A First Course in Physics · Ch. XVII, §§ 468–470 · Chapter 17: Nature and Transmission of Sound
54. Interference: beats and standing waves
Goal: Your student can explain beats as alternate reinforcement and cancellation, find the difference of two pitches from the beat rate, and explain nodes and loops in standing waves.
You'll need
- The book, open to these pages
- Two identical glasses or bottles and water, or a tone-generator app on two phones
- A jump rope or long rope tied to a doorknob
- A notebook and pencil
- A calculator (optional, for the problems)
What's in the book
Beats from two nearly equal tuning forks, and the number of beats per second equals the difference of their rates; standing waves in a tube of cork dust (Kundt's tube); nodes half a wave length apart; the row of balls showing waves passing through each other; nodes and loops in a shaken rope. Three problems follow.
The lesson, step by step
Hook question
SayTwo sounds together make more noise. Can two sounds ever add up to silence?
Read together
Read §§ 468–470. The book's rule:
the number of beats per second must be equal to the difference in the vibration numbers of the two forks.
Try it: hear beats
Play a 440 Hz tone on one phone and 443 Hz on another, held side by side: you hear a wah-wah-wah three times a second. Change one to 441: one beat a second. (Or tap two nearly equal glasses of water, adding a little water to one.) Musicians tune instruments by making the beats slow down and stop.
Try it: standing waves
Tie a rope to a doorknob, back up until it is fairly tight, and shake your end steadily. At just the right rates the rope settles into loops with still points (nodes) between them: one loop, then two, then three as you shake faster (Fig. 383). The nodes are half a wave length apart (§ 470).
Explain the idea
When two sounds are in step they add; when one is pushing forward as the other pulls back, they cancel (§ 468, Fig. 380). A wave and its own reflection running through each other make a standing wave with fixed nodes. Noise-cancelling headphones today use exactly this idea, making a sound wave that cancels the noise.
Say it back
Your student explains how to tell, from beats, which of two forks is the slower one (by adding wax).
Problems
Work problems 1–3 on book page 365.
Afterwards
Listen for beats when two people sing almost the same note, or when two engines or fans run at nearly the same speed.
Answers to the book's problems
1. 256 (it beats 4 times a second with the 260 fork, so it is 256 or 264; wax lowers it and the beats increase to 5, so it was the lower one). 2. 1130 vibrations per second (the wave length is 2 × 6 in. = 1 ft.). 3. 32 ft. per second (each loop is half a wave, so the wave length is 8 ft.; 4 × 8).
Worked out for this site from the scan. If a number in your copy differs, trust the book.