Chapter 5: Waves 21 The equation for a harmonic traveling plane wave is: u ( ± x, t )=ˆ u cos( ± k · ± x ± ω t + ϕ ) If waves reFect at the end of a spring this will result in a change in phase. A ±xed end gives a phase change of π / 2 to the reFected wave, with boundary condition u ( l )=0 . A lose end gives no change in the phase of the reFected wave, with boundary condition ( ∂ u/ ∂ x ) l =0 . If an observer is moving w.r.t. the wave with a velocity v obs , he will observe a change in frequency: the Doppler effect . This is given by: f f0 = v f-v obs v f . 5.2.2 Spherical waves When the situation is spherical symmetric, the homogeneous wave equation is given by: 1 v 2 ∂ 2 ( ru ) ∂ t 2-∂ 2 ( ru ) ∂ r 2 =0 with general solution: u ( r, t )= C 1 f ( r-vt ) r + C 2 g ( r + vt ) r 5.2.3 Cylindrical waves When the situation has a cylindrical symmetry, the homogeneous wave equation becomes: 1 v 2 ∂ 2 u ∂ t 2-1 r ∂ ∂ r ± r ∂ u ∂ r ² =0 This is a Bessel equation, with solutions which can be written as Hankel functions. ²or suf±cient large values
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