Noise-canceling headphones are amazing. They don’t just block the sounds from coming into your ears, but actively destroy them. They actually make more noise so that you end up hearing less noise. How does that work?

In this lesson, students use transformations of trigonometric functions to explore how sound waves can interfere with one another, and how noise-canceling headphones use incoming sounds to figure out how to produce that sweet, sweet silence.

### Students will

• Given the frequency of a pure tone, calculate the period of its sound wave
• Sketch the graph of a chord composed of two tones and estimate its period and frequency
• Describe any apparent relationship between a chord’s period and its perceived pleasantness
• Sketch the graph of a pure tone’s noise-canceling counterpart and construct multiple functions to describe it
• Develop general procedures for generating a noise-cancelling wave from a given pure tone
• Examine some difficulties in describing noise-canceling waves for chords
• Explain how noise-canceling headphones work and why they are not always completely effective

### Before you begin

Students should be familiar with the general form of sinusoids (i.e. f(x) = Asin(BxC) + D or f(x) = Acos(BxC) + D) and how changes in the parameters affect their graphs.

#### Common Core Standards

##### Mathematical Practices

• Graphing Technology

#### Shoutouts

Kyle Jones, Smithsonian; image courtesy of Patrik Theander (https://www.flickr.com/
photos/fabolous/8627113727/)