Beat (acoustics)/Differential equations

Proposed style of this section
The article should be accessible to students with one year of calculus. On the linear equations, I would like to start with two coupled pendulums as shown in the figure. I think the $$\dot x_j=dx_j/dt$$ would be best (with $$j=1,2.$$) The students might not know what matrices are, so write them in non-matrix form and just show them the matrices with a link to matrices on wikipedia (or wikibooks or wikiversity.)

We need a figure showing the equivalent of the linearly coupled linear pendulums to horizontally oscillating masses on spring. That way we can use k and m in solving the problem (instead of g and l for a simple pendulum)

The van der pol oscillator will give them a hint as to the complexity of nonlinear systems.

Linear

 * Linear differential equation: #Homogeneous_equation_with_constant_coefficients
 * Matrix differential equation
 * Characteristic equation (calculus)


 * General Mechanics/Coupled Oscillators
 * Circuit_Theory/Laplace_Transform

Nonlinear

 * Van der Pol oscillator
 * International Journal of Non-Linear Mechanics, 29, pp. 737-753, 1994 Averaging Method for Strongly Nonlinear Oscillators with Periodic Excitations. R. Valéry Roy, R. Val'ery Roy link1 link2]

[https://arxiv.org/ftp/arxiv/papers/0803/0803.1658.pdf Tsatsos, Marios. "the Van der Pol equation." arXiv preprint arXiv:0803.1658 (2008)]

[https://link.springer.com/content/pdf/10.1007/BF01320221.pdf King, H., U. Deker, and F. Haake. "Renormalized perturbation theory of the Van der Pol oscillator." Zeitschrift für Physik B Condensed Matter 36.2 (1979): 205-208.]

Other

 * Relaxation oscillator
 * Limit cycle
 * Hodgkin–Huxley model
 * Phase-locked loop
 * Arnold_tongue
 * Method_of_averaging
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