Lecture 23: Standing Waves and Superposition

Physics 2210Fall Semester 2014

Announcements

● Unit 21 “Simple and Physical Pendula” (Nov 24th)● HW Due 11/25th as usual

● No new material Wednesday November 26th. In-class discussion of problems for last midterm, final exam. ● Unit 22 “Harmonic Waves” Monday Dec 1st

● HW due Sunday December 7th at midnight● Exam 3 (covering thru unit 19) December 3rd

● Unit 23 “Standing Waves and Superposition”, Monday December 8th

● HW Due Tuesday 9th

● Course Review December 10th

● Final Exam (December 16th)

Announcements● Final exam time is Tues Dec 16th, 3:30-5:30 PM. ● Room JFB 101

● Source - http://registrar.utah.edu/academic-calendars/final-exams-fall2013.php

US

Prelecture Feedback

● This seems too abstract compared to previous sections.

● What happens when waves collide with different wavelengths or velocities? Is a standing wave still created?

● You should play the guitar in class.

Wave Equation

Example: A uniform string has mass 0.300 kg and length of 6.00 m. The string passes over a pulley and supports a 2 kg object. Find the speed of a pulse traveling on the string.

Checkpoint

Q: What happens when two waves “collide?”

A: They ADD together!

Superposition

Mechanics Lecture 23, Slide 9

For linear equations, if we have two separate solutions, A and B, then A + B is also a solution!

Ct At Bt

2 1

Superposition

CONSTRUCTIVEINTERFERENCE

DESTRUCTIVEINTERFERENCE

Mechanics Lecture 23, Slide 10

B2t

A1t

The wave equation we derived last time is linear. (It has no terms where

the variables x,t are squared.)2

2

22

2 1

dt

xd

vdy

xd

Can we predict this pattern mathematically?

Just add two cosines and remember the identity:

where and

cosLt cosHt

Beats

212

1 L 212

1 H

ttAtAtA HL coscos2)cos()cos( 21

Mechanics Lecture 23, Slide 11

Standing Waves

What happens when two waves having the same frequency but moving in the opposite direction meet?

Mechanics Lecture 23, Slide 12

Wave 1 Wave 2

Wave 1Wave 2

Wave 1Wave 2

zero (Nodes)

Superposition

)cos()cos(2)cos()cos( tkxAtkxAtkxA

Changingamplitude

Stationary wave

How it Works

How to make it:

Mechanics Lecture 23, Slide 13

www.youtube.com/watch?v=3mclp9QmCGshttp://

Mechanics Lecture 23, Slide 14

Checkpoint #2

Checkpoint #3

We just found that the frequency of a string scales

with its diameter if the tension is the same.

This is really what you find if you examine a

guitar:

dlow

dhigh

.050”.050”.012”.012”

Mechanics Lecture 23, Slide 18

Their diameters are about half of the adjacent strings.

.028 .052 .022 .042 .014 .030

The lighter string in these pairs is one octave higher (2x

frequency).

Different Example – 12 String Guitar

Mechanics Lecture 23, Slide 19

HW Example

● A guitar string with mass density m = 2.3 x 10-4 kg/m is L = 1.04 m long on the guitar. The string is tuned by adjusting the tension to T = 108.4 N.● With what speed do waves on the string travel?● What is the fundamental frequency for this string?● Someone places a finger a distance 0.164 m from

the top end of the guitar. What is the fundamental frequency in this case?

● To “down tune” the guitar (so everything plays at a lower frequency) how should the tension be adjusted?