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Computational Modeling and Visualization of Physical Systems with Python by Jay Wang

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Chapter 6

Oscillations and waves

Periodic motion is ubiquitous in nature, from the obvious like heart beats, planetary motion, vibrations of a cello string, and the fluttering of leaves, to the less noticeable such as the vibrations of a gong or shaking of buildings in the wind. Oscillations and periodicity are the unifying features in these problems, including nonlinear systems in Chapter 5. The motion is back and forth repeating itself over and over, and can cause waves in a continuous medium.

We will study periodic motion including oscillations and waves in this chapter. There are two motivating factors for studying these phenomena at this juncture. Physically, waves involve independent motion in both space and time. We can see waves moving along a string (space) and also feel the vibrations in time at a fixed point. We can better understand wave motion after understanding particle motion. Mathematically, waves are described by partial differential equations (PDE) with at least two independent variables such as space and time. We are equipped to tackle PDEs only after studying ODEs first in the previous chapters.

We begin with the discussion of a single, damped harmonic oscillator, and work our way to oscillations of small systems such as triatomic molecules. We then discuss the displacement of a string under static forces. Finally, simulations of waves on a string and on a membrane are discussed.

To solve these problems, we introduce matrix algebra and eigenvalue formulation ...

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