Q6 - boundary conditions are in fact satisFed: ( x = 0, t =...

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MAE 105 Quiz #6 (closed book and notes) Name:_________________________ Date: May 29, 2008 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 ( , t ) = 0, and initial conditions u ( x ,0) = x sin x , u t ( x ,0) = 1 0 for /4 < x <3 /4 otherwise . (a) (1 Point) Draw the ( x , t )-plane, and below the x -axis, sketch the initial condi- tions in two graphs, as discussed in the class and in your book. (b) (2 Point) Extend the initial conditions such that the boundary conditions are satisFed for all t > 0, if we use the general solution for the inFnite domain. Draw the characteristics that pass through the following points, write down the general solution for the inFnite domain, and show that, with the extended initial data, the
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Unformatted text preview: boundary conditions are in fact satisFed: ( x = 0, t = /2) ( x = 0, t = 3 /2) ( x = , t = /2) ( x = , t = 3 /2) (c) (2 Points) Draw the relevant characteristics and Fnd the values of: u ( /4, /4) u ( /4, /2) u ( /3, 4 /3 ) Note: The general solution for the inFnite 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....
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