Waves

1. Waves

2. Wave math I • f(x-vt) represents a positive moving wave at wave speed v.

3. Pure sine wave: one particular wave type • y = A sin(kx-wt) • What is k? Wave number, k=2p/l. • Does this formula have the y=f(x-vt) form? Yes! • To appreciate the physical significance of the wave formula with two variables (x and t) freeze one and look at the function. Freezing t is like taking a snap shot of the wave. Freezing x is like looking at one point on the wave as time passes.

4. Why are sine waves so important? • Sine waves are the fundamental building blocks out of which any wave shape can be constructed. • Mathematically they form a complete orthogonal set of functions. • Sine functions define the idea of frequency – only an infinite sine wave has a single pure frequency. All other waves have some combination of frequencies.

5. Fourier transforms and spectrum • We typically think of sounds and signals in the time domain; however, representing those signals in the frequency domain can give much more information than a time representation alone. • The process of converting from time to frequency is known as spectral analysis. As you learned in PHYS1600—the ear is a mechanical spectrum analyzer.

6. Fourier series and spectrum • Any periodic function can be broken down into the sum of harmonics of sine and cosine waves. • Building a signal up by adding sine (and cos) waves is called synthesis. Breaking a signal up to find the coefficients is called spectral analysis or Fourier analysis.

7. How can the coefficients be found? • How can we determine the an and bn values for a given function f(t)? • Orthogonality of the sine function m = n > 1 • m ≠ n

8. Coefficient formula • This shows that it is possible to find the coefficients • In practice, we will let the computer do this operation numerically (the Fast Fourier Transform).

9. Odd and Even functions • Odd and even refer to symmetry about zero. • Is the sine function odd or even? • What about cosine? • Consequence—odd functions can be made up of the addition of sine waves alone, even cosine • What about non-symmetric functions?

10. Quick question • Odd or even?

11. Next?

12. Finally

13. Odd, even or some combo • The combination of odd and even in most functions means that the Fourier transform consists of sines and cosines. But with complex exponential notation we already have both. The amplitude A can be complex—this mixes sines and cosines

14. Wave stuff you should remember! Note: k is called wave number, w is called angular frequency

15. Complex wave representation • Remember complex exponentials are just sine or cosine functions in disguise!

16. The Wave Equation • Just as for simple harmonic motion, all wave motion is described by a mathematical relation (technically a partial differential equation) • Every wave function y=f(x-vt) satisfies this equation.

17. Wave speed • The wave speed given by n=fl or by the wave equation is the wave speed for a pure sine wave of a given single frequency. This is called the phase velocity. • In audio range acoustics the speed of sound is essentially constant for all frequencies. • If the velocity changes with frequency then a pulse (many superposed sine waves) travels with a different velocity—the group velocity.

18. Wave properties • Superposition of waves • Interference • Diffraction • Reflection and refraction • Acoustic impedance concept

19. Superposition • Waves can occupy the same part of a medium at the same time without interacting. Waves don’t collide like particles. • At the point of overlap the net amplitude is the sum of all the separate wave amplitudes. Summing of wave amplitudes leads to interference. • Constructive versus destructive interference.

20. Superposition II • We use the additive property of superposition when we synthesize waveforms. We create a bunch of separate sine waves of different frequencies, amplitudes, and relative phases and just add them. • Note that when we make sound waves numerically in the computer we do not need to include kx term. Why not?

21. Diffraction • Bending of waves around objects and through openings. • Huygen’s principle—every point of a wave front becomes a point source for new wave fronts. • Transmission line matrix method demos.

22. Reflection and Refraction • http://webphysics.ph.msstate.edu/jc/library/24-2/simulation.html • http://www.sciencejoywagon.com/physicszone/lesson/otherpub/wfendt/huygens.htm • Refraction does not come up too much in acoustics.

23. Path length difference and phase • In many cases you can determine the existence of constructive or destructive interference by examining the path length difference between interfering waves. • Math to convert path length difference to phase difference • For pure constructive or destructive Df= mp • Constructive for m=0,2,4,6.. • Destructive for m=1,3,5,7…

24. Diffraction & interference example

25. Simple case—Lloyds mirror • What wavelengths will interfere destructively? (Assume no inversion on reflection)

26. Speaker enclosures and baffles • What is the purpose of a baffle? • Prevent destructive interference between front and back emitted waves from a speaker. • Why are circular baffles bad? • Baffle Step • 6 db difference between low frequencies and high frequencies; cross-overs.

27. Baffle Step • Low frequencies are diffracted in all directions • High frequencies are more directional in the forward direction

28. Circular baffle example • The dip is at about 460 Hz. Does this agree with a simple interference calculation? • Plot is relative to infinite baffle

29. What is the consequence of a circular baffle? Spectral hole.

30. Speaker placed off center in a rectangular baffle

31. Edge Diffraction

32. Pressure variation from a sphere • Normal incidence (q=0), high frequency, why the 6 dB rise? [y axis is in relative db]

33. Edge diffraction interference

34. Diffraction of sound around the head • Diffraction as a function of angle around head for three different frequencies. • Why the big variation with frequency?

35. Sound, pressure, and thermodynamics • Sound in air is the result of air molecule movement (displacement). • More air molecules in a given volume of space equals an increase in air pressure • Kinetic model of a gas—little molecules whizzing around banging into each other and the walls of the container • Ideal gas equation PV=nRT

36. Pressure and displacement Animation courtesy of Dr. Dan Russell, Kettering University

37. Physical model of gases • Air consists of mainly nitrogen (78%) molecules, along with oxygen (21%). • At room temp the average molecule is moving at about 400 m/s. • The average mass of a molecule is 5.4x10-26 kg • The average size of a molecule is 2x10-10 m • The average spacing between molecules is 30 x 10-10 m

38. What causes air pressure? • Pressure is caused by the reaction force of the collisions of gas molecules with any surface exposed to the gas. • Pressure increases with the number of gas molecules because there are more collisions. • Pressure increases with temperature (for same density of molecules) because the molecules are moving faster.

39. Ideal Gas Equation • PV=nRT • P – pressure (Nm-2) • V – volume (m3) • n – number of moles of gas • R – gas constant 8.31 Jmol-1K-1 • T – temperature in degrees Kelvin (K) • Isothermal versus adiabatic processes

40. Isothermal example • T is a constant. If n is a constant (R is always constant) then Right Hand side of equation is a constant • P1V1=P2V2 • If we reduce the volume the pressure rises • Big change in V use formula • Small DV we can show that

41. Adiabatic example • Adiabatic process—no heat flows so the temperature of the gas can vary. • Sound waves—the pressure variations happen so fast so that heat cannot be redistributed. Thus, sound pressure variations are adiabatic. • In a fixed volume of space through which a sound wave passes what factors in the ideal gas law are constant?

42. Adiabatic processes and sound • PVg=constant g depends on the gas involved usually 1.333 • We can show that for small changes • Look back at Helmholtz resonator derivation…

43. Sound is an adiabatic process • At the high and low pressure regions of a sound wave the temperature is slightly high and low respectively. • If very large amplitude sound waves can be formed the temperature difference can be used to make acoustic coolers (refrigerators). • Adiabatic nature sets speed of sound.

44. Relation between Displacement and Pressure Amplitude • Back in PHYS1600 we learned that displacement and pressure amplitude are p/2 (a quarter wavelength) out of phase. • Redo that old argument quickly. • Now we can also relate the relative amplitudes of pressure amplitude and displacement

45. Definition of the variables • p0 is the pressure amplitude of the wave. • r0 is the density of air (1.29 kg m-3) • w is the angular frequency • vs is the speed of sound in air • eo is the displacement amplitude

46. Review of Sound Pressure Level • You should be able to convert SPL to pressure amplitude. • You should be able to convert a pressure amplitude to a decibel value in SPL. • Example: What is the displacement amplitude of a 10 dB SPL pure tone at 1000 Hz? • Convert SPL to p0 • Use p0 and e0 formula

47. Acoustic impedance • Analogous quantity to electrical impedance. • Electrical impedance from Ohm’s law • Z=V/I • What is V? It is related to the “force” that pushes on the charges. • What is I? It is related to the velocity of the charges in the circuit. • Acoustic impedance: Zac=Force/Velocity

48. Strings • The two important physical parameters for a string are • m mass per unit length (kg/m) • T tension in the string (N) • Speed of wave, v, on a stretched string is given by

49. Review of standing wave resonances • Fundamental and harmonics • n is the harmonic number 1,2,3… • L is the string length • v is the wave velocity on the string ( )

50. String impedance • Impedance for a string: • Different forms of same equation depending on what parameters you know. • Why do the string as an example? Easiest to visualize in a reflection configuration.