Contextualization

When we talk about waves, the image of a beach with breaking waves usually comes to mind. However, the scope of waves goes far beyond that, being one of the most fundamental and fascinating parts of Physics. Waves are present in several aspects of everyday life and nature, such as the sound we hear, the light we see and even the radio and television signals we receive.

The wave equation, a profound concept in Physics, describes how waves behave. It is fundamental to understand many natural phenomena and has several applications, ranging from quantum physics to telecommunication engineering. The wave equation expresses how the propagation occurs in time and allows for the prediction of the wave behavior.

Surely you have witnessed the effect of throwing a stone into a puddle. If you paid close attention, you must have noticed that from the point where the stone hits the surface, a series of concentric circles starts to expand. This phenomenon, known as surface waves, is a clear illustration of how waves travel and is precisely what the wave equation seeks to describe.

Now, imagine how this knowledge can be applied! The wave equation has several practical applications, ranging from medical imaging diagnosis, through the acoustics of concert halls to weather forecasting. The manipulation and comprehension of waves open up endless possibilities and innovations. Dominating this topic is, therefore, essential for anyone interested in a career in science or engineering.

As study material, you may use the following references:

• Book "Physics for Scientists and Engineers" by Paul A. Tipler, Gene Mosca (Ed. LTC) - Chapter 16: Waves I
• Virtual Didactic Laboratory - USP - Waves and Propagation
• University Channel - Physics Class: Wave Equation
• Khan Academy - Series and classes on waves and sound

Remember, these materials are intended to support the comprehension of the theory behind wave equations and serve as the basis for your discussions and experiments.

Hands-on Activity

Activity Title: Dancing Waves

Project Goal

The goal of this project is to allow students to explore the creation of waves and understand how they behave. By the end of the project, students will be able to describe a wave in terms of time using the wave equation.

Project Description

Groups of students (3 to 5 students) will create a "wave pool". They will generate waves in water and observe how they behave. Students will document their observations and try to relate the behavior of the observed waves to the mathematical model of the wave equation.

Required Materials

• A large plastic or metal tray
• Water
• Smartphone or camera to document the experiment
• Ruler and stopwatch

Step by step

1. Fill the tray with water to a depth of about 2 centimeters.
2. Have a student tap the water from one end of the tray and observe the waves that form.
3. Document the experiment with videos and photos. Notice how the waves travel.
4. Check how the frequency and amplitude of the waves alter the wave behavior. Do this by changing the intensity and frequency of the taps on the water.
5. Use the ruler and stopwatch to measure the wave velocity. To do so, measure the time it takes for the wave to travel a certain distance.
6. Now, try to relate your observations to the wave equation.

Project Deliverables and Writing of the Report

Students must deliver a detailed report of the project. The document should be divided into four main sections:

1. Introduction: Contextualize the study of the wave equation and explain the goal of the project.
2. Development: Describe the theory of the wave equation. Explain the experiment performed and the methodology used, including the steps you followed, the materials used and any difficulties encountered along the way. Discuss the results obtained.
3. Conclusion: Summarize the main findings of the experiment. Highlight what you have learned throughout the project and how you have been able to connect theory and practical experiment.
4. References: List all the sources you used to learn about the wave equation and to carry out the experiment.

For each section, students must use the following guidelines to support their work:

• Introduction: Here, you must contextualize the study of the wave equation. Relating it to real-world examples can help explain the importance of this topic. Also, clarify the goal of the experiment.
• Development: In this section, explain in detail the experiment carried out. Talk about the methodology used, including all the steps followed and the materials that were necessary. Show the data collected and discuss the results obtained. This discussion should include a comparison between the waves observed in the experiment and the way waves are represented by the wave equation.
• Conclusion: In this part, summarize the main findings of the experiment. Highlight what you have learned throughout the project and how you have been able to connect theory and practice. Also, highlight the skills you developed throughout the project and how this contributed to your learning.
• References: Finally, in this section, you must list all the sources used to learn about the topic and to carry out the experiment. Remember to cite all sources correctly.