What is a wave? The one-dimensional wave equation
What is a wave? The one-dimensional wave equation
What is a wave? The one-dimensional wave equation
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2/3/14<strong>The</strong> <strong>one</strong>-<strong>dimensional</strong> <strong>wave</strong> <strong>equation</strong><strong>The</strong> <strong>one</strong>-<strong>dimensional</strong> <strong>wave</strong> <strong>equation</strong> for scalar (i.e., non-vector)functions, f:where v will be the velocity of the <strong>wave</strong>.<strong>The</strong> <strong>wave</strong> <strong>equation</strong> has the simple solution:where f (u) can be any twice-differentiable function.<strong>What</strong> about a harmonic <strong>wave</strong>?E = E 0cos k(x − ct)E 0 = <strong>wave</strong> amplitude (related to the energycarried by the <strong>wave</strong>).€k = 2π = angular <strong>wave</strong>numberλ = 2πν˜˜ ν(λ = <strong>wave</strong>length; = <strong>wave</strong>number = 1/λ)€Alternatively:E = E 0cos(kx − ωt)€Where ω = kc = 2πc/λ = 2πf = angularfrequency (f = frequency)€2
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