Nice superposition da same units correct 1 2 2 3 d

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Unformatted text preview: b identities: Wednesday, October 9, 2013 Wave equation 1 ∂2 ∂2 1d: ( 2 2 − 2 )ψ (t, x) = 0 v ∂t ∂x Linear 2nd order PDE :-)! Nice! Superposition! DA: Same units. Correct! 1 ∂2 2 3 d: ( 2 2 − ∇ )ψ (t, x) = 0 v ∂t ψ (t, x) = A cos(k (x − vt)) ψ (t, x) = A cos(k (x + vt)) ψ (t, x) = A cos(kx) cos(kvt) Wednesday, October 9, 2013 We’ll discuss 3d case later. This week, just 1d waves. Examples solutions of the 1d wave equation. Superpose for general solution (Fourier). (Aside: Fourier) Math statement: get general functions from a sum (superposition) of sin or cos functions. Physics application: get general solution of the wave equation from a superposition of waves of definite frequency and wavelength Wednesday, October 9, 2013 Wave equation, cont. 1 ∂2 ∂2 ( 2 2− )ψ (t, x) = 0 2 v ∂t ∂x Is solved by: ψ = ψR (x − vt) + ψL (x + vt) Arbitrary functions for right and left moving parts. E.g. right moving y (t, x) = A cos(kx − ω t) Velocity (speed) is the phase velocity: Wednesday, October 9, 2013 ω λ v= = k T Waves on a string y (t, x) x Acceleration in y direction, for fixed posit...
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