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. Wave stuff you should remember!

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

6. 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.

7. 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.

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

9. 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.

10. 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?

11. 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.

12. Reflection and Refraction • http://webphysics.ph.msstate.edu/jc/library/24-2/simulation.html • Refraction does not come up too much in acoustics.

13. 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 • Remember Df= mp • Constructive for m=0,2,4,6.. • Destructive for m=1,3,5,7…

14. Diffraction & interference example

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

16. 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?

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

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

19. Speaker placed off center in a rectangular baffle

20. Edge Diffraction

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

22. Edge diffraction interference

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

24. 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

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

26. 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

27. 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.

28. 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

29. 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

30. 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?

31. 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…

32. 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.

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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 ( )

39. 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.

40. Reflection at the junction between two strings—reflection formula • What values change as the wave travels from one medium to the next and which are the same? • Tension, mass per unit length, wave velocity, frequency, wavelength • What conditions must be met at the junction between two strings with different m?

41. Boundary conditions • At the junction the wave amplitudes must agree, otherwise the string comes apart! (frequency must be the same in each medium!) • Harder to see but the slopes at the junction must agree. The string must not “kink”. • How to calculate the expression for the reflection coefficient. Start by imagining the situation—incident wave goes along to junction where it is partially reflected and partially transmitted.

42. Reflection formula • Three waves amplitudes, incident A1, reflected B1, and transmitted A2 • Continuity of amplitude means that at the junction Incident+Reflected=Transmitted • We can choose to set the junction at x=0. • Time term is the same in all three and cancels.

43. Reflection formula II • Condition 1: A1+B1=A2 • Now continuity of slope is a bit more complicated, but leads to Condition 2: Z1(A1-B1)=Z2A2 • What is the reflection coefficient, r? r = B1/A1 • What is the transmission coefficient, t? t = A2/A1

44. Solving our two boundary condition equation for r and t gives: • Reflection • Transmission

45. What can we learn from r and t? • The transmitted pulse is never inverted. • The reflected pulse is inverted if Z2 > Z1. • Example: tied down end of a string has infinite Z. (Velocity is always zero independent of force) Thus for a wave from a string hitting a tied down end r=-1. The wave is inverted on reflection.

46. Animation • Light string to heavy string (low m to large m) Animation courtesy of Dr. Dan Russell, Kettering University

47. Impedance in pipes and ducts • Sound traveling in pipes and ducts where the wavelength is smaller than the dimensions of the duct form a one-dimensional wave system much like waves on a string. • The acoustic impedance of a pipe of cross-sectional area S is given by • Where r is the density of air (1.29 kg m-3) and vs is the speed of sound.

48. Reflection and transmission formulae are identical • Junction between two pipes leads to reflection given by

49. Acoustic pipe filters • A muffler shaped system of pipes acts to filter sound of particular frequencies. We can figure out which frequencies are reflected by combining inversion on reflection, path length differences, and interference.

50. Calculate the phase • We will look at the reflected signal. • Remember Z2>Z1 means inversion on reflection which is the same as a p phase shift. • For path length phase change Df is given by