lec13-2 - Wave equation in Cartesian coordinates The wave...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 7

lec13-2 - Wave equation in Cartesian coordinates The wave...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online