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

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

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

View Full Document

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

View Full Document
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 ω t , T ( t ) = cos ω t , X ( x ) = sin kx , and X ( x ) = cos kx • Here the angular frequency ω = kv , and k = 2 π/λ • From y ( x , t ) = X ( x ) T ( t ), we have four different basic solutions y ( x , t ) = sin kx sin ω t y ( x , t ) = sin kx cos ω t y ( x , t ) = cos kx sin ω t y ( x , t ) = cos kx cos ω 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

{[ snackBarMessage ]}

Page1 / 7

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

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

View Full Document
Ask a homework question - tutors are online