PHY 101 General Physics I (Oscillations, Waves I and II) 2017/18 academic session

Transcription

1 PHY 101 General Physics I (Oscillations, Waves I and II) 017/18 acadeic session Segun Fawole PhD (AMInstP) Dept. of Physics & Engr. Physics Obafei Awolowo University, Ile-Ife, Nigeria. eail: gofawole@oauife.edu.ng Office: G14F Physics Block (White House) 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 1

2 Reference text: Fundaentals of Physics Halliday, Resnick and Walker (10 th edition extended) Ch. 15, 16 & 17 (Oscillations, Waves I & II) 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II)

3 OSCILLATIONS Oscillations describe otions which are repetitive. An iportant property of oscillatory otion is its FREQUENCY, f, which describes the nuber of oscillations per second. The SI unit for frequency is the Hertz (Hz), where 1 Hz = 1 oscillation per second =1 s -1. Motion which regularly repeats is called periodic or haronic otion. 1 The period, T, is the tie to coplete one oscillatio n, where T. For SIMPLE HARMONIC MOTION, displaceent of x is is the AMPLITUDE ANGULAR π T f. particle the PHASE ANGLE FREQUENCY. displaceent and velocity of a (axiu value the otion, the particle t is called the PHASE of the otion. of x, is given t x cost x). which is at tie 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) by the x tie of dependence of t deterined by the 0. f the where 3

4 The velocity of a particle undergoing siple haronic otion can be found by differentiating the displaceent, x(t) with respect to tie. dx vt dt where x t dx cost v t v sin t x sin dt is called the VELOCITY AMPLITUDE. Note, in SHM, the agnitude of the velocity is greatest when the displaceent is sallest and vice versa, since cos(q )=sin(q+ / ) Illustration I The Velocity of Siple Haronic Motion An object undergoing siple haronic otion takes 0.5 s to travel fro one point of zero velocity to the next such point. The distance between those points is 36 c. Calculate the (a) period, (b) frequency, and (c) aplitude of the otion. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 4

5 The acceleration can be found by differentiating the velocity in SHM, dv at dt x a a t d x sin t x cos t dt is known as the acceleration aplitude. t xt, which is the signature equation for SHM In SIMPLE HARMONIC MOTION, the acceleration a(t), is proportional to the displaceent x(t), but opposite in sign, and the two quantities are related by the square of the angular frequency Illustration II The Acceleration of Siple Haronic Motion What is the axiu acceleration of a platfor that oscillates at aplitude.0 c and frequency 6.60 Hz? 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 5

6 Solved Illustrations x a Practice question A particle with a ass of 1.00 kg is oscillating with siple haronic otion with a period of 1.00 s and a axiu speed of 1.00x10 3 /s. Calculate (a) the angular frequency and (b) the axiu displaceent of the particle. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 6

7 The Force Law for Siple Haronic Motion Figure 1: A linear siple haronic oscillator. The surface is frictionless. The block oves in siple haronic otion once it has been either pulled or pushed away fro the x = 0 position and released. Its displaceent is then given: 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 7

8 Fro Newton' s which is then for a Siple Re - arranging, oscillator Therefore the PERIOD of which is nd Substituti ng for the law, HOOKE'S LAW. F Haronic a Motion, F SPRING CONSTANT, related to the strength of k k oscillatio n for a a -ω x linear oscillator kx we get the angular frequency,, for a siple the spring constant, k, by is haronic given as : T π ω k 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 8

9 Siple Haronic Motion is the otion which is described by a particle of ass subject to a force which is proportional to the displaceent of the particle but opposite in sign A LINEAR HARMONIC OSCILLATOR force F x Illustration 3, describes a syste where the is proportional to x (rather than soe other power of 1. An oscillator consists of a block of ass kg connected to a spring. When set into oscillation with aplitude 35.0 c, the oscillator repeats its otion every s. Find the (a) period, (b) frequency, (c) angular frequency, (d) spring constant, (e) axiu speed, and (f) agnitude of the axiu force on the block fro the spring. x). 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 9

10 Solved Illustration 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 10

11 Practice Questions 1. A siple haronic oscillator consists of a block of ass.00 kg attached to a spring of spring constant 100 N/. When t = 1.00 s, the position and velocity of the block are x = 0.19 and v = /s. (a) What is the aplitude of the oscillations? What were the (b) position and (c) velocity of the block at t = 0 s?. Two particles oscillate in siple haronic otion along a coon straight-line segent of length A. Each particle has a period of 1.5 s, but they differ in phase by π/6 rad. (a) How far apart are they (in ters of A) 0.50 s after the lagging particle leaves one end of the path? (b) Are they then oving in the sae direction, toward each other, or away fro each other? 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 11

12 3. In the figure below, a spring block syste is put into SHM in two experients. In the first, the block is pulled fro the equilibriu position through a displaceent d1 and then released. In the second, it is pulled fro the equilibriu position through a greater displaceent d and then released. Are the (a) aplitude, (b) period, (c) frequency, (d) axiu kinetic energy, and (e) axiu potential energy in the second experient greater than, less than, or the sae as those in the first experient? 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 1

13 Energy in Siple Haronic Motion The POTENTIAL ENERGY of 1 1 U The KINETIC ENERGY of recalling t kx k x cos t 1 1 K The MECHANICAL ENERGY, E, is defined E E U t v x sin t 1 U K k that for SHM, K 1 kx x cos t x sin t k the syste is ω a linear oscillator given as by 1, then since cos q sin q 1, cos kx t sin t is given This iplies that the echanical energy of an oscillator is a constant and it is tie independent. 1 by 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 13

14 Angular Siple Haronic Motion Figure shows an angular version of a siple haronic oscillator; the eleent of springiness or elasticity is associated with the twisting of a suspension wire rather than the extension and copression of a spring as we previously had. The device is called a torsion pendulu, with torsion referring to the twisting. A torsion pendulu is an angular version of a linear siple haronic oscillator. The disk oscillates in a horizontal plane; the reference line oscillates with angular aplitude θ. The twist in the suspension wire stores potential energy as a spring does and provides the restoring torque Fig. : A torsion pendulu 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 14

15 The torsion wire twists, thereby storing potential energy in the sae way that a copressed spring does in the linear siple haronic operator. The torsion wire also gives rise to the RESTORING TORQUE, τ. For angular siple q fro its given by rest position (at q τ -θ haronic, where is called By analogy with the siple otion, 0) causes a rotating the TORSION CONSTANT. haronic oscillator the disk through an angle RESTORING TORQUE case, the PERIOD of an angular siple haronic oscillator is given by T π I κ where I is the rotational inertia of the oscillating disc 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 15

16 Illustration 4 Fig. 3a shows a thin rod whose length L is 1.4 c and whose ass is 135 g, suspended at its idpoint fro a long wire. Its period T a of angular SHM is easured to be.53 s. An irregularly shaped object, which we call object X, is then hung fro the sae wire, as in Fig. 3b, and its period T b is found to be 4.76 s. What is the rotational inertia of object X about its suspension axis? 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 16

17 Solution to Illustration 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 17

18 The siple pendulu A siple pendulu, which consists of a particle of ass (called the bob of the pendulu) suspended fro one end of an unstretchable, assless string of length L that is fixed at the other end. F sinq lgsinq τ r F rf l g Fro Newton's nd law, I lg sin q I oent of inertia, angular acceleration of the pendulu at angular displaceent, q. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 18

19 Therefore I lgq lg I q lg I T I lg, I l T l g (no dependence) Illustration Suppose that a siple pendulu consists of a sall 60.0 g bob at the end of a cord of negligible ass. If the angle u between the cord and the vertical is given by θ=(0.08 rad) cos[(4.43 rad/s)t + ϕ], what are (a) the pendulu s length and (b) its axiu kinetic energy? 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 19

20 Solution to Illustration 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 0

21 Physical (Real) pendulu A real pendulu, usually called a physical pendulu, can have a coplicated distribution of ass. Does it also undergo SHM? If so, what is its period? For a physical pendulu with sall aplitude, we can write; T I gh For real pendulu I differs and depends on the shape of the pendulu. Taking the pendulu to be a unifor rod of length L, suspended fro one end. Fro the parallel axis theore, A physical pendulu. The restoring torque is hfg sinθ. When θ=0, center of ass C hangs directly below pivot point O. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 1

22 I l l 8π l since T g gh 3g T l 3g 3 Thus, by easuring L and the period T, we can find the value of g at the pendulu s location. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II)

23 Daped siple haronic otion If the otion of an oscillator is reduced as a result of an external force, the oscillator and its otion are described as daped. If of the daping force is the oscillatin g syste, where, b is a sign indicates Fro Newton's that this law, proportional then F net F d force opposes bv DAMPING CONSTANT. nd nd Thesolution for this to the The inus the otion. a kx bv d x dt kx, dx b dt velocity 0 order differential equation An idealized daped siple haronic oscillator. A vane iersed in a liquid exerts a daping force on the block as the block oscillates parallel to the x axis 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 3

24 is given by x bt t x e cos ' t where For b ω' k 0 (no daping) b ω' is the angular frequency of the daped oscillator k as in SHM If thedapingconstant is sall,i.e., b k, then ω'. which tells us that, like the aplitude, the echanical energy decreases exponentially with tie. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 4

25 Forced oscillation and resonance If there is an external force periodically pushing an oscillating syste (such as soeone pushing a swing), this is known as FORCED or DRIVEN OSCILLATION. Two angular frequencies are connected with a body undergoing forced oscillation: (i) NATURAL ANGULAR FREQUENCY ( ) of the syste, which is the frequency at which the syste would oscillate if it was disturbed and left to oscillate freely; and (ii) ANGULAR FREQUENCY OF THE EXTERNAL DRIVING FORCE ( d ) which is the angular frequency of the force causing the driven oscillations. If = d, the syste is said to be in resonance. If this condition is achieved, the velocity aplitude, v is axiised (and so approxiately is the displaceent, x ). 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 5

26 Practice Questions 1. A 5.00 kg object on a horizontal frictionless surface is attached to a spring with k = 1000 N/. The object is displaced fro equilibriu 50.0 c horizontally and given an initial velocity of 10.0 /s back toward the equilibriu position. What are (a) the otion s frequency, (b) the initial potential energy of the block spring syste, (c) the initial kinetic energy, and (d) the otion s aplitude?. For the daped oscillator syste shown in the figure below, with = 50 g, k = 85 N/, and b = 70 g/s, what is the ratio of the oscillation aplitude at the end of 0 cycles to the initial oscillation aplitude? 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 6

27 Waves - Introduction Waves involves the transfer of energy through space without the transfer of atter. That is in a wave, inforation and energy ove fro one point to another but no aterial object akes that journey. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 7

28 There are THREE types of waves: a. Mechanical waves* - These are governed by Newton s laws and can only exist within a ediu (such as taut string, water, air e.t.c) b. Electroagnetic waves These are assless objects which require no aterial ediu to travel in. All e waves travel through vacuu at the sae constant speed (the speed of light, c = 3 x 10 8 s -1 ). Exaples of EM waves are visible light, UV and IR radiation, radio-waves, x-rays and gaa-rays. The only difference between these waves is their wavelength and their ode of origin whether atoic, nuclear etc. c. Matter waves These are quantu descriptions of subatoic particles such as electrons, protons e.t.c. They are described by the de Broglie wavelength, dependent on the particle oentu. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 8

29 Transverse and Longitudinal waves One way to investigate wave otion is to look at the WAVEFORM, which describes the shape of the wave (i.e. y=f(x)). Alternatively, one can onitor the otion of a particular eleent of the wave ediu (e.g., a string) as function of tie (i.e., =f(t)). In cases where the displaceent of each eleent in an oscillating string is perpendicular to the direction of wave travel, the wave said to be TRANSVERSE (i.e. a transverse wave, such as waves on a string). By contrast, if the displaceent is parallel to the direction of otion of the wave (as in sound waves), the otion is described at LONGITUDINAL (i.e., transitted via a longitudinal wave such as sound). 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 9

30 Transverse and Longitudinal waves Motion of particles and wave 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 30

31 Wavelength and Frequency 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 31

32 Wavelength and frequency To copletely describe a wave on a string (and the otion of any eleent along its length) a function which describes the shape of the wave as a function of tie t, is required. This eans we need a function of the for, y = f (x,t), where y is the displaceent in the up-down direction and x is the position along the string. tie t, The aplitude (y ) is the agnitude of the axiu displaceent. The phase is the arguent of (kx-ωt ). As the wave passes through a string eleent at a position, x, its phase changes linearly with tie. For a sinusoidal wave, thedisplaceent function of along thestring is y( x, t) y sin kx t y, as a for an eleent at position x 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 3

33 Wavelength and Period 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 33

34 Angular wave nuber and angular frequency 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 34

35 Speed of a travelling waves If Δx dx WAVE SPEED v as the ratio of. If Δt dt fixed point on the oving wavefor which is the having to tie wave Re calling gives k travels The phase, kx ωt in the the sae displaceent x - direction, wavefor ust reain constant since const dx dx k - ω 0 v dt dt k π T and ω v T. we can define y, then the phase of y y differentiating we take a defined sin the f T the by kx t. with respect 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 35

36 Illustration 1. A wave travelling along a string is described by y(x,t) = sin(7.1x -.7t) where the nuerical constants are in the SI units. a. What is the aplitude of the wave? b. What is the wavelength, period and frequency of the wave? c. Calculate the transverse velocity, u and the transverse acceleration, a y of a string eleent at x=.5 c at t = 18.9 s. a. Recall: y(x,t) = y sin(kx ωt) aplitude = b. λ = π/k = (3.14)/7.1 = T = π/ω = (3.14)/.7 =.31 s f = 1/T = Hz 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 36

37 c. u = dy/dt = -ωy cos(kx ωt) = (-.7)(0.0037)cos{(7.1)(0.5) (.7)(18.9)} = /s transverse acceleration, a y = du/dt = -ω y sin(kx ωt) = (-.7) (0.0037)sin{(7.1)(0.5) (.7)(18.9)} = /s Practice Question A sinusoidal wave travelling in the positive direction has an aplitude of 0.15, a wavelength of 0.40 and a frequency of 8.0 Hz. The vertical position of an eleent of the ediu at t = 0 and x = 0 is also a. Find the wave nuber k, the period T, angular frequency and speed of the wave, v. b. Deterine the phase constant and write a general expression for the wave function 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 37

38 Wave speed on a stretched string For a wave to pass through a ediu (stretched string, water or etal), the part- icles in the ediu ust oscillate as the waves passes through. The ediu ust have ass (so the particles have K.E = 1 / v ) and elasticity (for potential energy = 1 / kx ). The ass and elasticity of the ediu deterine how fast the wave can travel through the ediu. A sall string eleent of length 5/07/018, fors a circular arc of radius R subtending an angle Ѳ. If a force, with a agnitude equal to the tension in the string pulls tangentially to the two ends, the horizontal coponents cancels. Δl v q q R Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 38 Dl O

39 The vertical coponentssu, providing the restoringforce F agnitudeof this force is, F Δl τ sin θ τθ τ R of The ass of the eleent Δ μdl the string. The string eleent oves in a circle, so Dl F a leads to R Iportant note: a, where µ is the linear density of μdl a v R v R The speed of a wave on an ideal stretched string only depends on the string s tension and linear density and not on the frequency of the waves. v 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 39

40 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) Energy and Power of a wave travelling along a string dk dt 1 dx μ. cos dt 1 y cos kx t μv y kx t

41 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 41

42 5/07/018 Wave equation The general differential equation that governs the travel of waves of all types is: t y x y Substituting v we have; 1 t y v x y Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 4

43 The principle of superposition for waves The principle of superposition states that when several effects occur siultaneously, their net effect is the su of the individual effects. Matheatically, this eans, y x, t y x, t y x, t ' 1 Overlapping waves add algebraically to produce a RESULTANT or NET WAVE. Note however, that overlapping waves do not in any way affect the travel of each other. Interference of waves If two sinusoidal waves of the sae wavelength and aplitude overlap, the resultant wave depends on the relative PHASES of the waves. If they are perfectly in phase they will add coherently (reinforce), doubling the displaceent observed for each waves. By contrast, if they are copletely out of phase (peaks of one wave atched by troughs of the other), they will copletely cancel out (annihilate each other) resulting in a flat string. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 43

44 x, t y sinkx t and y x, t y sinkx t If y1 1 These waves have the sae frequency deterined by ω, wavelength λ and aplitude y. They differ only by the phase constant f. Fro the principle of superposition, y' y' x, t y sin kx t y sin kx t Since, sin sin sin y sin cos kx t y sin kx t, cos sin x t y kx t aplitude of resultant wave y and phase angle. If waves interfere fully 'constructively'. If radians, the waves are copletely out of phase and interfere copletely, DESTRUCTIV ELY. cos 0, the two initial waves are 'in phase' the 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 44

45 Illustration What phase difference between two identical travelling waves, oving in the sae direction along a stretched string, resulting in the cobined wave having an aplitude 1.50 ties that of the coon aplitude of the two cobining waves? Express your answer in (a) degrees, (b) radians and (c) wavelengths (a). Let the phase difference be, resultant aplitude = 1.5 y = = 0.75 = (b) π rad = 180 = 1.45 rad y cos -1 cos o o (c) π rad = 1 wavelength 1.45 rad =0.3 wavelength o cos y 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 45

46 Phasor Waves can be represented in vector for using the idea of PHASORS. This is a vector whose aplitude is represented by the length which is equal to the agnitude of the wave and which rotates around the origin of a set of cartesian co-ordinates. The angular speed of the phasor about the origin is equal to the angular frequency, ω of the wave. As the phasor rotates about the origin, its projection, y 1 onto the vertical axis varies sinusoidally between +y and -y. y y, y 1 y,1 y =y 1 +y y y 1 y Two waves which travel along the sae string in the sae direction can be added using a PHASOR DIAGRAM. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 46

47 If be found x, t y,1 sin kx tand yx, t y, sin kx t ' ' ' of the for y x, t y sin kx t where y resultant is y y 1 ' y 1 using the PHASOR DIAGRAM. Adding and can vectorially the phasors x, tand yx, t at any instant, the agnitude of the resultant equals x, tand is the angle between the resultant and the phasor for y x, t. Phasors can be used to cobine waves even if they have different aplitudes Practice question 1. Two sinusoidal waves of the sae frequency travel in the sae direction along a string. If y,1 = 3.0 c, y, = 4.0 c, ϕ 1 = 0, and ϕ = π/ rad, what is the aplitude of the resultant wave?, the 1 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 47

48 Standing waves If two sinusoidal waves travel in opposite directions along a string, their su can be found using the principle of superposition. There are specific places along the resultant wave which DO NOT MOVE, known as NODES. Halfway between neighbouring nodes (the anti-nodes ) the aplitude of the resultant wave is axiised. Since the wave patterns do not ove in the x-direction, the wave patterns are called STANDING WAVES. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 48

49 If two sinusoidal waves of the sae aplitude and wavelength travel in opposite directions along a stretched string, their interference with each other produces a standing wave. The y 1 y' x, t y sin kx t and y x, t y sin kx t y' two cobining Recalling Fro the principle x, t y sin kx t y sin kx t x, t y sin kxcos t of waves sin sin which ake up the standing superposition, y' sin cos x, t y x, t y x, t. 1 wave are. The absolute value of [y sin(kx)] is the aplitude of oscillation at x. The aplitude varies with position for a standing wave. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 49

50 The aplitude is zero if sin aplitude is zero (i.e., nodes) occur at x n adjacent nodes areseparated by a distance of half kx 0.i.e., for integer n where kx nπ λ. For a standing waves, the wavelength. Siilarly, the axiu aplitude is y when sinkx 1 π λ x kx 1 n x n 1 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 50

51 Standing waves and resonance A standing wave can be set up by allowing a wave to be reflected at a boundary of a string. The interference of the original (incident) and reflected wave can interfere to give rise to a standing wave. (Note that for hard reflection, the reflection point ust be a fixed node.) If a taut string is fixed at both ends (such as in a guitar) and a continual sinusoidal wave is sent down fro one end, it will be subsequently reflected at the other end. The reflected wave and the next transitted wave will interfere. If ore waves are continually sent fro the generator, any such waves can add coherently. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 51

52 At certain frequencies, such behaviour results in STANDING WAVE PATTERNS on the string. Such standing waves RESONATE at fixed RESONANT FREQUENCIES. (Note that if the string is oscillated at a non-resonant frequency, a standing wave is NOT set up.) A standing This v is n n 1is called the fundaental ode or 'first haronic'. is The frequencies associated sybols, corresponds the wavespeed along called wave can be set up on a string the second haronic, f1,f, f3,, to resonance frequencies given f n the string. n of 3 the length L, by a wave third by with these odes are often given the f v n v L if λ where haronic and so on. L n 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 5

53 Illustrations 1. Two identical sinusoidal waves oving in the sae direction along a stretched string interfere with each other. The aplitude of each wave is 9.8 and the phase difference between the is 100 o. (a) What is the aplitude of the resultant wave due to the interference between these two waves? (b) What phase difference (in both radians and in fractions of wavelength) will give a resultant wave aplitude of 4.9? ' y y cos phase difference is given by cos cos ' Since y y cos 4.9 cos radians 9.8 single wavelength corresponds to, thus, in wavelengths, the rads.6 rads/ wavelength 0.4 wavelengt hs 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 53

54 . Two sinusoidal waves y 1 (x,t) and y (x,t) have the sae wavelength and travel together in the sae direction along a string. Their aplitudes are y 1, =4.0 and y, =3.0 and their phase constants are 0 and π / 3 respectively. What are the aplitude, y and phase constant b of the resulting wave? / 3 y y 1 y 5/07/018 Adding y' Adding y' Fro an aplitude The phase constant is h v y' y y the horizontal 1 the 1 cos sin 0 y cos 4 3cos vertical 0 y sin 0 3sin Pythagoras theore, the resultant wave has of y' coponents coponents tan x, t 6.1sin kx t 0.44rads rads Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 54

55 Sound waves Sound Waves: can be generally defined as longitudinal waves whose direction of particle oscillations (vibration) are parallel to the direction of travel through soe ediu (such as air). If a point source, P, eits sound waves, wavefronts and rays describe the direction of travel of the waves. Wavefronts correspond to surfaces over which the wave has the sae displaceent value. Rays are lines ray drawn perpendicular to wavefronts which indicate the direction of travel of the waves. Note that in real bodies, wavefronts spread out in 3 diensions in a P spherical pattern. Far fro the point source the wave- fronts can appear as planes or straight lines to an observer. wavefronts ray planes 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 55

56 Three Types of Sound Waves: -Audible waves lie within the range of sensitivity (0 Hz - 0 khz) of huan ear could be generated by usical instruents, huan voices, loudspeakers -Infrasonic waves have frequencies below the audible range (< 0 Hz). -Elephants counicate in this range with each other even with Kiloeters separation. -Ultrasonic waves have frequencies above the audible range (> 0 khz). It is used in edical iaging, huan ear cannot detect this at all but dogs can. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 56

57 Speed of sound The speed of any echanical wave depends on the physical properties of the ediu through which it travels. As a sound wave passes through air, we can associate a potential energy with periodic copressions and expansions of sall volue eleents. The BULK MODULUS is the property which deterines the volue change in a aterial when exposed to an external pressure (p=f/a). BULK MODULUS is defined by B Dp DV V where ΔV/V is the fractional change in volue produced by a change in pressure Δp.Since the signs of Δp and ΔV are always opposite, a inus sign is included to ake B a positive quanity. The speed of sound for a longitudin al wave in a ediu is give by v B 5/07/018 where is the density of the ediu. V air (0 o C)=343 s -1 V water (0 o C)=148 s -1 V steel = 5941 s -1 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 57

58 P is easured in force per unit area pascal (Pa) The speed of sound also depends on teperature and it is given by V V And V s speed of sound in air (331/s is air teperature in o s C. 1 T 73 c o C at 0 o C). T c 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 58

59 Intensity and sound level The sound waves eitted equally in all direction by a source is called a spherical wave and its Pave Pave given by : I A 4r where r is the radius of the sphere. intensity is The sound level is easured in decibel and defined log in the Io is I is In ters of threshold threshold equation takes care of the referenceintensity. intensity in W of of db, - pain (1.00 W The to which corresponds and is - hearing (1.00 x 10 threshold of ) 10 db - the W wide - ) range of 0 db I s I as: 10(dB)log 10 I o the huan ear can detect hearing equals 1x10 - W - easured in decibels db. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 59

60 Health hazards sets in by 90 db and be found In nearby jet air plane (150 db) high blood pressure, anxiety, nervousness due to noise pollution Siren (rock contact) (10 db) Bus+traffic (80 db) Vaccu cleaner (70 db) Noral conversation (50 db) Mosquito buzzing (40 db) Whisper (30 db). The huan ear is sensitive to frequencies between 0 KHz Rustling leaves (10 db) Threshold of hearing (0 db) happens to be the standard reference frequency in the acoustics 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 60

61 Doppler effect Doppler effect is the frequency change experienced when a source of sound wave oves towards and away fro a stationary observer. The Doppler effect describes how sound waves fro a point source (such as a car or train or star or galaxy!)) are apparently shifted in frequency for an observer which is oving relative to that source. O B S O A In a period T, source oves a distance x v s T v f s and wavelength is shortened by this aount such that ' sd But, thus OA hears the frequency : v f v s f 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 61

62 f ' f v ' ' v observer. However, v v f v v O s s f v f v v f Frequency heard when source is B s (increases in observed frequency). ' easures, oving and hear a towards reduced the ' ' v frequency f given by f f v v s (decreasein observed frequency) when a source oves s away. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 6

63 f ' is the air, Hence, the general expression for Doppler effect is : ' v f v where v o v o f v s is the observers and independent of air - fixed frae and v stationary, i.e., v or is 0). distance velocity if the detected frequency by the observer. v is v o is the relative speed of the source speed relative it were in otion the speed of the detector (or 'observer' ) sound relative to the sae air - fixed reference frae.(note in ost cases, either the source or the detector is o S v S through to an 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 63

64 When the otion of the detector or source are towards each other, the sign on its speed gives an UPWARD SHIFT IN FREQUENCY. When the otion of the detector or source are away fro each other the sign on its speed gives a DOWNWARD SHIFT IN FREQUENCY. The beat phenoenon is the difference between two sound frequencies which is used by usician to tune their instruents for sound frequencies f 1 and f, beat frequency is: f = f 1 f. 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 64

65 Illustration A rocket oves at a speed of 4 /s through stationary air directly towards a stationary pole while eitting sound waves at a source frequency of f =150Hz. (a) What frequency is easured by a detector attached to the pole? (b) If soe of the sound waves reflect fro the pole back to the rocket, what frequency f does the rocket detect for the echo? 1 v v D 343s 0 f ' f Hz. The - sign on botto 1 1 v v S 343s 4s gives an INCREASE in observed frequency for relative otion towards source. 1 1 v v D s s f ' f The sign on top gives Hz v vs s an INCREASE in observed frequency for relative otion towards source (pole). 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 65

66 5/07/018 Olusegun G. Fawole - PHY 101 (oscillations, waves I & II) 66