$4.(15$ pts$.)$ A single$-$expansion$-$chamber reaction muffler is to be designed to provide at least $15$ dB transmission loss for all frequencies between $1000$ and $2000$ Hz $.$ The smallest possible expansion$-$area ratio $\backslash ($ m$=$A$\_\{$M$\}/$ A $\backslash ),$ given $\backslash ($ A$\_\{$M$\}>\backslash )\backslash ($ A $\backslash ),$ compatible with this design objective is most desirable. What values of $\backslash ($ L $\backslash )($expansion chamber length$)$ and $\backslash ($ m $\backslash )$ would you select? $5.(20$ pts$.)$ Consider air in an infinitely long rigid$-$walled pipe of square cross section of dimension $17$ cm $.$ For the lowest four sets of modes,

$($a$)(10$ pts$.)$ find the cutoff frequencies

$($b$)(10$ pts$.)$ sketch the pressure distribution across the pipe.