Consider a Semi-Infinite, Elastic, Vibrating String, with Zero Initial Position $x = 0$

Consider a semi-infinite, elastic, vibrating string, with zero initial position and velocity, driven by a vertical force at $x = 0$ , so that $u ( 0 , t ) = f ( t )$ . Assume that $\lim _ { x \rightarrow \infty } u ( x , t ) = 0$ . The mathematical model for the deflection, $u ( x , t )$ , is

A) $\alpha ^ { 2 } \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( 0 , t ) = f ( t ) , u ( x , 0 ) = 0 , u _ { t } ( x , 0 ) = 0$
B) $\alpha ^ { 2 } \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 0 , t ) = f ( t ) , u ( x , 0 ) = f ( t ) , u _ { t } ( x , 0 ) = 0$
C) $\alpha ^ { 2 } \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = f ( t ) , u ( x , 0 ) = 0 , u _ { t } ( x , 0 ) = 0$
D) $\alpha ^ { 2 } \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial t } = 0 , u ( 0 , t ) = f ( t ) , u ( x , 0 ) = 0 , u _ { t } ( x , 0 ) = 0$
E) $\alpha ^ { 2 } \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( 0 , t ) = f ( t ) , u ( x , 0 ) = f ( t ) , u _ { t } ( x , 0 ) = 0$