Andy Buffler Department of Physics University of Cape Town

Size: px
Start display at page:

Download "Andy Buffler Department of Physics University of Cape Town"

Transcription

1 University of Cape Town Department of Physics PHY014F Vibrations and Waves Part Coupled oscillators Normal modes of continuous systems The wave equation Fourier analysis covering (more or less) French Chapters, 5 & 6 Andy Buffler Department of Physics University of Cape Town 1

2 Problem-solving and homework Each week you will be given a take-home problem set to complete and hand in for marks... In addition to this, you need to work through the following problems in French, in you own time, at home. You will not be asked to hand these in for marks. Get help from you friends, the course tutor, lecturer,... Do not take shortcuts. Mastering these problems is a fundamental aspect of this course. The problems associated with Part are: -, -3, -4, -5, -6, 5-, 5-8, 5-9, 6-1, 6-, 6-6, 6-7, 6-10, 6-11, 6-14 You might find these tougher: 5-4, 5-5, 5-6, 5-7

3 French page 0 The superposition of periodic motions Two superimposed vibrations of equal frequency x = A cos( ω t+ φ ) x = A cos( ω t+ φ ) 0 combination can be written as x= Acos( ω t+ φ) 0 Using complex numbers: ω t A β A 1 + φ 0 1 A φ φ j( 0t 1) z1 = Ae ω + φ 1 j 0t z = Ae ω + φ ( ) z = z1+ z { } 1 z= e A+ Ae 1 j( ω t+ φ ) j( φ φ ) Phase difference Then and φ = φ φ1 A = A + A + AA cos( φ φ ) ( ) Asin β = A sin φ φ 1 3

4 Superposed vibrations of slightly different frequency: Beats If we add two sinusoids of slightly different frequency we observe beats ω1 ω ω1+ ω cosω1t+ cosωt = cos t cos t cosω1t cosωt x 1 x ω1 and ω t French page x 1 +x t cosωt + cosω t 1 T beat = π ω ω 1 ω1 ω cos t 4

5 Combination of two vibrations at right angles x= A cos( ωt+ φ ) y = A cos( ω t+ φ )??? French page 9 Consider case where frequencies are equal and let initial phase difference be φ Write x= A cos( ω t) 1 0 and y = A cos( ω t+ φ) 0 Case 1 : φ = 0 x= A1cos( ω0t) y = A cos( ω t) 0 y A A = x Rectilinear motion 1 Case : φ = π x= A1cos( ω0t) y = A cos( ω t+ π ) = A sin( ω t) 0 0 x y 1 A + A = Elliptical path in clockwise direction 1 5

6 Combination of two vibrations at right angles Case 3 : φ x A cos( ω t) y = A cos( ω t+ π) = A cos( ω t) = π = y = A A 1 x Case 4 : φ 3π x= A cos( ω t) = 1 0 y = A cos( ω t+ 3π ) =+ A sin( ω t) 0 0 x y 1 A + A = Elliptical path in anticlockwise direction 1 Case 5 : φ = π 4 x= A1cos( ω0t) y = A cos( ω t+ π 4) 0 Harder to see use a graphical approach 6

7 Superposition of simple harmonic vibrations at right angles with an initial phase difference of π 4 7

8 Superposition of two perpendicular simple harmonic motions of the same frequency for various initial phase differences. 8

9 Abbreviated construction for the superposition of vibrations at right angles see French page 34. 9

10 Perpendicular motions with different frequencies: Lissajous figures See French page 35. Lissajous figures for ω = ω1 with various initial phase differences. φ = 0 π 4 π 3π 4 π 10

11 ω : ω1 1:1 Lissajous figures 1: 1:3 :3 3:4 3:5 4:5 5:6 φ = 0 π 4 π 3π 4 π 11

12 French page 11 Coupled oscillators When we observe two weakly coupled identical oscillators A and B, we see: x A t x B t these functions arise mathematically from the addition of two SHMs of similar frequencies so what are these two SHMs? These two modes are known as normal modes which are states of the system in which all parts of the system oscillate with SHM 1 either in phase or in antiphase.

13 Coupled oscillators A B x A x B t t 13

14 The double mass-spring oscillator k m x A m x B k Individually we know that For both oscillators: ω 0 = Now add a weak coupling force: mx A = kxa and mx B = kxb k m k m x A k c m x B k For mass A: mx A = kxa + kc( xb xa) k k or xa = ω0 xa +Ω ( xb xa) where ω 0 = c, Ω = m m 14

15 For mass A: For mass B: The double mass-spring oscillator x x x x A = ω0 A +Ω ( B A) x x x x B = ω0 B Ω ( B A) two coupled differential equations how to solve? Adding them: d ( x ) ( ) A + xb = ω0 xa + xb dt Subtract B from A: d ( x ) ( ) ( ) A xb = ω0 xa xb Ω xa xb dt Define two new variables: q1 = xa + xb q = xa xb called normal coordinates Then dq1 dt = ω q 0 1 and dq = ( ω + Ω ) q dt 0 15

16 The double mass-spring oscillator 3 The two equations are now decoupled dt Write dq dt dq1 = ω s f q = ω q 1 ω ω = ω s 0 f = ω0 + Ω s = slow f = fast which have the solutions: q = Ccos( ω t+ φ ) 1 s 1 q = Dcos( ω t+ φ ) f Since q1 = xa + xb and q = xa xb We can write A 1 1 = ( + ) x = ( q q ) x q q 1 and B 1 16

17 Then The double mass-spring oscillator 4 C D xa = cos( ωst+ φ1) + cos( ωft+ φ) C D xb = cos( ωst+ φ1) cos( ωft+ φ) So x A and x B have been expressed as the sum and difference of two SHMs as expected from observation. C, D, φ 1 and φ may be determined from the initial conditions. when x A = x B,then q = 0 there is no contribution from the fast mode and the two masses move in phase the coupling spring does not change length and has no effect on the motion ωs = ω0 when x A = x B,then q 1 = 0 there is no contribution from the slow mode the coupling spring gives an extra force each mass experiences a force ( k+ k c ) x giving k+ k ω c f = m 17 = ω 0 + Ω

18 The double mass-spring oscillator 5 symmetric mode antisymmetric mode mixed mode 18

19 The double mass-spring oscillator 6 We now have a system with two natural frequencies, and experimentally find two resonances. Amplitude Frequency 19

20 Pitch and bounce oscillator French page 17 d k k m L Pitching x A = x B x A Centre of mass stationary τ = I θ 1 1 kd( θd) = ml θ 1 θ = Two normal modes (by inspection): Bouncing x = x A B Restoring force = d x m = kx dt 6kd ml θ x B θ x A kx ω = bounce pitch ω = 6k d m L x B k m 0

21 N = 1π ω1 = ω0sin = ω0 1 ( + ) π ω = ω0sin = 3ω0 1 ( + ) 1

22 N = 3 N = 4

23 French page 136 fixed l N-coupled oscillators Tension T fixed y 1 3 p 1 p p+1 N Each bead has mass m consider transverse displacements that are small. α α p 1 p 1 3 p 1 p p+1 N Transverse force on p th particle: Fp = Tsinαp 1 + Tsinαp y y y y = T + T l l for small α p p 1 p+ 1 p 3

24 p N-coupled oscillators d yp yp yp 1 yp+ 1 yp dt l l F = m = T + T d yp ω ( ) 0yp ω 0 yp+ 1 yp = dt T where ω 0 =, p = 1, N ml a set of N coupled differential equations. Normal mode solutions: y = A sinωt Substitute to obtain N simultaneous equations or ( ) ω ω ( ) 0 Ap ω 0 Ap+ 1 Ap 1 p + + = 0 A + A ω + ω p+ 1 p 1 0 = Ap ω0 p 4

25 N-coupled oscillators 3 From observation of physical systems we expect sinusoidal shape functions of the form A = Csin pθ p Substitute into A + A ω + ω p+ 1 p 1 0 = Ap ω0 And apply boundary conditions A and 0 = 0 A N + 1 = 0 nπ find that θ = n = 1,, 3, N (modes) N + 1 There are N modes: and sin sin pn π ypn = Apn ωnt = Cn sinωnt N + 1 ω n = ω nπ 0 sin 1 ( N + ) 5

26 N-coupled oscillators 4 For small N: ω n = ω nπ 0 sin 1 ( N + ) ω n ω N+1 n 6

27 N-coupled oscillators 5 In many systems of interest N is very large and we are only interested in the lowest frequency modes. ω n ω 0 linear region For n << N : then ω n i.e. ωn n 0 n << N N+1 n nπ nπ sin = 1 1 ( N + ) ( N + ) nπ πω n + 1 N = ω0 = ( N ) for n << N 7

28 N-coupled oscillators 6 N coupled oscillators have N normal modes and hence N resonances response ω 8

29 Continuous systems 9

30 Continuous systems Consider a string stretched between two rigid supports x = 0 tension T String has mass m and mass per unit length x = L µ = ml Suppose that the string is disturbed in some way: x y The displacement y is a function of x and t : yxt (,) 30

31 Normal modes of a stretched string Consider the forces on a small length of string T θ + θ French page 16 θ T y x x + x Restrict to small amplitude disturbances then θ is small and y cosθ = 1 sinθ = tanθ = θ = x The tension T is uniform throughout the string. Net horizontal force is zero: Tcos( θ + θ) Tcosθ = 0 Vertical force: F = Tsin( θ + θ) Tsinθ Then y y F = T x+ x T x x x x 31

32 y y F = T x+ x T x x x Use dg gx ( + x) gx ( ) = dx x Then y F = T x x µ x y y = T x t x or ( ) giving Write Normal modes of a stretched string y µ y = x T t y 1 y = x v t One dimensional wave equation µ: mass per unit length Check: µ T has the dimensions 1 v Then v= T µ is the speed at which a wave propagates along the string see later 3

33 Normal modes of a stretched string 3 Look the standing wave (normal mode) solutions Normal mode: all parts of the system move in SHM at the same frequency Write: yxt (, ) = f( x)cosωt f( x) is the shape function substitute into wave equation y xt d f x (,) () = cos ωt x dx y (,) xt t ( ω ωt) = f( x) cos d f( x) 1 cos t = f( x)cos t dx ω ω ω v which must be true for all t then d f ω = f( x) dx v 33

34 Normal modes of a stretched string 4 d f ω = f( x) dx v which has the same form as the eq. of SHM: d x has general solution: x= Asin( ω0t+ φ) ω Thus we must have: f( x) = Asin x+φ v Apply boundary conditions: y = 0 at x = 0 and x = L dt = ω x 0 f (0) = 0 and f( L ) = 0 ω x = 0, f =0 : 0 = Asin 0 +Φ i.e. Φ=0 v ω ω x = L, f =0 : 0 = Asin L i.e. L= nπ v v n = 1,,3, 34

35 Normal modes of a stretched string 5 nπ v Write ω n = n = 1,,3, L xnπv nπx Therefore f( x) = Ansin = Ansin v L L shape function, or eigenfunction 1 ( π ) f( x) = Asin xl ( π ) f( x) = Asin xl 3 ( π ) f( x) = Asin 3 xl 4 ( π ) f( x) = Asin 4 xl 5 ( π ) f( x) = Asin 5 xl x = 0 x = L n = 1 n = n = 3 n = 4 n = 5 ω1 = πvl ω = πvl ω3 = 3πvL ω4 = 4πvL ω5 = 5πvL 35

36 Normal modes of a stretched string n = 1 n = n = 3 n = 4 36

37 37

38 Normal modes of a stretched string 6 Full solution for our standing waves: nπ x yxt (, ) = Ansin cosωnt L ω = The mode frequencies are evenly spaced: ωn = nω1 n nπ v L ω n ω 3 ω ω 1 (recall the beaded string) n This continuum approach breaks down as the wavelength approaches atomic dimensions also if there is any stiffness in the spring which adds an additional restoring force which is more pronounced in the high frequency modes. 38

39 Normal modes of a stretched string 7 All motions of the system can be made up from the superposition of normal modes nπ x yxt (, ) = Ansin cos( ωnt+ φn) n= 1 L with ω = n nπ v L Note that the phase angle is back since the modes may not be in phase with each other. 39

40 Whispering galleries best example is the inside dome of St. Paul s cathedral. If you whisper just inside the dome, then an observer close to you can hear the whisper coming from the opposite direction it has travelled right round the inside of the dome. 40

41 Longitudinal vibrations of a rod x x+ x section of massive rod French page 170 F 1 x + ξ x F + ξ + x+ ξ section is displaced and stretched by an unbalanced force ξ Average strain = ξ x Average stress = Y x Y : Young s modulus (stress) stress at x+ x = (stress at x) + x x 41

42 Longitudinal vibrations of a rod If the cross sectional area of the rod is α ξ F1 = αy ξ ξ and F x = αy + αy x x x ξ F1 F = αy x = ma x ξ αy x= ρα x x ξ t ξ ρ ξ or = x Y t ξ 1 ξ = x v t v = Y ρ 4

43 Longitudinal vibrations of a rod 3 Look for solutions of the type: ξ( xt, ) = f( x)cosωt ω where f( x) = Asin x+φ v Apply boundary conditions: one end fixed and the other free x = 0 : ξ (0, t) = 0 i.e. Φ= 0 ξ x = L : F = αy = 0 x ωl then cos = 0 v ω 1 or L= ( n ) π n = 1,,3, v 1 1 ( n ) πv ( n ) π Y The natural angular frequencies ωn = = L L ρ 43

44 x = 0 x = L n = 1 ω = 1 π L Y ρ n = ω = 3π L Y ρ n = 3 ω = 3 5π L Y ρ n = 4 n = 5 44

45 Normal modes for different boundary conditions Simply supported Clamped one end Free both ends Clamped both ends n = 1 n = n = 3 45

46 l The elasticity of a gas French page 176 A ρ, p Bulk modulus: K dp = V dv m 1 Kinetic theory of gases: Pressure p = 3 ρvrms = vrms 3Al E 1 If Ek = mvrms then p = k 3A l Now move piston so as to compress the gas work done on gas: W = pa l = Ek Ek l l 5 l p = E ( ) k = pa l p = p 3A l 3Al 3A l 3 l Then ( ) giving p 5 Kadiabatic = V = p V 3 and K v = = ρ p ρ 46

47 French page 174 Sound waves in pipes A sound wave consists of a series of compressions and rarefactions of the supporting medium (gas, liquid, solid) In this wave individual molecules move longitudinally with SHM. Thus a pressure maximum represents regions in which the molecules have approached from both sides, receding from the pressure minima. wave propagation 47

48 Longitudinal wave on a spring 48

49 Standing sound waves in pipes t = 0: Pressure p p 0 Flow velocity u 0 x x t = T : p x u x 49

50 Standing sound waves in pipes Consider a sound wave in a pipe. At the closed end the flow velocity is zero (velocity node, pressure antinode). At the open end the gas is in contact with the atmosphere, i.e. p = p 0 (pressure node and velocity antinode). Open end pressure node p Closed end pressure antinode p 0 u 0 velocity antinode velocity node 50

51 Standing sound waves in pipes 3 Pipe closed at both ends Pipe open at both ends Pipe open at one end L nλ nv = = f nv f = L nπ v ω n = L L ( 1) λ ( 1) n n v = = 4 4 f ( n 1) v f = 4L ( n 1) πv ωn = 51 L

52 Sound Audible sound is usually a longitudinal compression wave in air to which the eardrum responds. Velocity of sound (at NTP) ~ 330 m s -1 By considering the transport of energy by a compression wave, can show that P = π f ρavs m where A is cross sectional area of air column and s m is maximum displacement of air particle in longitudinal wave P Then intensity = = π f ρvsm unit: W m - A 5

53 Sound The human ear detects sound from ~10-1 W m - to ~1 W m - use a logarithmic scale for I : I Intensity level or loudness : β = 10log10 decibels I0 where I 0 = reference intensity = 10-1 W m - 53

54 Musical sounds Waveforms from real musical instruments are complex, and may contain multiple harmonics, different phases, vibrato,... Pitch is the characteristic of a sound which allows sounds to be ordered on a scale from high to low (!?). For a pure tone, pitch is determined mainly by the frequency, although sound level may also change the pitch. Pitch is a subjective sensation and is a subject in psychoacoustics. The basic unit in most musical scales is the octave. Notes judged an octave apart have frequencies nearly (not exactly) in the ratio :1. Western music normally divides the octave into 1 intervals called semitones... which are given note names (A through G with sharps and flats) and designated on musical scales. 54

55 Musical sounds... Timbre is used to denote tone quality or tone colour of a sound and may be understood as that attribute of auditory sensation whereby a listener can judge that two sounds are dissimilar using any criteria other than pitch, loudness or duration. Timbre depends primarily on the spectrum of the stimulus, but also on the waveform, sound pressure and temporal characteristics of the stimulus. One subjective rating scale for timbre (von Bismarck, 1974) dull sharp compact scattered full empty colourful colourless 55

56 Two dimensional systems y French page 181 Consider an elastic membrane clamped at its edges Δx Δy x the membrane has mass per unit area σ, and a surface tension S which gives a force SΔl perpendicular to a length Δl in the surface The forces on the shaded portion are SΔy SΔx SΔy SΔx 56

57 Two dimensional systems If the membrane is displaced from the z = 0 plane then a cross section through the shaded area shows: θ SΔy z SΔy θ + θ x x + x x looks exactly like the case of the stretched string. z The transverse force on the element will be S y x x And if we looked at a cross section perpendicular to this the transverse force will be z S x y y 57

58 Two dimensional systems 3 The mass of the element is σ x y. Thus z z z S y x+ S x y = σ x y x y t or z z σ z + = x y S t a two dimensional wave equation with the wave velocity v = S σ 58

59 Two dimensional systems 4 Look for normal mode solutions of the form: zxyt (,, ) = f( xg ) ( y)cosωt z d f = g y t ( )cosω x dx z d g = ( )cos f x ω t y dy z = f( xgy ) ( ) cos t ( ω ωt) d f dx d g g( y)cos ωt+ f( x)cosωt = dy ω f( xgy ) ( )cosωt v 1 d f 1 d g ω i.e. + = f dx g dy v In a similar fashion to the 1D case, find n1π x nπ y f( x) = An sin 1 and g( y) = Bn sin L x L y 59

60 Two dimensional systems 5 n1πx nπy then zxyt (,, ) = Cnn sin sin cos 1 ωn 1, nt L x L y where the normal mode frequencies are ω n, n 1 1 = + L x L y nπv n πv e.g. for a membrane having sides 1.05L and 0.95L then ωn 1, n πv n1 n = + L

61 Normal modes of a rectangular membrane 1,1 up down,1, 3,1 3, 61

62 Normal modes of a circular membrane 1,0,0 3,0 1,1,1, 6

63 Modes of vibration of a 38 cm cymbal. The first 6 modes resemble those of a flat plate... but after that the resonances tend to be combinations of two or more modes. 63

64 Normal modes of a circular drum 64

65 Chladni plates 65

66 Soap films 66

67 Holographic interferograms of the top and bottom plates of a violin at several resonances. 67

68 Holographic interferograms of a classical guitar top plate at several resonances. 68

69 Holographic interferograms showing the vibrations of a 0.3 mm thick trombone driven acoustically at 40 and 630 Hz. 69

70 Time-average hologram interferograms of inextensional modes in a C 5 handbell 70

71 n v L 3 v L v L 1 v L Normal modes of a square membrane f 4,3 area per point = f n, n 1 v L one point per normal mode 1 nv nv = + L L 0 0 v 1 L v L 3 v L 4 v L 5 v 0 L Normal modes having the same frequency are said to be degenerate n 1 71

72 Normal modes of a square membrane for large n 1 and n n df area = 1 ( π f ) df 4 f area per mode = v L Number of modes with frequencies between f and (f + df) = = 1 4 n 1 ( π f ) df π L f df v L v 7

73 French page 188 Three dimensional systems Consider some quantity Ψ which depends on x, y, z and t, e.g. the density of air in a room. In three dimensions: Ψ Ψ Ψ 1 Ψ + + = x y z v t which can be written: Ψ= 1 v Ψ t The solutions for a rectangular enclosure: ω n, n, n 1 3 nπv n πv n3π v L x L y Lz 1 = + + πv and for a cube: ω n1 n n = n + n + n 3 L,,

74 How many modes are there with frequencies in the range f and (f + df)? Set up an imaginary cubic lattice with spacing v L n f df Three dimensional systems n 1 and consider positive frequencies only. Volume of shell = Volume per mode = 1 (4 π f ) df 8 3 v L n 3 Number of modes with frequencies between f and (f + df) = = 1 8 (4 f ) π 4 π L f v 3 3 df df L v 3 74

75 Three dimensional systems 3 Number of modes with 4 πv f df frequencies between f and (f + df) = 3 v holds for any volume V provided its dimensions are much greater than the wavelengths involved. need to multiply by a factor of when dealing with electromagnetic radiation ( polarization states) Ultraviolet catastrophe for blackbody radiation Equipartition theorem: in thermal equilibrium each mode has an average energy kt in each of its two energy stores 1 B Hence, energy density of radiation in a cavity: 4 πv f df 1 µ df = 3 ( kt ) µ c 8 π f or µ = kt 3 c experiment!? f 75

76 Planck was able to show, effectively by assuming that energy was emitted an absorbed in quanta of energy hf, that the average energy of a cavity mode was not kt but hf hf kt e 1 Then where Planck s constant h = J K -1 µ df = 8 π c f df 3 e hf hf kt Planck s law 1 energy density no. of modes in range f to f +df average energy per mode which agrees extremely well with experiment. 76

77 77

78 French page 189 Introduction to Fourier methods We return to our claim that any physically observed shape function of a stretched string can be made up from normal mode shape functions. f( x) x nπ x i.e. f( x) = Bn sin n= 1 L a surprising claim? first find Bn 1 multiply both sides by sin n π x L and integrate over the range x = 0 to x = L L L n1πx n1πx nπ x f ( x)sin dx = sin Bn sin dx L L n= 1 L

79 Fourier methods L L 1 1 ( )sin sin n sin 0 0 n= 1 nπx nπx nπ x f x dx = B dx L L L If the functions are well behaved, then we can re-order things: L L 1 1 ( )sin n sin sin 0 n= 1 0 nπx nπx nπ x f x dx = B dx L L L [n 1 is a particular integer and n can have any value between 1 and.] L Integral on rhs: ( ) π ( + ) π π n n x n n πx sin sin = cos cos L L L L L n1 x n x 1 dx 1 1 dx

80 Both (n 1 + n) and (n 1 n) are integers, so the functions ( n1 ± n) π x cos L on the interval x = 0 to L must look like from which it is evident that ( ± ) L 1 cos 0 0 n n π x dx = L Fourier methods 3 Except for the special case when n 1 and n are equal then ( ) n n π x = L 1 cos 1 and L 0 cos ( ) n1 n π x dx L L = 80

81 L Fourier methods 4 Thus all the terms in the summation are zero, except for the single case when n 1 = n i.e. L n1π x 1 n1 n πx n1+ n πx f ( x)sin dx = B n cos cos dx 1 L L L L = B 0 0 n 1 ( ) ( ) i.e. L n1π x Bn = f ( x)sin dx 1 L L 0 We have found the value of the coefficient for some particular value of n 1 the same recipe must work for any value, so we can write: L nπ x Bn = f ( x)sin dx L L 0 81

82 Fourier methods 5 The important property we have used is that the functions 1 sin n π x and sin n π x L L are orthogonal over the interval x = 0 to x = L. i.e. L 0 n1π x nπ x dx sin sin L L = 0 if n1 n L if n1 = n Read French pages

83 Fourier methods 6 The most general case (where there can be nodal or antinodal boundary conditions at x = 0 and x = L) is A0 nπx nπx f( x) = + Ancos + Bnsin n= 1 L L where L nπ x An = f ( x)cos dx L L 0 L nπ x Bn = f ( x)sin dx L L 0 83

84 Fourier methods 7 One of the most commonly encountered uses of Fourier methods is the representation of periodic functions of time in terms of sine and cosine functions π Put Ω= T This is the lowest frequency in f() t clearly there are higher frequencies by the same method as before, write A0 πnt πnt f( t) = + Ancos + Bnsin n= 1 T T A 0 = + n= 1 ncos Ω+ nsin Ω T A n t B n t Bn = f t nωt dt T where A f ( t)cos( n t) dt and ( )sin( ) n 0 f() t = Ω T T T 0 84 t

85 Waveforms of... a flute a clarinet an oboe a saxophone 85

86 Fast Fourier transform experiments, 10 March

87 Fast Fourier transform experiments, 10 March

88 Fast Fourier transform experiments, 10 March

89 Fast Fourier transform experiments, 10 March

90 Fast Fourier transform experiments, 10 March

91 Odd functions An odd periodic function f( t) = f() t where T < t < T f() t f() t f() t t t t can be expressed as a sum of sine functions f( t) = Bn sin nωt n= 0 T Bn = f ( t)sin( nωt ) dt T 0 Ω= π T 91

92 Even functions An even periodic function f( t) =+ f() t where T < t < T f() t f() t f() t t t t can be expressed as a sum of cosine functions A f t = + A nωt ( ) 0 n= 1 n cos T An = f ( t)cos( nωt ) dt T 0 Ω= π T 9

93 Fourier methods Example Find Fourier coefficients for the case: f() t 1-1 This is an odd function: f( t) = f() t T Bn = f ( t)sin( nωt ) dt T 0 T T t f( t) = Bn sin nωt = (1) sin ( n t) dt ( 1) sin ( n t) dt T Ω + T Ω 0 T 1 T 1 = (1) cos ( nω t) + ( 1) cos ( nωt) T nω T nω n= 1 0 T 93 T

94 Ω= π T Fourier methods Example cont. 1 1 B = n n n n nπ + nπ [ cos π cos 0] [ cos π cos π ] For even n: cos nπ = cos nπ = 1 Beven = 0 For odd n: cos nπ = 1 and cos nπ = B = odd n ( 1 1) (1 1) nπ + nπ + = nπ n f( t) = sin Ω+ t sin3ω+ t sin5 Ω+ t... π

95 Fourier sums Example 3 terms T T 0 terms 4 terms 50 terms 8 terms 00 terms 95

96 Fourier sums Example 1 terms 0 T 8 terms 3 terms 0 terms 4 terms 50 terms 96

97 Fourier sums Example terms 0 T 0 terms 4 terms 50 terms 8 terms 00 terms 97

98 Time domain Frequency spectrum Fourier transforms t t f 1 7 f 1 f f t f t f t t f1 3 f1 5 f1 7f1 f1 3 f1 5 f1 7f1 f f 98

99 99

100 100

Wave Equation in One Dimension: Vibrating Strings and Pressure Waves

Wave Equation in One Dimension: Vibrating Strings and Pressure Waves BENG 1: Mathematical Methods in Bioengineering Lecture 19 Wave Equation in One Dimension: Vibrating Strings and Pressure Waves References Haberman APDE, Ch. 4 and Ch. 1. http://en.wikipedia.org/wiki/wave_equation

More information

PHYSICS 149: Lecture 24

PHYSICS 149: Lecture 24 PHYSICS 149: Lecture 24 Chapter 11: Waves 11.8 Reflection and Refraction 11.10 Standing Waves Chapter 12: Sound 12.1 Sound Waves 12.4 Standing Sound Waves Lecture 24 Purdue University, Physics 149 1 ILQ

More information

XI PHYSICS [WAVES AND SOUND] CHAPTER NO. 8. M. Affan Khan LECTURER PHYSICS, AKHSS, K. https://promotephysics.wordpress.

XI PHYSICS [WAVES AND SOUND] CHAPTER NO. 8. M. Affan Khan LECTURER PHYSICS, AKHSS, K. https://promotephysics.wordpress. XI PHYSICS M. Affan Khan LECTURER PHYSICS, AKHSS, K affan_414@live.com https://promotephysics.wordpress.com [WAVES AND SOUND] CHAPTER NO. 8 OSCILLATORY MOTION A motion in which an object moves to and fro

More information

-Electromagnetic. Waves - disturbance that propagates through space & time - usually with transfer of energy -Mechanical.

-Electromagnetic. Waves - disturbance that propagates through space & time - usually with transfer of energy -Mechanical. Waves Waves - disturbance that propagates through space & time - usually with transfer of energy -Mechanical requires a medium -Electromagnetic no medium required Mechanical waves: sound, water, seismic.

More information

Physics 123 Unit #3 Review

Physics 123 Unit #3 Review Physics 123 Unit #3 Review I. Definitions and Facts longitudinal wave transverse wave traveling wave standing wave wave front wavelength wave number frequency angular frequency period crest trough node

More information

Andy Buffler Department of Physics University of Cape Town

Andy Buffler Department of Physics University of Cape Town University of Cape Town Department of Physics PHY014F Vibrations and Waves Part 3 Travelling waves Boundary conditions Sound Interference and diffraction covering (more or less) French Chapters 7 & 8 Andy

More information

WAVES( SUB) 2. What is the property of the medium that is essential for the propagation of mechanical wave? Ans: Elasticity and inertia

WAVES( SUB) 2. What is the property of the medium that is essential for the propagation of mechanical wave? Ans: Elasticity and inertia WAES( SUB). What is meant by a wave? Ans: The disturbance set up in a medium is known as a wave. What is the property of the medium that is essential for the propagation of mechanical wave? Ans: Elasticity

More information

Vibrations and Waves Physics Year 1. Handout 1: Course Details

Vibrations and Waves Physics Year 1. Handout 1: Course Details Vibrations and Waves Jan-Feb 2011 Handout 1: Course Details Office Hours Vibrations and Waves Physics Year 1 Handout 1: Course Details Dr Carl Paterson (Blackett 621, carl.paterson@imperial.ac.uk Office

More information

SIMPLE HARMONIC MOTION

SIMPLE HARMONIC MOTION WAVES SIMPLE HARMONIC MOTION Simple Harmonic Motion (SHM) Vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium TYPES OF SHM THE PENDULUM

More information

PHY 103: Standing Waves and Harmonics. Segev BenZvi Department of Physics and Astronomy University of Rochester

PHY 103: Standing Waves and Harmonics. Segev BenZvi Department of Physics and Astronomy University of Rochester PHY 103: Standing Waves and Harmonics Segev BenZvi Department of Physics and Astronomy University of Rochester Sounds of the Universe NASA/JPL, September 2016 2 Properties of Waves Wavelength: λ, length

More information

SoundWaves. Lecture (2) Special topics Dr.khitam Y, Elwasife

SoundWaves. Lecture (2) Special topics Dr.khitam Y, Elwasife SoundWaves Lecture (2) Special topics Dr.khitam Y, Elwasife VGTU EF ESK stanislovas.staras@el.vgtu.lt 2 Mode Shapes and Boundary Conditions, VGTU EF ESK stanislovas.staras@el.vgtu.lt ELEKTRONIKOS ĮTAISAI

More information

Chapter 11 Vibrations and Waves

Chapter 11 Vibrations and Waves Chapter 11 Vibrations and Waves 11-1 Simple Harmonic Motion If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic.

More information

Copyright 2009, August E. Evrard.

Copyright 2009, August E. Evrard. Unless otherwise noted, the content of this course material is licensed under a Creative Commons BY 3.0 License. http://creativecommons.org/licenses/by/3.0/ Copyright 2009, August E. Evrard. You assume

More information

Standing Waves If the same type of waves move through a common region and their frequencies, f, are the same then so are their wavelengths, λ.

Standing Waves If the same type of waves move through a common region and their frequencies, f, are the same then so are their wavelengths, λ. Standing Waves I the same type o waves move through a common region and their requencies,, are the same then so are their wavelengths,. This ollows rom: v=. Since the waves move through a common region,

More information

PHYS-2020: General Physics II Course Lecture Notes Section VIII

PHYS-2020: General Physics II Course Lecture Notes Section VIII PHYS-2020: General Physics II Course Lecture Notes Section VIII Dr. Donald G. Luttermoser East Tennessee State University Edition 4.0 Abstract These class notes are designed for use of the instructor and

More information

Longitudinal Waves. waves in which the particle or oscillator motion is in the same direction as the wave propagation

Longitudinal Waves. waves in which the particle or oscillator motion is in the same direction as the wave propagation Longitudinal Waves waves in which the particle or oscillator motion is in the same direction as the wave propagation Longitudinal waves propagate as sound waves in all phases of matter, plasmas, gases,

More information

Sound. Speed of Sound

Sound. Speed of Sound Sound TUNING FORK CREATING SOUND WAVES GUITAR STRING CREATING SOUND WAVES Speed of Sound Sound travels at a speed that depends on the medium through which it propagates. The speed of sound depends: - directly

More information

Oscillatory Motion SHM

Oscillatory Motion SHM Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A

More information

Content of the course 3NAB0 (see study guide)

Content of the course 3NAB0 (see study guide) Content of the course 3NAB0 (see study guide) 17 November diagnostic test! Week 1 : 14 November Week 2 : 21 November Introduction, units (Ch1), Circuits (Ch25,26) Heat (Ch17), Kinematics (Ch2 3) Week 3:

More information

Transverse Wave - Only in solids (having rigidity), in liquids possible only on the surface. Longitudinal Wave

Transverse Wave - Only in solids (having rigidity), in liquids possible only on the surface. Longitudinal Wave Wave is when one particle passes its motion to its neighbour. The Elasticity and Inertia of the medium play important role in the propagation of wave. The elasticity brings the particle momentarily at

More information

Chap 12. Sound. Speed of sound is different in different material. Depends on the elasticity and density of the medium. T v sound = v string =

Chap 12. Sound. Speed of sound is different in different material. Depends on the elasticity and density of the medium. T v sound = v string = Chap 12. Sound Sec. 12.1 - Characteristics of Sound Sound is produced due to source(vibrating object and travels in a medium (londitudinal sound waves and can be heard by a ear (vibrations. Sound waves

More information

Oscillations Simple Harmonic Motion

Oscillations Simple Harmonic Motion Oscillations Simple Harmonic Motion Lana Sheridan De Anza College Dec 1, 2017 Overview oscillations simple harmonic motion (SHM) spring systems energy in SHM pendula damped oscillations Oscillations and

More information

Physics 121, April 3, Equilibrium and Simple Harmonic Motion. Physics 121. April 3, Physics 121. April 3, Course Information

Physics 121, April 3, Equilibrium and Simple Harmonic Motion. Physics 121. April 3, Physics 121. April 3, Course Information Physics 121, April 3, 2008. Equilibrium and Simple Harmonic Motion. Physics 121. April 3, 2008. Course Information Topics to be discussed today: Requirements for Equilibrium (a brief review) Stress and

More information

WAVES & SIMPLE HARMONIC MOTION

WAVES & SIMPLE HARMONIC MOTION PROJECT WAVES & SIMPLE HARMONIC MOTION EVERY WAVE, REGARDLESS OF HOW HIGH AND FORCEFUL IT CRESTS, MUST EVENTUALLY COLLAPSE WITHIN ITSELF. - STEFAN ZWEIG What s a Wave? A wave is a wiggle in time and space

More information

2 u 1-D: 3-D: x + 2 u

2 u 1-D: 3-D: x + 2 u c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience 2013-14 Onde 1 1 Waves 1.1 wave propagation 1.1.1 field Field: a physical quantity (measurable, at least in principle) function

More information

Wave Motion and Sound

Wave Motion and Sound Wave Motion and Sound 1. A back and forth motion that repeats itself is a a. Spring b. Vibration c. Wave d. Pulse 2. The number of vibrations that occur in 1 second is called a. A Period b. Frequency c.

More information

Maxwell s equations and EM waves. From previous Lecture Time dependent fields and Faraday s Law

Maxwell s equations and EM waves. From previous Lecture Time dependent fields and Faraday s Law Maxwell s equations and EM waves This Lecture More on Motional EMF and Faraday s law Displacement currents Maxwell s equations EM Waves From previous Lecture Time dependent fields and Faraday s Law 1 Radar

More information

Chapter 7 Hooke s Force law and Simple Harmonic Oscillations

Chapter 7 Hooke s Force law and Simple Harmonic Oscillations Chapter 7 Hooke s Force law and Simple Harmonic Oscillations Hooke s Law An empirically derived relationship that approximately works for many materials over a limited range. Exactly true for a massless,

More information

PREMED COURSE, 14/08/2015 OSCILLATIONS

PREMED COURSE, 14/08/2015 OSCILLATIONS PREMED COURSE, 14/08/2015 OSCILLATIONS PERIODIC MOTIONS Mechanical Metronom Laser Optical Bunjee jumping Electrical Astronomical Pulsar Biological ECG AC 50 Hz Another biological exampe PERIODIC MOTIONS

More information

Lecture #4: The Classical Wave Equation and Separation of Variables

Lecture #4: The Classical Wave Equation and Separation of Variables 5.61 Fall 013 Lecture #4 page 1 Lecture #4: The Classical Wave Equation and Separation of Variables Last time: Two-slit experiment paths to same point on screen paths differ by nλ-constructive interference

More information

Honors Classical Physics I

Honors Classical Physics I Honors Classical Physics PHY141 Lecture 31 Sound Waves Please set your clicker to channel 1 Lecture 31 1 Example Standing Waves A string of mass m = 00 g and length L = 4.0 m is stretched between posts

More information

BASIC WAVE CONCEPTS. Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, Giancoli?

BASIC WAVE CONCEPTS. Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, Giancoli? 1 BASIC WAVE CONCEPTS Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, 9.1.2 Giancoli? REVIEW SINGLE OSCILLATOR: The oscillation functions you re used to describe how one quantity (position, charge, electric field,

More information

Travelling and Standing Waves

Travelling and Standing Waves Travelling and Standing Waves Many biological phenomena are cyclic. Heartbeats Circadian rhythms, Estrus cycles Many more e.g. s Such events are best described as waves. Therefore the study of waves is

More information

Chapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx

Chapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx When the mass is released, the spring will pull

More information

CHAPTER 4 TEST REVIEW

CHAPTER 4 TEST REVIEW IB PHYSICS Name: Period: Date: # Marks: 74 Raw Score: IB Curve: DEVIL PHYSICS BADDEST CLASS ON CAMPUS CHAPTER 4 TEST REVIEW 1. In which of the following regions of the electromagnetic spectrum is radiation

More information

(Total 1 mark) IB Questionbank Physics 1

(Total 1 mark) IB Questionbank Physics 1 1. A transverse wave travels from left to right. The diagram below shows how, at a particular instant of time, the displacement of particles in the medium varies with position. Which arrow represents the

More information

APMTH 105 Final Project: Musical Sounds

APMTH 105 Final Project: Musical Sounds APMTH 105 Final Project: Musical Sounds Robert Lin, Katherine Playfair, Katherine Scott Instructor: Dr. Margo Levine Harvard University April 30, 2016 Abstract Strings, drums, and instruments alike are

More information

SURFACE WAVES & DISPERSION

SURFACE WAVES & DISPERSION SEISMOLOGY Master Degree Programme in Physics - UNITS Physics of the Earth and of the Environment SURFACE WAVES & DISPERSION FABIO ROMANELLI Department of Mathematics & Geosciences University of Trieste

More information

Waves and the Schroedinger Equation

Waves and the Schroedinger Equation Waves and the Schroedinger Equation 5 april 010 1 The Wave Equation We have seen from previous discussions that the wave-particle duality of matter requires we describe entities through some wave-form

More information

G r a d e 1 1 P h y s i c s ( 3 0 s ) Final Practice exam

G r a d e 1 1 P h y s i c s ( 3 0 s ) Final Practice exam G r a d e 1 1 P h y s i c s ( 3 0 s ) Final Practice exam G r a d e 1 1 P h y s i c s ( 3 0 s ) Final Practice Exam Instructions The final exam will be weighted as follows: Modules 1 6 15 20% Modules

More information

f 1/ T T 1/ f Formulas Fs kx m T s 2 k l T p 2 g v f

f 1/ T T 1/ f Formulas Fs kx m T s 2 k l T p 2 g v f f 1/T Formulas T 1/ f Fs kx Ts 2 m k Tp 2 l g v f What do the following all have in common? Swing, pendulum, vibrating string They all exhibit forms of periodic motion. Periodic Motion: When a vibration

More information

Electrodynamics HW Problems 06 EM Waves

Electrodynamics HW Problems 06 EM Waves Electrodynamics HW Problems 06 EM Waves 1. Energy in a wave on a string 2. Traveling wave on a string 3. Standing wave 4. Spherical traveling wave 5. Traveling EM wave 6. 3- D electromagnetic plane wave

More information

Partial Differential Equations

Partial Differential Equations Partial Differential Equations Xu Chen Assistant Professor United Technologies Engineering Build, Rm. 382 Department of Mechanical Engineering University of Connecticut xchen@engr.uconn.edu Contents 1

More information

The Schrödinger Equation in One Dimension

The Schrödinger Equation in One Dimension The Schrödinger Equation in One Dimension Introduction We have defined a comple wave function Ψ(, t) for a particle and interpreted it such that Ψ ( r, t d gives the probability that the particle is at

More information

PHY 103 Impedance. Segev BenZvi Department of Physics and Astronomy University of Rochester

PHY 103 Impedance. Segev BenZvi Department of Physics and Astronomy University of Rochester PHY 103 Impedance Segev BenZvi Department of Physics and Astronomy University of Rochester Midterm Exam Proposed date: in class, Thursday October 20 Will be largely conceptual with some basic arithmetic

More information

Chapter 15 SIMPLE HARMONIC MOTION

Chapter 15 SIMPLE HARMONIC MOTION Physics Including Human Applications 309 Chapter 15 SIMPLE HARMONIC MOTION GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define

More information

single uniform density, but has a step change in density at x = 0, with the string essentially y(x, t) =A sin(!t k 1 x), (5.1)

single uniform density, but has a step change in density at x = 0, with the string essentially y(x, t) =A sin(!t k 1 x), (5.1) Chapter 5 Waves II 5.1 Reflection & Transmission of waves et us now consider what happens to a wave travelling along a string which no longer has a single uniform density, but has a step change in density

More information

Solution The light plates are at the same heights. In balance, the pressure at both plates has to be the same. m g A A A F A = F B.

Solution The light plates are at the same heights. In balance, the pressure at both plates has to be the same. m g A A A F A = F B. 43. A piece of metal rests in a toy wood boat floating in water in a bathtub. If the metal is removed from the boat, and kept out of the water, what happens to the water level in the tub? A) It does not

More information

This Week. Waves transfer of energy and information. sound (needs an elastic medium)

This Week. Waves transfer of energy and information. sound (needs an elastic medium) This Week Waves transfer of energy and information sound (needs an elastic medium) Standing waves Musical instruments, guitars, pianos, organs Interference of two waves tuning a piano, color of oil films

More information

Chapter 1. Harmonic Oscillator. 1.1 Energy Analysis

Chapter 1. Harmonic Oscillator. 1.1 Energy Analysis Chapter 1 Harmonic Oscillator Figure 1.1 illustrates the prototypical harmonic oscillator, the mass-spring system. A mass is attached to one end of a spring. The other end of the spring is attached to

More information

EM Waves. From previous Lecture. This Lecture More on EM waves EM spectrum Polarization. Displacement currents Maxwell s equations EM Waves

EM Waves. From previous Lecture. This Lecture More on EM waves EM spectrum Polarization. Displacement currents Maxwell s equations EM Waves EM Waves This Lecture More on EM waves EM spectrum Polarization From previous Lecture Displacement currents Maxwell s equations EM Waves 1 Reminders on waves Traveling waves on a string along x obey the

More information

C. points X and Y only. D. points O, X and Y only. (Total 1 mark)

C. points X and Y only. D. points O, X and Y only. (Total 1 mark) Grade 11 Physics -- Homework 16 -- Answers on a separate sheet of paper, please 1. A cart, connected to two identical springs, is oscillating with simple harmonic motion between two points X and Y that

More information

Introduction to Vibration. Mike Brennan UNESP, Ilha Solteira São Paulo Brazil

Introduction to Vibration. Mike Brennan UNESP, Ilha Solteira São Paulo Brazil Introduction to Vibration Mike Brennan UNESP, Ilha Solteira São Paulo Brazil Vibration Most vibrations are undesirable, but there are many instances where vibrations are useful Ultrasonic (very high

More information

PHY 103 Impedance. Segev BenZvi Department of Physics and Astronomy University of Rochester

PHY 103 Impedance. Segev BenZvi Department of Physics and Astronomy University of Rochester PHY 103 Impedance Segev BenZvi Department of Physics and Astronomy University of Rochester Reading Reading for this week: Hopkin, Chapter 1 Heller, Chapter 1 2 Waves in an Air Column Recall the standing

More information

AP physics B - Webreview ch 13 Waves

AP physics B - Webreview ch 13 Waves Name: Class: _ Date: _ AP physics B - Webreview ch 13 Waves Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A large spring requires a force of 150 N to

More information

Good Vibes: Introduction to Oscillations

Good Vibes: Introduction to Oscillations Good Vibes: Introduction to Oscillations Description: Several conceptual and qualitative questions related to main characteristics of simple harmonic motion: amplitude, displacement, period, frequency,

More information

Modern Physics. Unit 1: Classical Models and the Birth of Modern Physics Lecture 1.2: Classical Concepts Review of Particles and Waves

Modern Physics. Unit 1: Classical Models and the Birth of Modern Physics Lecture 1.2: Classical Concepts Review of Particles and Waves Modern Physics Unit 1: Classical Models and the Birth of Modern Physics Lecture 1.: Classical Concepts Reiew of Particles and Waes Ron Reifenberger Professor of Physics Purdue Uniersity 1 Equations of

More information

Normal modes. where. and. On the other hand, all such systems, if started in just the right way, will move in a simple way.

Normal modes. where. and. On the other hand, all such systems, if started in just the right way, will move in a simple way. Chapter 9. Dynamics in 1D 9.4. Coupled motions in 1D 491 only the forces from the outside; the interaction forces cancel because they come in equal and opposite (action and reaction) pairs. So we get:

More information

Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is

Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring

More information

Born of the Sound Wave Equation

Born of the Sound Wave Equation Corso di Laurea in Fisica - UNITS ISTITUZIONI DI FISICA PER IL SISTEMA TERRA Born of the Sound Wave Equation FABIO ROMANELLI Department of Mathematics & Geosciences University of Trieste romanel@units.it

More information

Physics 9 Fall 2009 Homework 12 - Solutions

Physics 9 Fall 2009 Homework 12 - Solutions Physics 9 Fall 009 Homework 1 - s 1. Chapter 1 - Exercise 8. The figure shows a standing wave that is oscillating at frequency f 0. (a) How many antinodes will there be if the frequency is doubled to f

More information

Lab 11 - Free, Damped, and Forced Oscillations

Lab 11 - Free, Damped, and Forced Oscillations Lab 11 Free, Damped, and Forced Oscillations L11-1 Name Date Partners Lab 11 - Free, Damped, and Forced Oscillations OBJECTIVES To understand the free oscillations of a mass and spring. To understand how

More information

AP Physics Problems Simple Harmonic Motion, Mechanical Waves and Sound

AP Physics Problems Simple Harmonic Motion, Mechanical Waves and Sound AP Physics Problems Simple Harmonic Motion, Mechanical Waves and Sound 1. 1977-5 (Mechanical Waves/Sound) Two loudspeakers, S 1 and S 2 a distance d apart as shown in the diagram below left, vibrate in

More information

VII. Vibrations, Waves and Sound. Concept Review. Conflicting Contentions. 1. Vibrating Strings 2. The Speed of Sound in Metal 3.

VII. Vibrations, Waves and Sound. Concept Review. Conflicting Contentions. 1. Vibrating Strings 2. The Speed of Sound in Metal 3. VII. Vibrations, Waves and Sound Concept Review Conflicting Contentions 1. Vibrating Strings 2. The Speed of Sound in Metal 3. Wave Collisions Qualitative Reasoning 1. Piano Wires 2. Organ Pipes 3. Out

More information

Sound. p V V, where p is the change in pressure, V/V is the percent change in volume. The bulk modulus is a measure 1

Sound. p V V, where p is the change in pressure, V/V is the percent change in volume. The bulk modulus is a measure 1 Sound The obvious place to start an investigation of sound recording is with the study of sound. Sound is what we call our perception of the air movements generated by vibrating objects: it also refers

More information

(Total 1 mark) IB Questionbank Physics 1

(Total 1 mark) IB Questionbank Physics 1 1. A transverse wave travels from left to right. The diagram below shows how, at a particular instant of time, the displacement of particles in the medium varies with position. Which arrow represents the

More information

b) (6) With 10.0 N applied to the smaller piston, what pressure force F 2 (in newtons) is produced on the larger piston?

b) (6) With 10.0 N applied to the smaller piston, what pressure force F 2 (in newtons) is produced on the larger piston? General Physics I Exam 4 - Chs. 10,11,12 - Fluids, Waves, Sound Nov. 17, 2010 Name Rec. Instr. Rec. Time For full credit, make your work clear to the grader. Show formulas used, essential steps, and results

More information

Simple Harmonic Motion

Simple Harmonic Motion 3/5/07 Simple Harmonic Motion 0. The Ideal Spring and Simple Harmonic Motion HOOKE S AW: RESTORING FORCE OF AN IDEA SPRING The restoring force on an ideal spring is F x k x spring constant Units: N/m 3/5/07

More information

Kutztown Area School District Curriculum (Unit Map) High School Physics Written by Kevin Kinney

Kutztown Area School District Curriculum (Unit Map) High School Physics Written by Kevin Kinney Kutztown Area School District Curriculum (Unit Map) High School Physics Written by Kevin Kinney Course Description: This introductory course is for students who intend to pursue post secondary studies.

More information

Q ( q(m, t 0 ) n) S t.

Q ( q(m, t 0 ) n) S t. THE HEAT EQUATION The main equations that we will be dealing with are the heat equation, the wave equation, and the potential equation. We use simple physical principles to show how these equations are

More information

Chapter 20 Traveling Waves. Chapter Goal: To learn the basic properties of traveling waves. Slide 20-2

Chapter 20 Traveling Waves. Chapter Goal: To learn the basic properties of traveling waves. Slide 20-2 Chapter 20 Traveling Waves Chapter Goal: To learn the basic properties of traveling waves. Slide 20-2 Chapter 20 Preview Slide 20-3 Chapter 20 Preview Slide 20-5 result from periodic disturbance same

More information

cos(θ)sin(θ) Alternative Exercise Correct Correct θ = 0 skiladæmi 10 Part A Part B Part C Due: 11:59pm on Wednesday, November 11, 2015

cos(θ)sin(θ) Alternative Exercise Correct Correct θ = 0 skiladæmi 10 Part A Part B Part C Due: 11:59pm on Wednesday, November 11, 2015 skiladæmi 10 Due: 11:59pm on Wednesday, November 11, 015 You will receive no credit for items you complete after the assignment is due Grading Policy Alternative Exercise 1115 A bar with cross sectional

More information

PHY205H1F Summer Physics of Everyday Life Class 7: Physics of Music, Electricity

PHY205H1F Summer Physics of Everyday Life Class 7: Physics of Music, Electricity PHY205H1F Summer Physics of Everyday Life Class 7: Physics of Music, Electricity Noise and Music Musical Sounds Pitch Sound Intensity and Loudness Quality Fourier Analysis Electrical Forces and Charges

More information

The Trumpet: Demystified Using Mathematics

The Trumpet: Demystified Using Mathematics Jeff Ward The Trumpet: Demystified Using Mathematics Abstract Presented, is a model of how a trumpet works, starting from the simplest tube and increasing the complexity until faced with a realistic model.

More information

Fourier Series. Fourier Transform

Fourier Series. Fourier Transform Math Methods I Lia Vas Fourier Series. Fourier ransform Fourier Series. Recall that a function differentiable any number of times at x = a can be represented as a power series n= a n (x a) n where the

More information

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.

More information

Lecture Presentation Chapter 14 Oscillations

Lecture Presentation Chapter 14 Oscillations Lecture Presentation Chapter 14 Oscillations Suggested Videos for Chapter 14 Prelecture Videos Describing Simple Harmonic Motion Details of SHM Damping and Resonance Class Videos Oscillations Basic Oscillation

More information

SURFACE WAVE DISPERSION PRACTICAL (Keith Priestley)

SURFACE WAVE DISPERSION PRACTICAL (Keith Priestley) SURFACE WAVE DISPERSION PRACTICAL (Keith Priestley) This practical deals with surface waves, which are usually the largest amplitude arrivals on the seismogram. The velocity at which surface waves propagate

More information

1 (2n)! (-1)n (θ) 2n

1 (2n)! (-1)n (θ) 2n Complex Numbers and Algebra The real numbers are complete for the operations addition, subtraction, multiplication, and division, or more suggestively, for the operations of addition and multiplication

More information

dt r r r V(x,t) = F(x,t)dx

dt r r r V(x,t) = F(x,t)dx Quantum Mechanics and Atomic Physics Lecture 3: Schroedinger s Equation: Part I http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh Announcement First homework due on Wednesday Sept 14 at the beginning

More information

21.55 Worksheet 7 - preparation problems - question 1:

21.55 Worksheet 7 - preparation problems - question 1: Dynamics 76. Worksheet 7 - preparation problems - question : A coupled oscillator with two masses m and positions x (t) and x (t) is described by the following equations of motion: ẍ x + 8x ẍ x +x A. Write

More information

4. Sinusoidal solutions

4. Sinusoidal solutions 16 4. Sinusoidal solutions Many things in nature are periodic, even sinusoidal. We will begin by reviewing terms surrounding periodic functions. If an LTI system is fed a periodic input signal, we have

More information

Brenda Rubenstein (Physics PhD)

Brenda Rubenstein (Physics PhD) PHYSICIST PROFILE Brenda Rubenstein (Physics PhD) Postdoctoral Researcher Lawrence Livermore Nat l Lab Livermore, CA In college, Brenda looked for a career path that would allow her to make a positive

More information

4.5. Applications of Trigonometry to Waves. Introduction. Prerequisites. Learning Outcomes

4.5. Applications of Trigonometry to Waves. Introduction. Prerequisites. Learning Outcomes Applications of Trigonometry to Waves 4.5 Introduction Waves and vibrations occur in many contexts. The water waves on the sea and the vibrations of a stringed musical instrument are just two everyday

More information

Simple Harmonic Motion ===============================================

Simple Harmonic Motion =============================================== PHYS 1105 Last edit: May 25, 2017 SMU Physics Dept. Simple Harmonic Motion =============================================== Goal To determine the spring constant k and effective mass m eff of a real spring.

More information

Chapters 10 & 11: Energy

Chapters 10 & 11: Energy Chapters 10 & 11: Energy Power: Sources of Energy Tidal Power SF Bay Tidal Power Project Main Ideas (Encyclopedia of Physics) Energy is an abstract quantity that an object is said to possess. It is not

More information

Chapter 4. Oscillatory Motion. 4.1 The Important Stuff Simple Harmonic Motion

Chapter 4. Oscillatory Motion. 4.1 The Important Stuff Simple Harmonic Motion Chapter 4 Oscillatory Motion 4.1 The Important Stuff 4.1.1 Simple Harmonic Motion In this chapter we consider systems which have a motion which repeats itself in time, that is, it is periodic. In particular

More information

SECTION A Waves and Sound

SECTION A Waves and Sound AP Physics Multiple Choice Practice Waves and Optics SECTION A Waves and Sound 1. Which of the following statements about the speed of waves on a string are true? I. The speed depends on the tension in

More information

4 The Harmonics of Vibrating Strings

4 The Harmonics of Vibrating Strings 4 The Harmonics of Vibrating Strings 4. Harmonics and Vibrations What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school... It is my task

More information

Oscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance

Oscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance Oscillations Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance 1 Revision problem Please try problem #31 on page 480 A pendulum

More information

Copyright 2009, August E. Evrard.

Copyright 2009, August E. Evrard. Unless otherwise noted, the content of this course material is licensed under a Creative Commons BY 3.0 License. http://creativecommons.org/licenses/by/3.0/ Copyright 2009, August E. Evrard. You assume

More information

The Physics Behind the Cosmic Microwave Background

The Physics Behind the Cosmic Microwave Background The Physics Behind the Cosmic Microwave Background Without question, the source of the most precise information about the universe as a whole and about its early state is the cosmic microwave background

More information

df(x) = h(x) dx Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation

df(x) = h(x) dx Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation Chemistry 4531 Mathematical Preliminaries Spring 009 I. A Primer on Differential Equations Order of differential equation Linearity of differential equation Partial vs. Ordinary Differential Equations

More information

TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students.

TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students. 5 TEST 2 This test is on the final sections of this session's syllabus and should be attempted by all students. Anything written here will not be marked. Formulae and data E = hc " " = neµ = ne2 # m N

More information

Wheel and Axle. Author: Joseph Harrison. Research Ans Aerospace Engineering 1 Expert, Monash University

Wheel and Axle. Author: Joseph Harrison. Research Ans Aerospace Engineering 1 Expert, Monash University Wheel and Axle Author: Joseph Harrison British Middle-East Center for studies & Research info@bmcsr.com http:// bmcsr.com Research Ans Aerospace Engineering 1 Expert, Monash University Introduction A solid

More information

1.50 m, and a speed of 750 km/hr. What is the distance between adjacent crests of these waves? A) 9000 m B) 32,400 m C) 2500 m D) 9000 km E) 32,400 km

1.50 m, and a speed of 750 km/hr. What is the distance between adjacent crests of these waves? A) 9000 m B) 32,400 m C) 2500 m D) 9000 km E) 32,400 km Exam Physics 3 -TTh - Fall 2016 Name Email Perm# Tel # Remember to write all work in your Bluebook as well as put the answer on your Scantron MULTIPLE CHOICE. Choose the one alternative that best completes

More information

1. Data analysis question.

1. Data analysis question. 1. Data analysis question. The photograph below shows a magnified image of a dark central disc surrounded by concentric dark rings. These rings were produced as a result of interference of monochromatic

More information

Physics 240: Worksheet 24 Name:

Physics 240: Worksheet 24 Name: () Cowboy Ryan is on the road again! Suppose that he is inside one of the many caerns that are found around the Whitehall area of Montana (which is also, by the way, close to Wheat Montana). He notices

More information

EFFECTS OF PERMEABILITY ON SOUND ABSORPTION AND SOUND INSULATION PERFORMANCE OF ACOUSTIC CEILING PANELS

EFFECTS OF PERMEABILITY ON SOUND ABSORPTION AND SOUND INSULATION PERFORMANCE OF ACOUSTIC CEILING PANELS EFFECTS OF PERMEABILITY ON SOUND ABSORPTION AND SOUND INSULATION PERFORMANCE OF ACOUSTIC CEILING PANELS Kento Hashitsume and Daiji Takahashi Graduate School of Engineering, Kyoto University email: kento.hashitsume.ku@gmail.com

More information

2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements

2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements 1 Historical introduction The Schrödinger equation for one-particle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical

More information