Consider the Vibrations of a Circular Membrane of Radius 2 $1 - \sin ( \pi r / 2 )$

Multiple Choice

Consider the vibrations of a circular membrane of radius 2 clamped along the circumference, with an initial displacement of $1 - \sin ( \pi r / 2 )$ and an initial velocity of zero. The mathematical model for this situation is

A) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$
B) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$
C) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } = - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi / 2 ) , u _ { t } ( r , 0 ) = 0$
D) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } = - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u _ { t } ( r , 0 ) = 0$
E) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$