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Unformatted text preview: Wave equation in Cartesian coordinates The wave equation is given in one spatial dimension 2 u x 2 = 1 v 2 2 u t 2 We again use separation of variables u ( x , t ) = X ( x ) T ( t ), and then we can write the wave equation as, X 00 X = 1 v 2 T 00 T = k 2 Hence we wind up with two Helmholtz equations to solve, X 00 + k 2 X = 0 T 00 + k 2 v 2 T = 0 Wave equation in Cartesian coordinates, continued We find solutions T ( t ) = sin t , T ( t ) = cos t , X ( x ) = sin kx , and X ( x ) = cos kx Here the angular frequency = kv , and k = 2 / From y ( x , t ) = X ( x ) T ( t ), we have four different basic solutions y ( x , t ) = sin kx sin t y ( x , t ) = sin kx cos t y ( x , t ) = cos kx sin t y ( x , t ) = cos kx cos t We can have linear combinations of solutions of this kind depending on the boundary conditions and initial conditions (superposition principle!) Example: Waves on a string with fixed ends Imagine a string (e.g. a guitar string) fixed at the ends, so thatImagine a string (e....
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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.
 Spring '03
 Staff

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