2nd & 3th N.U.T.S. Workshops Gulu University Naples FEDERICO II University 4 – Waves 2nd & 3th NUTS Workshop ( Jan 2010) Wave Motion Waves are everywhere: Earthquakes, vibrating strings of a guitar, light from the sun; a wave of heat, of bad luck, of madness… Something moving, passing by, bringing a change and then going away, sometimes without a trace… Some waves are man-made: radio waves, mexican wave, microwaves, annoying sound waves of a physics lecture, crime wave rap music blaring out of an audio system in a car, on the crest of a wave, …. Waves Appl. 4- Waves 2 2nd & 3th NUTS Workshop ( Jan 2010) Wave Definition A wave is a travelling disturbance that transports energy but not matter. Mechanical waves require • • • Some source of disturbance A medium that can be disturbed Some physical connection between or mechanism through which adjacent portions of the medium influence each other All waves carry energy and momentum 4- Waves 3 2nd & 3th NUTS Workshop ( Jan 2010) Free Harmonic Oscillations & Sinusoidal Motion y (t ) A sin( t ) A sin( 2ft ) y height of the object with respect to its equilibrium position; A amplitude of the oscillations; ω 2πf angular frequency (in rad/s); f = 1/T regular frequency (in Hertz or cycles per second or s-1); T period of oscillations (in seconds) 4- Waves 4 2nd & 3th NUTS Workshop ( Jan 2010) Wavelength Waves are characterised by the same variables as the oscillation: amplitude, frequency, period, energy… But they have much more, because they propagate in space… l 4- Waves The two basic new parameters are: wavelength and wave speed 5 2nd & 3th NUTS Workshop ( Jan 2010) How to Calculate the Speed of a Traveling Wave? Some poles that are placed a wavelength l apart. The oscillations at the poles are always in phase. Time taken by a crest to travel between two consecutive poles = Period of the oscillation =T (exactly the time between two consecutive crests at a same pole!). l The wave speed, v, can be calculated as: v λ T 4- Waves 6 λf 2nd & 3th NUTS Workshop ( Jan 2010) Wave Motion (a travelling wave) l NO direct connection between the wave speed: v T and the speed of the oscillating material particles: v y (t ) 4- Waves dy λ A cos(t ) dt 7 2nd & 3th NUTS Workshop ( Jan 2010) How to Calculate the Frequency of a Travelling Wave? v l Blue light has shorter v f lf wavelength than red light; what about their frequencies? l T Sound and light are very different types of waves (cfr. later) When they have the same wavelength: which has the higher frequency? Free Harmonic Waves: Sinusoidal waves where the crests move with a constant speed, while the material elements oscillate harmonically. A sin( 2ft .....) or A cos( 2ft .....) What is missing here ?? 4- Waves 8 2nd & 3th NUTS Workshop ( Jan 2010) Transverse Waves material velocity wave speed Electric Field Transverse wave – material elements (medium) move (or variable Electric and Magnetic Field change) perpendicularly to the direction of wave propagation 4- Waves 9 2nd & 3th NUTS Workshop ( Jan 2010) Longitudinal Waves In a longitudinal wave the particle displacement is parallel to the direction of wave propagation. The animation shows a one-dimensional longitudinal plane wave propagating down a tube. The particles do not move down the tube with the wave; they simply oscillate back and forth about their individual equilibrium positions. Pick a single particle and watch its motion! The wave is seen as the motion of the compressed region (i.e. it is a pressure wave), which moves from left to right. 4- Waves 10 2nd & 3th NUTS Workshop ( Jan 2010) Transverse vs. Longitudinal Wave Both propagate from left to right, but cause disturbances in different directions, Dy and Dx. wavelength, l Dx(t ) Ax cos(t ) wavelength, l 4- Waves 11 2nd & 3th NUTS Workshop ( Jan 2010) Waves on a Spring 4- Waves 12 2nd & 3th NUTS Workshop ( Jan 2010) Harmonic Waves are not the only Possible Type of Waves! A wave can also have the shape of a propagating impulse. True for both transverse and longitudinal waves. 4- Waves 13 2nd & 3th NUTS Workshop ( Jan 2010) Wave Train An harmonic wave and a pulse are extreme cases. Intermediate case = a wave train = a finite duration sinusoidal 4- Waves 14 2nd & 3th NUTS Workshop ( Jan 2010) Mathematical Description of an Harmonic Wave y x l • Features to incorporate: at any space point the wave produces harmonic oscillations as: y(x) = Aycos(ωt+φ) ω angular frequency , φ initial phase If we “freeze” the wave in time, an harmonic function results in space: y(x) = Aycos(kx+φ) what is k ? If we freeze the wave and move 1 wavelength λ along it, the same level of disturbance y has to be found Therefore, it must be kλ=2π so that y ( x l ) A y cos( k ( x l ) ) A y cos( kx 2 ) y ( x ) 4- Waves 15 2nd & 3th NUTS Workshop ( Jan 2010) Time and Space in Harmonic Waves y ( t ) A y cos( t ) y ( x ) A y cos( kx ) phase k l 2 k 2 / l k is measured in m-1. What is its meaning? tells us every how many times per meter it is going to happen k / 2 to have a crest, freezing the time. / 2 f tells us every how many times per second there is a crest if the angular frequency position is frozen and the wave propagates y x l k is pretty much the same for space as is for time! k behaves like a spatial frequency and is usually called the “wave number” 4- Waves 16 2nd & 3th NUTS Workshop ( Jan 2010) Waves: Space and Time 1 x l y y ( t ) A y cos( t ) 2 / T T is period in time y ( x ) A y cos( kx ) k 2 / l l is period in space How do we combine the two equations (in time and in space)? 4- Waves 17 2nd & 3th NUTS Workshop ( Jan 2010) Waves Time and Space 2 At one specific point in space, x0, kx0 At one specific instant t0 , = t0 2 / T 2 f k 2 / l l v ; k f 2 f v kv v y ( x , t ) A y cos( kx t ) A y cos( kx k v t ) A y cos[ k ( x v t )] A crest at a point, where Position of the crest k ( x vt ) 0 x vt It is moving with wave speed v !!! 4- Waves 18 2nd & 3th NUTS Workshop ( Jan 2010) Harmonic Waves: Summary y ( x , t ) A y cos( kx t ) equation of a harmonic wave y ( x , t ) A y cos[ k ( x v t )] the same equation, in a form emphasizing propagation along x axis and wave speed v y ( x , t ) A y cos[ k ( x v t )] what would this one stand for? - v is changed to + v , the wave is propagating in the negative x direction, from right to left according to usual convention In this case location of a crest is given by cos[ k ( x v t )] 1 x vt 0 4- Waves x vt 19 2nd & 3th NUTS Workshop ( Jan 2010) An Example The figure shows a simple harmonic wave at t = 0, and later at t = 2.6 s. Write a mathematical description of the wave. 4- Waves 20 2nd & 3th NUTS Workshop ( Jan 2010) Wave Fronts Wave front is a continuous line or a surface connecting nearby wave crests. Plane waves Wave fronts = flat surfaces for plane light, radar or sound waves propagating in one direction without spreading Wave fronts = straight lines for ripples on water surface at shore line. “Straight” waves 4- Waves wave fronts 21 2nd & 3th NUTS Workshop ( Jan 2010) Spherical Wave Fronts Wave fronts = spherical surfaces for spherical waves originating from a point source and propagating in 3D space. Wave fronts = circles for waves on water surface originating from a point source. 4- Waves 22 2nd & 3th NUTS Workshop ( Jan 2010) Wave Intensity Wave intensity is the wave power per unit area: I = P/A Plane wave: its intensity remains constant because the wave front area remains constant Spherical wave: its intensity decreases with the distance, r, from the wave source I = P/4r2 4- Waves 23 2nd & 3th NUTS Workshop ( Jan 2010) Wave Interference When two (or more) waves of the same kind propagate through the same space region a composite wave results. (wave interference) It is constructive, when the waves reinforce each other. It is destructive, when they reduce each other’s amplitude. Usually the disturbances (displacements) the waves produce are added algebraically. This is called superposition principle. 4- Waves 24 2nd & 3th NUTS Workshop ( Jan 2010) Superposition of Pulses Constructive Interference Destructive Interference 4- Waves 25 2nd & 3th NUTS Workshop ( Jan 2010) Electromagnetic Waves Electromagnetic waves propagate along a direction perpendicular to electric and magnetic field, with a speed c=3x108 m/s in the vacuum wavelength and frequency are connected as: l 4- Waves c f 26 2nd & 3th NUTS Workshop ( Jan 2010) Electromagnetic Spectrum increasing frequency 4- Waves increasing wavelength 27 2nd & 3th NUTS Workshop ( Jan 2010) E.M. Waves: Frequency and Wavelength in the Vacuum 4- Waves 28 2nd & 3th NUTS Workshop ( Jan 2010) Rays and Wave Fronts In the wave formulation of optics, the mathematical model of thin beams (rays) correspond to lines perpendicular to the wave fronts 4- Waves 29 2nd & 3th NUTS Workshop ( Jan 2010) Light Refraction: Frequency and Wavelength As light travels from one medium to another, its frequency does not change • • Both the wave speed and the wavelength do change The wavefronts do not pile up, nor are created or destroyed at the media interface, so ƒ must stay the same v1 f1 f 2 l1 l2 v1 v2 l1 v2 l2 c / n1 c / n2 n2 n1 So: 4- Waves 30 2nd & 3th NUTS Workshop ( Jan 2010) Refractive Index and Speed of the E.M. Wave When an E.M. Plane wave θ1 traveling through a transparent medium encounters an interface with another transparent medium θ1 θ2 the refraction phenomenon can be described with the mathematical model of rays n2 θ2 corresponding to lines perpendicular to the wave Refraction Index n = c1/c2 fronts (plane) If c2 < c1 (medium 2 is more dense n1 than medium 1) the refracted light beams bends toward the normal to the interface 4- Waves 31 2nd & 3th NUTS Workshop ( Jan 2010) Wavelengths and Colours We see colour when waves of different wavelengths enter our eyes! Light with wavelength of 650 nm appears red when it enters a viewers eye Light with wavelength of 520 nm appears green when it enters a viewers eye Light with wavelength of 470 nm appears blue when it enters a viewers eye 4- Waves 32 2nd & 3th NUTS Workshop ( Jan 2010) Two Different Wavelengths and Colours What happens when two or more waves with different wavelengths reach your eye? Light with both wavelengths 650 nm and 520 nm appears yellow when it enters a viewers eye Light with only wavelength 580 nm ALSO appears yellow when it enters a viewers eye (A DEEPER YELLOW THAN FOR THE CASE ABOVE) 4- Waves 33 2nd & 3th NUTS Workshop ( Jan 2010) What is White Light? Light which is a mixture of all wavelengths of the visible light spectrum It appears WHITE when it reaches your eye No single wavelength (monochromatic) wave appears white when it reaches your eye! 4- Waves 34 2nd & 3th NUTS Workshop ( Jan 2010) Colours of NON Luminous Objects The colour of an object depends on the range of visible light wavelength that it absorbs The absorbed range depends on the chemical properties of the substances composing the objects The light coming from the object and reaching the eye does not have the absorbed range The perceived colour depends on the NOT absorbed range 4- Waves 35 2nd & 3th NUTS Workshop ( Jan 2010) Color of the Sun from under Water The Sun looks yellow, since its radiation intensity has a maximum at l = 550 nm, which is yellow light. Wavelength of this yellow light in water will be l’ = l / nwater = l/1.33 = 413 nm, corresponding to violet light. Is the Sun going to look violet from under water? NO! The only thing that matters is the wavelength inside your eye, which is defined by n of eye vitreous humor. f eye f air 4- Waves l eye l air n air n eye 36

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# 4- Waves - Istituto Nazionale di Fisica Nucleare