CSUEB Physics 1200 Lecture 2 & 3 II. Oscillations & Waves Updated 2012 Apr 4 Dr. Bill Pezzaglia

Outline A.Harmonic Oscillators B.Mersenne’s Laws C.Wavespeed & Resonant Modes 2

A. Harmonic Oscillators 1.Equilibrium 2.Periodic Motion 3.Frequency 3

1. Equilibrium Equilibrium: a system which is not changing with time (no net force). There are 3 types: a)Neutral equilibrium: boring case, if you move the object to another position it will just sit there. 4

b. Unstable Equilibrium if you displace system slightly it changes drastically, e.g. a ball perched on top of a steep hill 5

c. Stable Equilibrium if you displace the system, there is a “restoring force” which opposes the change Strength of restoring force increases with displacement 6

2. Periodic Motion a) Oscillations: A system displaced from stable equilibrium will oscillate about the equilibrium point The motion is “periodic” (repeats in time) The time for one cycle is called the “period” 7

Galileo: (1581) showed the period of oscillation depends only upon gravity “g” and length “L” of the string: Period is INDEPENDENT of: –Mass on end of string –Size (“amplitude”) of oscillation b. Pendulums 8 Acceleration of gravity on earth: g=9.8 meters/second 2. Gravity on moon is 1/6 as strong, so pendulum will go slower!

i.Hooke’s Law: if you squash a spring by distance x, it will give a restoring force F proportional to x: ii.Spring Constant “k” tells the stiffness of the spring. iii.Period of Oscillation for mass m on spring: c. Springs 9

3. Frequency a). Definition: Frequency is the rate of vibration Units: Hertz=“cycles per second” Relation to Period: 10

b. Frequency is “Pitch” 1600 Scraper across grooved board produces notes (relates frequency of vibration to pitch of sound) Mathematical Discourses Concerning Two New Sciences (1638) most lucid of the frequency equivalence Made sound waves visible by striking a wine glass floating in water and seeing the vibrations it made on the water’s surface. First person to accurately determine frequency of musical pitch was probably Joseph Sauveur ( ) 11 Galileo Galilei ( )

c. Toothed Wheels & Sirens 1819 Cagnaird de la Tour’s siren used to precisely measure frequency of sound (disk with holes spun, air blown across holes) 1830 Savart uses card against moving toothed wheel to equate frequency and vibration Measures the lowest pitch people can hear is about 16 to 20 Hertz 12

B. Mersenne’s Laws 1.Frequency and String Length 2.Frequency and Tension 3.Frequency and String Mass 13

1. Frequency & Length a). Consider a string under tension “plucking” the string causes it to vibrate 14

1b. Frequency and String Length Pythagoras of Samos ( BC) found that if you put your finger midway on the string, the string would sing an octave higher (i.e. double the frequency). Fundamental Frequency One octave higher (double frequency) is half the wavelength 15

1c. Frequency inversely proportional to length Frequency is inversely proportional to string length More generally: Frequency is inversely proportional to wavelength 16

2. Frequency and String Tension Vincenzo Galilei, the father of Galileo Galilei, was an Italian lutenist, composer, and music theorist. Vincenzo determined: To double the frequency of a violin string, one must quadruple the tension! Hence: 17 Vincenzo Galilei ( )

3. Frequency and Mass a. Mersenne (1630) states: Frequency is inversely proportional to the diameter d of the string Putting it all together: 18 ( ) Marin Mersenne “The Father of Acoustics”

b. Guitar String Diameters For guitar, all strings same length, and want tensions the same, so to get different frequencies, must vary diameters of strings StringDiameterFreq E mm330 Hz B G D A E E2 is 2 octaves lower than E4 Or (1/4) the frequency Hence diameter is nearly 4x bigger! 19

3c. Frequency and Mass Mersenne (1630) further states: Frequency is inversely proportional: to the root of the mass Or to the root of the mass density  Putting it together, (  =Mass per length) 20

3d. Guitar String Masses For guitar, all strings same length, and want tensions the same, so to get different frequencies, the masses of strings must be different Stringgm/cmFreq E Hz B G D A E E2 is 2 octaves lower than E4 Or (1/4) the frequency Hence mass bigger by nearly a factor of 16 21

C. WaveSpeed & Modes 1.Wavespeed Formula 2.Harmonic Modes 3.Melde’s Experiment 22

1. Wavespeed Frequency “f” : oscillations per second (Hertz) Wavespeed “v”: is frequency  wavelength v = f 23

1b Speed of various waves Blue Light Frequency 6x10 14 Hertz Wavelength 500 nm Speed: 3x108 meters/sec Sound: Middle C Frequency 260 Hertz Speed 340 metres per second (1,115 ft/s). This is 1,236 kilometres per hour (768 mph) 24

1b. Standing Waves Standing wave is really the sum of two opposing traveling waves (both at speed v) Makes it easy to measure wavelength 25

2. Harmonic Modes Daniel Bernoulli (1728?) shows string can vibrate in different modes, which are multiples of fundamental frequency (called “Harmonics” by Sauveur) 26 n=1f 1 n=2f 2 =2f 1 n=3f 3 =3f 1 n=4f 4 =4f 1 n=5f 5 =5f 1

2b. Wavelengths of Harmonic Modes The wavelength of n-th mode is: 27

3. Melde’s Experiment Fixed vibrator frequency f (probably 120 Hertz) vary tension F by changing hung mass Adjust until get standing wave Measure wavelength Calculate: v=f 28

3b. Results from Melde’s Experiment If tension is constant: wavespeed of standing waves on string is independent of the mode (all same speed!) Velocity v depends upon tension F and mass density  (mass per unit length) 29

References 1.Old 10 min movie on sound

Notes 1.Who came up with 2.Demo: Spring, Pendulum 3. Comb or other similar device 31

ideas 1.Demo of savart wheel? 2.More on Melde’s experiment (demo)? 3.More on resonance? 4.Did not explain resonant modes in terms of wavelength. 32