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

question 40

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) \(\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0\)

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Definitions:

Behavioral Perspective

A view in psychology focusing on observable behaviors and the ways in which they are learned through interactions with the environment.

Operant Conditioning

A method of learning that occurs through rewards and punishments for behavior, thereby influencing the likelihood of that behavior occurring again.

Negative Consequences

Unfavorable results or outcomes that arise as a result of an action, decision, or behavior, potentially causing harm or disadvantage.

Negative Reinforcement

A behavioral principle where the removal of an unfavorable event or outcome following a desired behavior increases the likelihood of that behavior being repeated in the future.