Music, Math, and Motion

with Dr. Arun Chandra & Dr. E.J. ZitaThe Evergreen St. College

Fall week 3

Tuesday 14 Oct. 2008

Overview

• Syllabus for the next two weeks

• Contact:– Radio waves– 21 cm line and its frequency– Estimate the chance of extraterrestrial intelligence

• Loy Ch.8:– Easier way to derive frequencies– Springs, strings, and pipes– Resonant wavelengths and Q factor

Syllabus for the next two weeks

Contact

– Radio waves– 21 cm line and its frequency– Estimate the chance of extraterrestrial intelligence

Radio waves and EM spectrum

21-cm line of Hydrogen

http://abyss.uoregon.edu/~js/ast122/lectures/lec22.html

21 cm line and its frequency

What would be the frequency of radio waves of wavelength = 21 cm?

8 8

8

8 9

13*10 3*103*5*10

10.215

15*10 1.5*10

c f

mc s sf Hzm

f Hz Hz

Circular polarization

http://www.nsm.buffalo.edu/~jochena/images/circular3.gif

Estimate possibility of ETI

Estimate possibility of ETI

Estimate possibility of ETI

http://www.activemind.com/Mysterious/Topics/SETI/drake_equation.html

Loy Ch.8: Vibrating Systems

– Easier way to derive frequencies– Springs, strings, and pipes– Resonant wavelengths and Q factor

Simple Harmonic Motion (SHM)

Loy p.130

distance 2

22

1

angular displacement s r

sspeed v r r

t tcircumference

vtime period T

angular frequency fT

frequency f whereT periodT

s

Kinematics of oscillations

2

2

( ) cos ( )

xsin ( )

cos ( ) cos ( )

displacement x t A t

speed rateof changeof displacement

v A tt

acceleration rateof changeof speed

va A t A t

t

a x

Easier way to derive frequencies

Compare to p.241-241 in Loy (harder, but same result) for SPRING OSCILLATION:

Easier way to derive frequencies

F = ma

-k x = m 2 x

k = m 2

2 = k/m

= k/m = 2 f

2

kmf

Frequency of air in piston

PAk

l

What instruments are like this?

What is the spring constant k of air?

Recall that F = -kx, so the units of k are those of F/x, or (N/m).

Recall that Pressure = Force/Area, so P = F/A or F = PA.

For air,

where

= a constant and l = the length of the chamber in which air is trapped.

2

kmf

Springs, strings, and pipes

Nodes = still points at equilibrium

Antinodes = points of maximum oscillation

Springs, strings, and pipes

How many waves fit in the length L?

Use that to find the resonant wavelengths n and resonant frequencies fn.

Springs, strings, and pipes

How many waves fit in the length L?

Use that to find the resonant wavelengths n and resonant frequencies fn.

Conical pipes

Displacement must vanish at opening, therefore

Resonance: Q factor

rfQf

Sharpness of resonance

Galloping Gertie

http://www.youtube.com/watch?v=j-zczJXSxnw