lec8-1 - Periodic functions simple harmonic oscillator...

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Periodic functions: simple harmonic oscillator Recall the simple harmonic oscillator (e.g. mass-spring system) d 2 y dt 2 + ω 2 0 y = 0 Solution can be written in various ways: y ( t ) = Ae i ω 0 t y ( t ) = A cos ω 0 t + B sin ω 0 t The constants of integration A and B depend on the initial conditions ( t = 0). The y ( t ) is periodic with period (or periodicity) τ 0 = 2 π ω 0 Chapter7: Fourier series
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Periodic functions: wave motion If I take a snapshot of a wave (e.g. a water wave), I find y ( x ) = A sin 2 π x λ = A sin kx where A is the amplitude and λ is the wavelength and k is the wave number . The period (or periodicity) in this case is λ = 2 π k If we consider time-dependence with angular frequency ω , y ( x , t ) = A sin( kx - ω t ) If v is a constant (i.e. independent of λ ), then v = ω k and y ( x , t ) = A sin 2 π λ ( x - vt ) Chapter7: Fourier series
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More on periodic functions Consider again y ( x , t ) = A sin 2 π λ ( x - vt ) I At fixed t , y ( x ) is a periodic function of x with periodicity λ What about more general cases: I Might have periodic y ( x ) with periodicity L Chapter7: Fourier series
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More on periodic functions Consider again y ( x , t ) = A sin 2 π λ ( x - vt ) I At fixed t , y ( x ) is a periodic function of x with periodicity λ I At fixed x , y ( t ) is a periodic function of t with periodicity 2 π ω = λ v What about more general cases: I Might have periodic y ( x ) with periodicity L I Might have more complex pattern, not described by y ( x ) = sin 2 π x L or y ( x ) = cos 2 π x L Chapter7: Fourier series
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More on periodic functions Consider again y ( x , t ) = A sin 2 π λ ( x - vt ) I At fixed t , y ( x ) is a periodic function of x with periodicity λ I At fixed x , y ( t
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lec8-1 - Periodic functions simple harmonic oscillator...

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