Students create multiple drawings for how standing waves form on a string.

Musical instruments have overtones (called tonality) due to the formation of multiple standing waves from the instrument.

In this lesson students explain why they observe multiple peaks, or overtones, when I play a single note on a guitar string. Through an exploration of the different Standing Waves that can form on a single string, students make the logical connection to argue that it is the different standing waves occurring on the same string at the same time that produces these overtones. This builds off the previous lesson, Design Your Own Instrument, where students used the fundamental frequency to determine the lengths of the strings or tubes. Students must also apply the wave equation which they learn in Wave Hello.

To complete this activity, students apply CCSS Math Practice 3: Construct viable arguments and critique the reasoning of others and Math Practice 7: Look for and make use of structure. Students also apply NGSS Science Practice 1: Asking questions, Science Practice 4: Analyzing and interpreting data and Science Practice 6: Constructing explanations. This lesson is in the context of performance standard HS-PS4-1: Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling in various media.

I have an Android smart phone with a frequency analyzer app that shows the formation of overtones. I also have a students bring in instruments to demonstrate how the overtones form when they play a single note.

15 minutes

To begin the lesson , I hand out Multiple Standing Waves Worksheet for the students to work on. I instruct them to complete only the front for now. The purpose of the sheet is to have students explore that a single length of string can have many different standing waves form on it. This leads neatly to the demonstration when I play a what appears to be a single note on an instrument and show them that multiple frequencies come from that single plucked string.

While the Students Discuss the worksheet, I circulate the room and give support where needed. My worksheet is well designed and students find the task relatively easy to complete, especially after learning about standing waves on a previous lesson. This is seen on the following Student Work - Multiple Standing Waves samples.

Once most of the students are done with the front, I use the document camera to display my Android phone that is running a sound frequency analyzer called Spectral Audio. The analyzer displays frequency verses amplitude and I have it in logarithmic mode, zoomed into the frequencies from 0 to 3000 Hz. This gives the clearest view of the overtones. I pluck a string on my guitar and make it clear that I am playing one note. I ask the students to then complete the back of the Multiple Standing Waves Worksheet. My hope is that students will be able to provide a solution similar to Multiple Standing Wave Solutions.

15 minutes

Most of the students make the connection between the appearance of overtones and the ability for multiple standing waves to form on a string. I see this as I circulate the room, so I don't collect this sheet but instead have students reflect on their own work. I display examples of exemplary student work on the document camera so that everyone can see and correct any misunderstandings they may have.

Now it is time to apply the mathematics behind the concept of overtones using the Overtones Power Point. I give a short 15 minute lecture and students take notes. The lecture starts with the formation of standing waves on a length of string and explains how the wave equation, wave velocity = frequency * wavelength, is manipulated so that you have the length of string rather than the wavelength of the wave. This manipulation is applied several times for the different size standing waves that can form on a single length of string. The rule that is stressed is that you must have a node at both ends of the string as the string cannot vibrate at the points where it is attached to the instrument. The PowerPoint goes through a similar explanation for how multiple standing waves form in a tube open at both ends and a tube open at one end.

By the end of the presentation, students should be able to explain why any one note on a musical instrument have a fundamental frequency and several overtones. Next they apply that understanding to several word problems about musical instruments.

20 minutes

At the end of the lecture, I hand out the Instruments Graphic Organizer which is an overview of the formulas for the three different types of instruments (string, tube open at both ends, tube open at one end). It has several blank boxes that students are to fill in. This can be used as a reference sheet and helps students organize the how the formulas apply to the fundamental frequency and overtones for the 3 types of instruments. Students have 5 minutes to fill in the blanks at which point I display the Instruments Graphic Organizer-Filled In. They can check their answers against the solution sheet and correct any mistakes they have.

I want students to fortify the concept of overtones, so I give them the Instrument Problems handout. This activity has students work through a series of questions about the overtones, harmonics and fundamental frequency of various instruments. I circulate the room with the Instrument Problem Solutions in hand to give support to students. At the end of class most students have not finished this worksheet, so they complete it for homework.