$10\mathrm{.}10\mathrm{.}3.$ A plane wave of pressure amplitude P is traveling to the right in a pipe of cross$-$ sectional area Si$.$ A narrower pipe of area S$2$ and infinite length is attached to the right end of the pipe. $($a$)$ What must be the ratio of the two areas if the transmitted pressure amplitude in the second pipe is to be $1\mathrm{.}5$ that of the incident wave in the first pipe? $($b$)$ If the smaller pipe is cut off at a distance of a quarter$-$wavelength from the junction of the two pipes and then covered with a rigid cap, derive an expression giving the pressure amplitude at the cap in terms of that of the incident wave in the large pipe. $($c$)$ What is the ratio of these two pressures for the area ratio determined in $($a$)?$