1. 3. Energy in Waves Updated May 13, 2012 1

2. Index A.Waves and Amplitude 1.Amplitude and Particle Velocity 2.Impedance (of a spring) 3.Specific (acoustic) Impedance B.Energy in Waves 1.Energy 2.Power & Intensity 3.Decibels C.Propagation of Waves 1.Inverse Square Law 2.Decibels and Distance 3.Sound Pressure Level D.References 2

3. A. Waves & Amplitude 1. Amplitude and Particle Velocity The maximum speed of the molecules in a sound wave is related to the maximum displacement (amplitude “A”) by the frequency “ f ” : 3

4. 2. Impedance of a Spring Impedance “Z” relates the (maximum) force of a spring “F” to the (maximum) oscillation speed of the mass “v”. Hooke’s Law relates force, spring constant “k” to maximum stretch, or amplitude “A” Substitute for amplitude (previous page) Substitute for frequency : Yields impedance: 4

5. 3. Impedance & Acoustic Velocity The relation between the wave pressure “P” and the oscillating speed “v” of the molecules. The “Specific” Acoustic Impedance “z” can be calculated from the medium’s density “  ” and coefficient of stiffness “B” Units: “rayl” (Pascal  sec/meter) z=420 rayls for air at room temp 5

6. B. Energy in Waves 1. Energy Definition a)Kinetic Energy of molecule (where “v” is the “particle speed”, NOT the wavespeed “c”): b)Energy Density (energy per unit volume) where  is mass density of gas in kg/m 3. 6

7. B.1.c Energy Density c) Rewrite equation substitute  =z/c substitute v=P/z –“z” is the impedance, –“c” is the wavespeed –“P” the amplitude of the pressure wave. For a “loud” P=1 Pascal wave, energy density is only 3x10 -6 J/m 3 7

8. 2. Power & Intensity Power is rate of using energy. Units=Watts=Joule/sec Intensity is the amount of power per unit area: Intensity measured in Watts/square meter [e.g. sunlight is 1400 Watts/meter 2 ] The ear can hear as small as 10 -12 Watts/m 2 Intensity is proportional to the square of the pressure amplitude Minimum ear can hear is 2x10 -5 Pascals 8

9. 2b SPL: Sound Pressure Level Intensity I (Watts/m 2 ) is the energy density “u” times the wavespeed c. Hence Intensity can be written: So, a ratio of intensities is proportional to ratio of squares of the pressures in a media 9

10. 3a. Decibels: Fechner’s Law 1860 Fechner’s Law As stimuli are increased by multiplication, sensations increase by addition (Sensation grows as the logarithm of the stimulus) Example: A 10x bigger intensity sound is “heard” as only 2x bigger by the ear 10 Gustav Theodor Fechner (1801-1887)

11. 3b. Which sounds half as loud as first? Reference: http://www.phys.unsw.edu.au/jw/dB.html 11

12. 3c. Decibel Scale The decibel is a logarithmic scale A multiplicative factor of 10x in intensity is +10 db 0 dbis threshold of hearing 1 dbis just noticeable difference 15 dbis a whisper 60 dbis talking 120 dbis maximum safe level 150 dbis jet engine (ear damage) 180 dbstun grenade 12 ================== Power RatiodB ___________________ 0.5-3 10 2+3 5+710 2013 5017 10020 100030 1000040 ==================

13. 3d. dBSPL Hence we can write the “decibels” in terms of ratio of pressure to the threshold pressure P 0 =2x10 -5 Pa. The result is called “dBSPL” (decibels of sound pressure level) to distinguish it from “regular” dB (which is based upon sound intensity where I 0 =10 -12 Watts/m 2 ). In this derivation we assumed that the Impedance of air did not change with the amplitude of the pressure wave. This is only approximately true, so a meter reading dBSPL might not get exactly the same answer as a meter reading sound intensity dB, which is why we distinguish between them. 13

14. 3e. Threshold of Hearing At 1000 Hertz, human can hear a pressure displacement of 2x10 -5 Pascals 14

15. C. Propagation of Waves 1. Inverse Square Law (a) Alkindus (al-Kindi 801- 873), Based upon optics of Euclid, knew that light rays are scattered in a cone with the light source as apex, hence PROBABLY knew that the intensity of light drops off in proportion to the increase in the surface area (i.e. square of the distance) 15

16. C1b. Inverse Square Law 16

17. C1c. Inverse Square Law 17 Apparent Luminosity drops off inversely proportional to squared distance. Sun at Jupiter (5x further away than earth) would appear 1/25 as bright. Kepler knew this Gravity follows this So does sound!

18. C.2 Inverse Square law and dB Factor in 10 in distance is 1/100 the intensity of sound Corresponds to drop of 20 dB. 18 DistanceIntensitydB 11120 100.01100 0.000180 10000.000,00160 10,00010 -8 40 100,00010 -10 20 1,000,00010 -12 0

19. C.3b Sound Pressure and Distance Intensity drops with square of distance (total power L 0 ) Intensity is proportional to square of amplitude (i.e. pressure amplitude) Hence, pressure amplitude drops proportional to inverse of distance Amplitude of displacement will then also be inversely proportional to distance 19

20. C.3c Sound Pressure and Distance Factor in 10 in distance is 1/100 the intensity of sound But only 1/10 the amplitude in pressure 20 DistanceIntensityPressure (PA) 111 100.010.1 1000.0001.01 10000.000,001.001 10,00010 -8 10 -4 100,00010 -10 10 -5 1,000,00010 -12 10 -6

21. C3d Monopole Radiating 21

22. D. Notes/References/Links http://en.wikipedia.org/wiki/Sound_pressure_level What is a decibel (audio demonstrations at): http://www.phys.unsw.edu.au/jw/dBNoFlash.html http://www.phys.unsw.edu.au/jw/dBNoFlash.html http://www.phys.unsw.edu.au/jw/dB.html 22

23. Things to do Need a dB vs dBSPL comparison table Need a dBSPL vs P table Need a distance vs dB and dBSPL table Slide 18 (threshold of hearing vs frequency) belongs in next topic lecture topic. Delete from here.

24. Table of displacement vs dB for air at 1000 Hz Relate particle displacement to impedance Example, 60 dB corresponds to what pressure displacement? P^2=ZI What velocity? V^2=I/z What amplitude at 1000 Hz?

25. fix The calculation of minimum pressure does not match minimum intensity. Is there issues about rms vs peak running around?