Quiz7 - boundary conditions are in fact satisfied x = 0 t...

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MAE 105 Quiz #7 (closed book and notes) Name:_________________________ Date: June 7, 2007, Time: 3:35 to 3:55pm Consider the wav e equation 2 u t 2 - 2 u x 2 = 0, t >0 ,0 < x < π , with the boundary conditions u (0, t ) = u ( π t ) = 0, and initial conditions u ( x ,0) = x sin x , u t ( x ,0) = 1 0 for π x <3 π otherwise . (a) (1 Point) Drawthe ( x , t )-plane, and belowthe x -axis, sketch the initial condi- tions in twographs, as discussed in the class and in your book. (b) (2 Point) Extend the initial conditions such that the boundary conditions are satisfied for all t >0 ,ifweuse the general solution for the infinite domain. Draw the characteristics that pass through the following points, write down the general solution for the infinite domain, and showthat, with the extended initial data, the
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Unformatted text preview: boundary conditions are in fact satisfied: ( x = 0, t = /2) ( x = 0, t = 3 /2) ( x = 0, t = 4 /3 ) ( x = , t = /2) ( x = , t = 3 /2) ( x = , t = 4 /3 ) (c) (2 Points) Draw the relevant characteristics and find the values of: u ( /4, /4) u ( /4, /2) u ( /4, 4 /3 ) u (4 /5, /4) u (4 /5, /2) u (4 /5, 4 /3 ) Note: The general solution for the infinite domain is given by: u ( x , t ) = 1 2 ( f ( x-ct ) + f ( x + ct )) + 1 2 c x + ct x-ct ∫ g ( x ) d x , where c is the wav e speed and f ( x ) and g ( x ) are the prescribed initial displace-ment and velocity respectively.-2-...
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This note was uploaded on 05/21/2008 for the course MAE 105 taught by Professor Neiman-nassat during the Spring '07 term at UCSD.

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Quiz7 - boundary conditions are in fact satisfied x = 0 t...

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