## Description

This project has three parts, the first part to be done at home, the last two parts in class. In the home part we recall how to solve second order, linear, differential equations with constant coefficients and simple source functions. In the first in-class part we analyze the solutions found at home and try to understand what is a resonant solution, the beats phenomenon, and how both arise. The last in-class part is to apply the beats phenomenon to send an audible tone across the Atlantic Ocean using radio waves.

The project really begins with the in-class parts. We start analyzing the graph of the solutions found at home as the driver frequency approaches the natural frequency. This is when students {em should understand resonance and discover beats}. They should realize that beats actually happens when the solution is the difference of two cosine functions with similar frequencies. Finally, in the last part of the project students should use the insight learned about beats to transmit an audible pulses through large distances using radio waves.

The second and third parts are supposed to be done in class, with the students working in groups of three, maybe four. The students need to open an interactive graph which graphs the different solutions the students should have found at home. The interactive graph has a free parameter---the driver frequency---that can be varied in a certain range, from zero to close to the natural frequency. Students can see what the resonant solution looks like, what the non-resonant solution looks like, and how the non-resonant solution approaches the resonant one as the driver frequency approaches the natural frequency.

The interactive graph does not require any paid software, such as Mathematica or Maple. The interactive graph uses a free software that works in any browser, MathStudio. MathStudio is a computational software for iPhone, iPad, Apple Watch, Mac and any web browser. It feels like a light version of Mathematica or Maple. Since it works in any web browser, students do not need to install additional software in their computers. The link itself, provided later on in this project, contains all the instructions in Javascript needed to create the code we run. The browser itself performs the calculations. And the calculations are not intensive; they can be carried out in any browser, in any laptop, and in any cellphone.

The third part of the project is a simple application of the beats phenomenon containing only trivial algebraic calculations. The idea is to rediscover how people sent the first Morse code audible tones across long distances using radio waves. Simple algebraic calculations show that to receive radio waves at the human hearing frequencies requires a 100 km long antenna. That is why people used carrier waves of high frequency---high enough to be received by a 1 meter antenna---so that they have beats on the human audible range.

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