This lesson, students design their own musical instrument using the wave equation, which they learn in Wave Hello, as well as the formation of standing waves which is covered in Standing Waves and Standing Waves Demonstrations. Students apply this understanding to the formation of standing waves in a musical instrument. Students have the choice to select a stringed instrument or a tube (wood wind or brass). For now, we focus on the fundamental frequency of the musical notes produced based on the length of the string or tube.
Students apply Math Practice 4: Model with mathematics and NGSS Science Practice 2: Developing and using models and Science Practice 5: Using mathematics and computational thinking as they apply the wave equation to the standing wave formation in an instrument. The NGSS 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 is also an important part of this lesson.
I have a guitar, a slide whistle and Boomwackers for this class. The Boomwackers are a variety of different length tubes that produce a C-major scale and are used so that, through observation, students see that there is a relationship between the length of an instrument and the musical note or frequency that it produces. I also play a few videos from YouTube on my projector. For this I use a good pair of speakers so the students can hear the music.
The class starts with 8 student volunteers who demonstrate that there is a relationship between the length of a tube and the musical note it produces. They line up shoulder to shoulder in size order of the Boomwacker they have. Playing the notes up and down, it becomes clear to all students that it is the length of the tube that determines the frequency of the note sounded.
To have fun, I ask if there is a volunteer conductor who would like to play a simple song. If no students volunteer, then I conduct something like Twinkle Twinkle Little Star.
I then get out a guitar and pluck a string to sound a note. I ask the students how I can change the frequency of the note. Suggestions using include things like "Turn the tuning peg" or "Press the string down on a fret". I agree with correct suggestions but focus on the pushing a fret down which changes the effective length of the string. Finally, I play the slide whistle to reinforce the idea that the length of the tube determines the frequency of the note. Students inevitably smile when I play the slide whistle.
Once students hear the changes in frequencies based on the changes in the length of the instrument, I move onto the Instrument Physics Power Point. This Power Point has several animations that do not work in preview mode and should be downloaded for full functionality.
Through this lecture, students takes notes while I show them the mathematical relationship between the standing waves that form in a musical instrument and the length of the instrument. A standing wave is required for a musical note to form. The standing wave has a specific wavelength based on the length of the tube which is wavelength = 2*length for a stringed instrument and a tube open at both ends. For a tube closed at one end, the relationship is wavelength = 4*length because is has a node at one end and an anti-node at the other. Once we have these relationships, we plug them into the wave formula to determine the fundamental frequencies.
Once we are done with the lecture, I tell students that now...(dramatic pause) they are going to design their own musical instruments with the might of their amazing imaginations and their newly learned physics prowess. To open up their thinking, I play a few YouTube videos. One shows a stringed instrument with strings that are several meters in length, the Earth Harp. The second instrument shows an expertly played PVC pipe instrument.
I show a few minutes of each video, again to ignite their imaginations and what is possible, at which point I hand out Standing Waves in a Musical Instrument worksheet so that students can practice the application of the ideas just presented.
As students complete the Instrument Physics worksheet, they bring it up to me. I don't tell this to the students but the answers to the worksheet are on the bottom of the second page. I check their work and if everything is correct, I give them the Design Your Own Instrument activity. Here students have to create their own instrument with application of the physcis formulas learned earlier. Students have to choose a frequency range and from that determine the lengths of their strings.
The part of the activity that students like best is that they have to imagine a history for their instrument. Why was it created, who plays it, etc. This is based on their instruments purpose;they can choose their frequency range.
For instance, one student decides to mix tubes and strings to create the "flutar" which is used to summon physics teachers from around the world. Another student creates the Excalibow. It plays subsonic and supersonic notes and is wielded by a giant the size of Florida (there is no requirement that their instrument be based in reality!).
Near the end of the lesson, I call up students to share their completed instruments with the class. They use my document camera to display their design and explain their instruments purpose and history. Students applaud each others creativity and ability to apply physics to a creative purpose. I collect the students' work as they exit and use an MS-Excel file, Instrument Calculation Reference, to assess their calculations. The spreadsheet has a range of lengths for each of the different instrument designs and makes correcting much easier.