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identities: Wednesday, October 9, 2013 Wave equation
1d: ( 2 2 − 2 )ψ (t, x) = 0
∂x Linear 2nd order PDE
:-)! Nice! Superposition! DA: Same units. Correct!
3 d: ( 2 2 − ∇ )ψ (t, x) = 0
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.
( 2 2−
)ψ (t, x) = 0
∂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
T Waves on a string
y (t, x) x
Acceleration in y direction,
for ﬁxed posit...
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- Fall '09