FrequenPy

frequenpy is a high-precision physics engine dedicated to the study and visualization of standing waves.

Wave theory

In this section I will briefly explain the systems available for simulation, according predictions of wave theory. This results have been (and can be) demonstrated experimentally.

String loaded with with N masses oscillating transversally.

A flexible elastic string with tension T is loaded with N identical particles, each of mass m, equally spaced a distance a apart. Let us hold the string fixed at two points, one at a distance a to the left of the first particle and the other at a distance a to the right of the Nth particle.

According to the theory, the movement of each of the masses in the vertical direction can be decomposed into a superposition of N normal modes modes of oscillation. That way, the $y$ position of the particle n as a function of time is

$$y_n(t) = \sum_{p=1}^N A_p \sin(k_p n a) \cos(\omega_p t + \theta_p).$$

Where $A_p$ and $\theta_p$ depend on the initial conditions, $k_p$ will depend on the boundary conditions and $\omega_p$ will have the form

$$\omega_p = 2 \omega_0 \sin\left(\frac{p \pi}{2(N + 1)}\right) \qquad, \qquad \omega_0 = \sqrt{\frac{T}{ma}}.$$

There are as many normal modes as there are degrees of freedom (masses) in the system. In each natural mode p, all masses in the system oscilate at the same frequency $\omega_p$ and pass through the equilibrium position at the same time. The first mode, p=1, corresponds to the lowest frequency (called fundamental) and each subsequent mode will have a frequency higher than the previous one. Any movement of the string, as strange as it may be, can be expressed as a superposition of those N normal modes (some will contribute more than others to the final movement).

As the number of masses gets higher and highter ($N \rightarrow \infty$), we approximate to the continuous system (a vibrating string - no discrete masses). In this simulation, you can use N = 30 to see a continuous effect.

Installation

To install FrequenPy, just run:

pip install frequenpy

Usage

Once installed, just run:

frequenpy

This will prompt the following help:

(.venv) $ frequenpy
usage: FrequenPy [-h] {loaded_string} ...

Welcome to FrequenPy! High-precision physics engine dedicated to the study of standing waves.

positional arguments:
  {loaded_string}  Choose a system to simulate
    loaded_string  Transverse oscillations on a string loaded with masses.

options:
  -h, --help       show this help message and exit

Enjoy!

If you pass loaded_string as an argument:

(.venv) $ frequenpy beaded_string
usage: FrequenPy beaded_string [-h] --masses  [--modes  [...]] [--boundary BOUNDARY] [--speed SPEED] [--save]

Transverse oscillations on a string loaded with masses.

options:
  -h, --help           show this help message and exit

required arguments:
  --masses             Number of masses.

optional arguments:
  --modes  [ ...]      Normal modes to combine. Ex: "1 2 3" (default: [1]).
  --boundary BOUNDARY  Boundary conditions: 0 (fixed), 1 (free), or 2 (mixed) (default: 0).
  --speed SPEED        Animation speed. Can be a float number (default: 1).
  --save               Save the animation in mp4 format (default: False).

Example: frequenpy loaded_string --masses 3 --modes 1 2 3 --speed 0.1 --boundary 0

Remember that for system of N masses there are N normal modes. You can pass only one of them or a combination of several, e.g. "2 6 3". The order doesn't matter.

TODO

• Interactive GUI to be able to play more easily with all the parameters of the system.
• Plot each individual normal mode that is contributing to the movement.
• Loaded String:
  • Allow changing damping and tension as parameters.
  • Allow initial conditions to generate more arbitrary and crazy movements of the string, like picking the string with your mouse and realease it from some position.