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Nonlinear Physics with Maple for Scientists and Engineers by Richard H. Enns

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By Richard H. Enns

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D. For 0 = 179 0, to what term in the series would one have to go to have a 1% correction to the linear term? Problem 1-11: Small time behavior For the falling badminton bird (Example 1-4), determine the analytic power law behavior of y(t) for small t by Taylor expanding about t = and using the convert command. Keep only the first non-vanishing term in the expansion . ° • Problem 1-12: Visualizing the small angle approximation Plot sin(x) versus x and the series expansions of sin(x) to 5th, 7th, 9th, and 11 order in x on the same graph over the range x = 0...

In the animation Example 1-6, replace the speed c = 1 in the nth Fourier component with c = 1/(1 +an) where a is a positive constant. Animate the new wave form, choosing different numerical values for the parameter a. Describe what happens to the input square pulse wave form as time evolves and relate it to the concept of dispersion. +Problem 1-24: Viscous drag Suppose that in the classical mechanical Example 1-7, the mass ml experienced a viscous drag force F = -p,v(t), where v(t) is the velocity and p, the positive drag coefficient.

Tan (x + y), b. sin(2x), c. cos(7x), d. 2 cos((x + y)/2) cos((x - y)/2), e. 4cos3 (x) - 3cos(x), f. Problem 1-20: Sums Use the sum command to evaluate the following sums, simplifying the answers where possible: n a. Lk2 , k=1 n b. L(2k-l), k=1 n c. , k=1 00 d. , k=O 00 e. Ll/(2k-l)2. Problem 1-21: Beats Consider the addition of two waves whose resultant displacement is given by y = Al sin (~: (x - vlt) + 81) + A2 sin (~: (x - V2t) + 82) where A is the amplitude, A the wavelength, x the spatial coordinate in the direction of wave propagation, v the speed, t the time, and 8 a phase factor.

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