Construction of a saw tooth wave from harmonic waves L 2 Length of the string v 1 Wave velocity on the string A0 1 Overall amplitude of the wave xa 15 Location of the kink m 20 Number of harmonics to include in the sum n 2L n k n 2 n Wavelength of the n th harmonic a n 4 A0 n v k n Wave number and frequency of the n th harmonic L 1 sin k n xa xa L xa n2 2 h x t n a n cos n t sin k n x Fourier amplitude of the n th standing wave n th standing wave m f x t h x t n Sum m harmonics n 1 Note this is similar to the waveform on a bowed violin string but somewhat simpler On a violin string the peak of the sawtooth will not be always at the same amplitude but will follow the shape of the fundamental vibration n 1 2 m t x 0 05 L FRAME 40 Range of values for plotting Time at which to plot The FRAME variable is incremented as 0 1 2 when making an animation Amplitudes for 10 harmonics Sum of all 10 harmonics 1 0 8 0 6 a n cos n t f x t 0 0 4 0 2 1 0 0 2 4 6 8 10 0 0 5 n 1 x Plots of the first 5 harmonics 1 h x t 1 0 5 h x t 2 h x t 3 h x t 5 t 0 0 h x t 6 0 5 1 0 0 5 1 x 1 5 2 1 5 2
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