Test Review Sound Chapters 15,16,17 & 18

1. Test Review SoundChapters 15,16,17 & 18 PHYS 2326-29

2. Simple Harmonic MotionChapter 15 PHYS 2326-23

3. Concepts to Know • Displacement • Restoring Force • Hooke’s Law • Spring Constant • Amplitude • Period • Frequency • Angular Frequency • Circle of Reference

4. Hooke’s Law • F = -kx • Restoring force always directed towards equilibrium • Newton’s second law F=ma ma=-kx, a = -kx/m or –(k/m) x k is the spring constant, m the mass and a is the acceleration

5. Amplitude • x(t) = Acos(ωt+Φ) • A is the amplitude or peak displacement • Since cos varies between +/-1.0, A determines just how far the object moves away from the equilibrium point • Note A doesn’t affect how x changes in time and the angle doesn’t change the peak amplitude

6. Velocity & Acceleration • dx/dt = - ωAsin(ωt+Φ) = v • d(dx/dt)/dt =- ω2Acos (ωt+Φ) = a • Since sin varies between +/- 1 so • ωA becomes the peak velocity • Note while Amplitude (position) doesn’t depend upon ω, velocity & acceleration do

7. Period & Frequency • ω is the angular frequency and it is normally in radians / second when t is in seconds. Since a cycle is complete for 2π radians, then one can have a frequency f • f = ω/ 2π or ω = 2πf f is normally in Hertz, Hz • The time it takes to complete a cycle is T, the period. In mks units – T is in seconds • f = 1/T or T = 1/f

8. Energy • Total energy is the sum of kinetic, Ek or K, and potential energy, Ep or U. • Ek = ½ mv2 • E = K + U • From chapter 7.8 The work done within a system by a conservative force equals the decrease in potential energy of the system • Since F=-kx • U=1/2 kx2

9. Total Energy

10. Waves Chapter 15 PHYS 2326-24

11. Transverse Wave • A wave that vibrates the medium perpendicular to the direction of travel Examples • Ocean waves – rise and fall • Electromagnetic waves • A rope or a slinky shifted up and down or sideways • Earthquake S waves (secondary waves)

12. Longitudinal Waves • Sound waves compressing the air • A slinky (or spring) being compressed or expanded • Earthquake P waves (primary waves) • Some waves have both longitudinal and transverse components

13. Periodic Waves • Waves that repeat with the same waveform • A nonperiodic wave is often called a pulse

14. Sinusoidal Wave • A sinusoidal wave is described by a sine or cosine function.

15. Wavelength & Period y Variation at a point x over time T A t λ y Variation at a time over x A x

16. Equation of a Wave

17. Wave Number The wave number or angular wave number is defined as k= 2π / λ Remember the angular frequency is ω = 2π / T = 2 π f Using these our wave eqn turns into 16.10 y = A sin(kx – ωt) Also, our speed v = ω/k = f λ NOTE: this k is not the spring constant

18. Wave Velocity • Wave Velocity or Phase Velocity is the speed that the phase such as the crest or trough of the wave travels through the medium

19. Particle Velocity • The velocity of a particle (real or imagined) as it transmits the wave • This motion may be longitudinal as in sound or transverse as a a guitar string on ocean wave. The equation is for a velocity in the y direction, regardless of whether the wave is moving in the y direction

20. Partial Differential Wave Equation • The linear wave equation 16.27 gives a complete description of the wave motion including the wave speed • Partials are used since the function is of both t and x • This is good for traveling waves such as transverse displacement on a string or longitudinal displacement from equilibrium for soundwaves

21. Waves Chapter 16 & 17 PHYS 2326-25

22. Concepts to Know • Mass per Unit Length • Bulk Modulus • Young’s Modulus • Ratio of Heat Capacities • Ideal Gas • Pressure • Gas Constant • Molecular Mass • Absolute Temperature

23. Transverse Wave A wave on a string is usually transverse. Examples include plucking or strumming a guitar or banjo. Eqn 16.18 provides the basis for transverse waves on a string

24. General Expression • All mechanical wave speeds follow a general expression • For a string the elastic property was the tension T and the inertial property of the medium was the mass per unit length • See section 12.4 for review information

25. Bulk ModulusSound Waves in Liquid or Gas (fluid) • Speed of sound in a medium depends upon the compressibility and density of the medium B = bulk modulus (elastic property) ρ = density (inertial property)

26. Young’s ModulusSound Waves in a Solid For longitudinal waves in a solid material Y = Young’s modulus (elastic property) ρ = density (inertial property)

27. Ideal Gas Law • Chapter 19.5 is the ideal gas law – this is covered in the other semester • PV = nRT (eqn 19/8) • P = pressure in pascals (N/m^2) • R = universal gas constant = 8.314 J/mol*k or 0.08206 L*atm / mol*K for Volume in liters and pressure in earth atmospheres • n = number of moles of gas • T = absolute temperature degrees kelvin

28. PV = nRT = N/Na RT where • N = number of molecules • Na = Avogadro’s number 6.022E+23 atoms/mole • PV = N Kb T • Kb = Boltzman’s constant R/Na = • 1.38E-23 J/K

29. Longitudinal Wavein an Ideal Gas • It is a pressure wave • B – bulk modulus = change in pressure / fractional change in volume • for an ideal gas – relationship (for adiabatic condition) is pVγ = constant • γ = cp/cv = specific heat at constant pressure divided by the specific heat at constant volume adiabatic = activity where no heat enters or leaves Bad = γp so • M = molecular mass (28.8 avg. for air)

30. Speed of Sound in AirAssuming Ideal Gas v = 348m/s or 1148 ft/s The eqn below 17.1, v=(331) sqrt(1+Tc/273) m/s comes from this

31. Energy in the Wave(String) • Chapter 16.5 – energy transfer per unit time in a string by a sinusoidal wave

32. Sound Pressure Energy • Like the transverse wave in a string, sound waves also have energy. • Rather than an transverse amplitude A we have a longitudinal displacement smax

33. Sound Pressure Energy • Like the transverse wave in a string, sound waves also have energy. • Rather than an transverse amplitude A we have a longitudinal displacement smax For a solid we have:

34. Interference of Waves Chapter 18 PHYS 2326-26

35. Concepts to Know • Boundary Conditions • Principle of Superposition • Standing Waves • Traveling Waves • Nodes • Antinodes • Interference • Destructive Interference

36. Concepts to Know • Constructive Interference • Normal Modes • Fundamental Frequency • Overtones • Harmonics • Fourier Analysis (Harmonic Analysis) • Open Pipe • Closed Pipe • Resonance

37. Phase • The difference in phase between two tones of the same frequency arriving from slightly different paths is Δθ = 2πΔx/λ If two speakers are driven by the same source there will be a difference in phase associated with the difference in distance and will depend upon the wavelength

38. Standing WavesChapter 18.2 • Given two waves, one going left, the other going right, we have:

39. Standing Waves • Note the equation for the standing wave doesn’t have (kx-ωt) so it is not a traveling wave • It is an oscillation pattern with a stationary outline • 2Asinkx is an amplitude that varies with position • cosωt is simple harmonic motion oscillating at an angular frequency ω

40. Nodes & Antinodes • Notice that has points where the amplitude is 0, kx=0,π, 2π, 3π and remember that k = 2 π/λ For x=0, λ/2, 3 λ/2,… =nλ/2, n= 0,1,2,3, … These are NODES • In between these nodes are points where 2Asinkx are maximum( sine = +/- 1). These occur at kx = π/2, 3 π/2, 5 π/2, …so that x = λ/4, 3 λ/4, 5 λ/4,… = n λ/4 for n=1,3,5, … These are called ANTINODES

41. Nodes & Antinodes Distance between adjacent antinodes = λ/2 Distance between adjacent nodes = λ/2 Distance between a node and an antinode = λ/4

42. Standing Waves on Fixed String A • possible wavelengths λn = 2L/n • fn = v/ λn = n v/2L n=1,2,3, … • these are quantized freq. since v= sqrt(T/μ) fn = (n/2L)sqrt(T/μ) Fundamental or first harmonic f1 = (1/2L)sqrt(T/μ) fn = n f1 N N L=1λ/2 n=1 A A L=2λ/2 N N n=2 N A A A N N N n=3 N L=3λ/2 L

43. Resonance • A system can oscillate in one or more normal modes. If a periodic force is applied to the system, the amplitude of the resulting motion is greatest when the frequency of the applied force is equal to one of the natural frequencies of the system

44. Periodic Sound WavesChapter 17.2 • Eqns 17.2 and 17.3 are for the displacement wave and the pressure wave • These are 90 degrees out of phase with the displacement wave being a cosine function and the pressure wave being a sine function • Pressure variations are a maximum when the displacement is 0 and the displacement wave is maximum where the pressure is a minimum

45. Air Columns(Pipes) • May be open or closed end • Closed end pipe is rigid – forms a pressure antinode • An open ended pipe essentially forms a displacement antinode (maximum variation) and a pressure node

46. Open Pipe Fundamental or first harmonic is ½ wavelength resonance λ1 = 2L, f1 = v/ λ1 = v/2L Second harmonic λ2 = L, f2 = v/ λ2 = v/L = 2 f1 Third harmonic λ3 = 2/3 L, f3 = 3v/2L = 3 f1

47. Closed Pipe Fundamental or first harmonic is ¼ wave resonant λ1 = 4L, f1 = v/ λ1 = v/4L Second harmonic – doesn’t exist (no even ones) Third harmonic λ3 = 4/3 L, f3 = 3v/4L = 3 f1 Fifth harmonic λ5 = 4/5 L, f5 = 5v/4L = 5 f1

48. Harmonics & Overtones • Harmonics are multiples of the fundamental frequency • Overtones are higher order frequencies above the fundamental frequency that are often harmonics but not always A harmonic (beyond first) is an overtone An overtone may not be a harmonic (integer multiple of the primary frequency) NOTE: The first overtone is the 2ndharmonic

49. Sound Waves Chapter 17 PHYS 2326-26

50. Concepts to Know • Sound • Perception • Pressure Amplitude • Intensity • Intensity Level • Decibels