Outline. Oscillations & Waves. 1. Equilibrium. A. Harmonic Oscillators. b. Unstable Equilibrium. 1. Equilibrium. 2. Periodic Motion. 3.

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1 CSUEB Physics CSUEB Physics 1200 Oscillations & Waves Outline A. Harmonic Oscillators 2 B. Waves Updated 2012 August2 C. Wave Phenomena Dr. Bill Pezzaglia A. Harmonic Oscillators 3 1. Equilibrium 4 1. Equilibrium 2. Periodic Motion 3. Frequency 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. b. Unstable Equilibrium 5 c. Stable Equilibrium 6 if you displace system slightly it changes drastically, e.g. a ball perched on top of a steep hill if you displace the system, there is a restoring force which opposes the change Strength of restoring force increases with displacement 1

2 a) Oscillations: 2. Periodic Motion A system displaced from stable equilibrium will oscillate about the equilibrium point The motion is periodic (repeats in time) 7 b. Pendulums 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 T 2 L g 8 The time for one cycle is called the period Acceleration of gravity on earth: g=9.8 meters/second 2. Gravity on moon is 1/6 as strong, so pendulum will go slower! c. Springs i. Hooke s Law: if you squash a spring by distance x, it will give a restoring force F proportional to x: F kx 9 3. Frequency a). Definition: Frequency is the rate of vibration 10 ii. Spring Constant k tells the stiffness of the spring. iii. Period of Oscillation for mass m on spring: T 2 m k Units: Hertz= cycles per second Relation to Period: 1 f T b. Frequency is Pitch 11 c. Toothed Wheels & Sirens 12 Galileo Galilei ( ) 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 ( ) 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 2

3 B. Waves 1. Wave Modes 2. Structure of a wave 3. Wavespeed Types of Waves (a) Longitudinal ( p waves) travel through gas, liquid and solids. Particle motion in direction of wave (b) Transverse ( s waves) travel only in solids. Particle motion perpendicular to direction of wave. 14 1c Two Basic Wave Types (Andrija Mohorovičić 1909) 15 1d Liquid Core 16 (pressure waves) Richard Dixon Oldham (July 31, 1858 July 15, 1936) was a British geologist who, in 1906, argued that the Earth must have a molten interior as S waves were not able to travel through liquids nor through the Earth's interior. S Shadow Zone (shear waves) S waves can only travel in solids, and are slower P can travel in solid, liquid and gas! (c) Surface Waves 17 (c) Light Waves 18 Ocean Waves are Cycloid Waves. Particles travel in circles. Light is a wave of electric and magnetic phenomena that can travel through empty space (no medium!) Rayleigh Waves ( R wave) are similar, but exist in solids (more damaging in earthquakes than p or s waves) 3

4 (d) Gravity Waves? Not discovered yet, but we are looking for them. They would distort space as they travel by! The Structure of a Wave a) Nodes: no displacement 20 b) Antinode: maximum displacement (c) Wave parameters Wavelength (measured in meters) 21 2e. Sound is displacement of pressure Rarefaction: the trough of the wave where pressure (density) is low Compression: the peak of the wave where pressure (density) is high 22 (d) Amplitude (loudness) is the maximum wave displacement 3a. Wavespeed Frequency f : oscillations per second (Hertz) Wavespeed c : is frequency x wavelength c = f 23 3b. Wave speed vs Displacement Speed a) Wavespeed: is how fast the nodes (or antinodes) are moving. In air, all sounds move the same speed whether loud, soft, high or low. b) Particle (Displacement) Speed: is how fast the particles in the medium are moving, which can be quite different! This is related to the intensity of the sound (soft sound they move slowly and not very far) 24 c) In a standing wave, the particles are moving but the wave is not moving at all! 4

5 3c. Speed of sound is finite 25 3d Speed of sound depends upon medium! 26 Leonardo da Vinci ( ) A bell far away, will be heard to resonate in response to a bell ringing, after a delay in time Did he measure the speed of sound? (some references say yes). Robert Boyle ( AD) (1640) classic experiment on the sound radiation by a ticking watch in a partially evacuated glass vessel provided evidence that air is necessary, either for the production or transmission of sound. 3e Measurement of Speed of Sound 27 3f Speed of Sound in Water (1826) 28 William Derham ( ) First to accurately measure speed of sound (in air) (341 meters/second) Newton used his value in the Principia (1686), although it was 16% higher than the value Newton theoretically calculated. Ion Lake Geneva, Switzerland, Jean- Daniel Colladen, a physicist, and Charles-Francois Sturm, a mathematician, measured speed to be 5x faster than in air. In their experiment, the underwater bell was struck simultaneously with ignition of gunpowder on the first boat. The sound of the bell and flash from the gunpowder were observed 10 miles away on the second boat. The time between the gunpowder flash and the sound reaching the second boat was used to calculate the speed of sound in water. Speed did NOT depend upon frequency! 3g. What Wavespeed does NOT depend upon 29 3h. Theory of Speed of Sound 30 Gassendi: demonstrated speed of sound is independent of pitch by comparing measurements from cannon and rifle (no dispersion ) Speed of sound (in air) does not depend on wavelength or frequency Speed of sound depends upon the properties of the medium through which it travels: B is coefficient of stiffness (springiness, or Bulk Modulus ) measured in Pascals (the unit of pressure, Newton/meter 2 ) density (kilograms per cubic meter) c B Speed of sound (in air) does not depend upon amplitude (loudness) Pierre Gassendi ( ) c is wavespeed in meters/second 5

6 3i. Speed of Sound in Gas, Liquid and Solid 31 3j. Velocities of P and S waves in different layers of the Earth 32 Note generally speed in solid > speed in liquid > speed in gas Item B (Pa) (kg/m 3 ) V (m/s) Air Water Ice Iron k. Speed in Gas 33 3l: Speed of Sound in Air 34 For a gas, formula can be rewritten: T Temperature (degrees Kelvin) m molecular mass (kg/mole) Heavier gas slower (CO 2 ) Lighter gas faster (Helium) c Fudge Factor (7/5 for diatomic molecules) R gas constant Joules/ K Mole RT m Rewrite the speed of sound in air in simpler form: c T Temperature (degrees Celsius) Speed at 0C is m/s m s Speed of sound increases approximately by 0.18% for every 1C increase. Hot day (30C) speed would be 5.3% faster T C. Wave Phenomena 1. Mersenne s Laws 2. Harmonics 3. Doppler Effect 35 1a. 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 Frequency is hence inversely proportional to string length L 1 1 f L 36 6

7 1b. Mersenne s Laws (1630) 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), or can rewrite in terms of density and string diameter d 37 1c. Guitar String Diameters For guitar, all strings same length L, and want tensions the same, so to get different frequencies, must vary diameters d of strings String Diameter Freq E mm 330 Hz B G D A E v F 2 d F ( ) Marin Mersenne The Father of Acoustics To double the frequency of a fixed length string, would need to double velocity, which requires 4 the tension! E2 is 2 octaves lower than E4 Or (1/4) the frequency Hence diameter is nearly 4x bigger! f 1 Ld F 2a. Harmonic Modes Daniel Bernoulli (1728?) shows string can vibrate in different modes, which are multiples of fundamental frequency (called Harmonics by Sauveur) 39 2b. Wavelengths of Harmonic Modes The wavelength of n-th mode is: 2L n n 40 n=1 f 1 1 2L n=2 f 2 =2f 1 n=3 f 3 =3f 1 n=4 f 4 =4f 1 L 2 2L 3 3 L 4 2 n=5 f 5 =5f 1 2c. Harmonic Series The musical notes of harmonic series Doppler Effect (approximate formula) 1842 Christian Doppler shows detected frequency f d depends upon: f s frequency of source fd fs 1 v relative speed between detector and source c velocity of sound in medium So if moving 10% speed of sound towards you, the frequency will be increased 10% 42 v c Reference: Sound: 7

8 3c Shock Waves If travel faster than speed of sound, the waves form a cone of angle c sin Mach Number v 43 References & Notes Old 10 min movie on sound Ruben s Tube Demo: Mythbusters: Dispersion of sound waves Part 1: Part 2: Echo Tube (shows dispersion of sound) References 45 References Donald E. Hall, Musical Acoustics (3rd edition) [Brooks/Cole 2002] Rossing, Moore & Wheeler, The Science of Sound (3rd ed, Addison Wesley 2002) Fletcher & Rossing, The Physics of Musical instruments (2 nd ed, Springer 1998) Olson, Music, Physics & Engineering (2 nd ed, Dover 1967) 8