$1.(20$ pts$.)$ A spherical loudspeaker of radius $10$ cm vibrates in an air at a frequency of $100$ Hz with a displacement amplitude $1$ mm $.$

$($a$)(10$ pts $.)$ Derive an expression for the acoustic pressure by the pulsating sphere and determine the SPL generated by that vibration at a distance $10$ m from the center of the loudspeaker.

$($b$)(10$ pts $.)$ Derive an expression for the acoustic power radiated by the loudspeaker as a function of frequency. How much power is radiated at the frequency of $100$ Hz $?$