as functions of $x,z$ and $t$ and the given constants $R,k,d,$

and $.$ Use the expressions for the constants A and B you determined in part $($a$)$ of Prob. $2$ of

HW# $3.$ Suggestion: Use the identities: $sin(2)=2sin()cos()$ and $cos{h}^{2}z-sin{h}^{2}z=1,$

in order to make some of these expressions more compact! $(18$ points$)$

$($b$)$ Apply the expressions you determiend in part $($a$)$ at $x=0$ and $z=-\frac{d}{3}$ and for the same

values of the constants as those given or determined in part $($d$)$ of Prob. $2$ of HW# $3.$ Provide

the values of the maximum magnitudes for the unsteady and for the convective terms of the

horizontal particle acceleration $($at $z=-\frac{d}{3}\frac{?}{\text{b}\text{a}\text{r}}($ and $x=0)).$ Which term $($the unsteady or the

convective$)$ has larger maximum magnitude? Provide a fully annotated, computer generated

graph which shows the unsteady, the convective, and the total horizontal acceleration $($at

$z=-\frac{d}{3}$ and $x=0)$ vs$.$ t for $0tT.(18$ points$)$