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Lecture 10: Phasors and Standing Waves
February 25, 2025
Reading Assignment
- Read: 14.7-14.8; Supplementary Reading Ch 1.1-1.2
- Study: Ex 14.6; Supp Fig 1.4
- Ignore: Eq 14.13 (it's not wrong - just too specialized to be useful)
Objectives
- (Continuing objective) Describe applications of the concepts of induction, waves, and light to everyday “real life” situations.
- Given a specific expression for a traveling transverse or longitudinal harmonic wave in sinusoidal form, determine amplitude, angular frequency, frequency, wavenumber, wavelength, period, phase shift and wavespeed.
- Correctly sketch phasor diagrams to represent an oscillating system at various times.
- Calculate the beat period and frequency for two signals differing slightly in frequency.
- For standing wave modes on strings or in air columns in pipes: sketch the wave pattern, identify nodes and antinodes, and determine wavelength, wave-number, frequency or wave speed.
Homework
- Wednesday's Assigned Problems: A32, A33, A34, A92, A112, Problem X10; CH 14: 34, 35; Supp CH 1: 1, 2
- Monday's Hand-In Problems from Lecture 10:
A35, A37; CH 14: 36, 70; Supp CH 1: 3
Note: this is only the first half of the hand-in set.
Lecture Materials
- Click here for the Lecture overheads. Answers: CT1 - 4; CT2 - 2; CT3 - 3; CT4 - 4; CT5 - 3
- Spreadsheet showing the addition of two waves.
- Demonstration of beats and rotating phasors. Download the file and then right click on it and select “open”.
Videos of example problems
To see the problem statement, click on the link below. To play the video example, click on the underlined words "Video Demonstration" near the top of the page with the problem statement.- Standing waves on a string.
- Organ pipe that is (a) open on both ends, and (b) open on one end and closed on the other.
- Drawing phasor diagrams
Pre-Class Entertainment
- Cap In Hand - The Proclaimers
- How Sweet It Is (To Be Loved by You) - Marvin Gaye
- Oye Como Va - Santana
- Hand In My Pocket - Alanis Morissette
- Happy - The Rolling Stones
Assigned Problems Guide
- A32: relatively quick problem with your slinky. Try to get a measurement of the traveling wave speed.
- A33: medium long. Setting up standing waves on the slinky and timing the requency, then comparing to your calculated frequency.
- A34: medium quick. Thinking about frequency with the slide whistle.
- A92: medium. Literally reading numbers off a graph and adding them together, and then plotting the new numbers. Sets up hand-in problem A93, which shows that phasor addition works!
- A112: medium long. Calculating standing wave frequencies for your whirly tube, and then comparing them to sounds generated with those frequencies.
- X10: medium. Working with a standing wave plot on Wolfram Alpha, very similar to the hand-in problem X9 that you did for traveling waves.
- 14-34: medium quick. Use the technique we discussed in class to find the wavelength and then $v=\lambda f$ to find the frequency.
- 14-35: medium long. Similar to #34, but you're not given values or even a symbol for the length of the string or the wave speed. Assume they are $v$ and $L$ and work them out symbolically for the various cases, and relate them back to the fundamental.
- Supp 1-1: medium. Drawing a bunch of phasor diagrams for an oscillator at different times.
- Supp 1-2: medium long. You need to pull the information from the problem statement to figure out the phasor parameters (amplitude, phase angle at different times).