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Advanced Higher Physics Quanta and Waves Student Booklet II http://cdn.phys.org/newman/gfx/news/hires/2014/1-tourbillonetbulle.jpg Page 0 Dick Orr Wave Phenomena What Is Wave Motion? When a wave is moving through a medium, energy is transferred from one position in the medium to another position in the medium, with no net transport of mass. What Does This Mean? Consider the example of water waves where each water particle moves at right angles to the direction of travel of the wave. This is illustrated below: Direction of the water particles Direction of the wave As the wave travels, each water particle is displaced by the same distance perpendicular to the direction of the wave. The result is that the water particles themselves do not travel with the wave. What is seen is the surface of the water bobbing up and down (in the absence of any wind or tide). Page 1 Dick Orr Wave definitions – You already know these, allegedly, make sure you do now!!! Wavelength (symbol λ units metres, m). It is the distance between two successive points on a wave in phase. Amplitude (symbol A, units metres, m) It is measured from the centre line to the crest or trough. It is a measure of how much energy a wave carries. Frequency (symbol f, unit hertz, Hz) This is how many waves are produced each second. This is the same as the number of waves that pass a point in one second. Period (symbol T unit second s) This is the time to produce one wave. Speed (symbol v, unit metres per second, m/s) This is the distance a wave travels in one second. Wave equation v=f×λ Page 2 Dick Orr Intensity of A Wave By definition the intensity of a wave is directly proportional to its amplitude squared. i.e. I A2 This means that (therefore I = constant x A2) I1 I2 constant A12 A22 where I is the intensity measured Watts per metre squared (Wm-2) A is the amplitude measured in metres (m) Example If the intensity of a wave is 45 Wm-2 when the amplitude is 2cm, what is the new intensity of the wave when the amplitude increases to 4cm. I α A2 I1 I2 A12 A22 45 I2 22 42 45 42 I2 22 I2 180 Wm2 Page 3 Dick Orr Travelling Wave A travelling or progressive wave consists of a disturbance moving from a source to surrounding places as a result of which energy is transferred from one point to another with no net transfer of mass. There are two types of travelling wave: 1. The diagram shows a wave on a slinky, any transverse wave has the plane of vibration at right angles to the direction of energy transfer. Transverse Wave The diagram shows a wave on a slinky, any longitudinal wave has the plane of vibration in the direction of energy transfer. Waves can be either mechanical or electromagnetic Mechanical waves are disturbances that occur in materials that have mass and elasticity. e.g. water waves Electromagnetic waves are disturbances that consist of varying electric or magnetic fields. e.g. light and microwaves Page 4 Dick Orr Wave Superposition The principle of superposition states: If a number of waves are travelling through a medium, then the resulting disturbance at a point in the medium is the algebraic sum of the individual disturbances produced by each wave at that point. As a consequence of this, waves can pass through one another without being affected in any way. For example if two stones are dropped into a pool of calm water, two sets of circular waves are produced. These two waves pass through each other, but at any particular point in time the disturbance at that point is the algebraic sum. e.g. if a crest of one wave meets a trough of another wave with equal amplitude then the water will remain calm at the point that they meet. And when they meet…….. Page 5 Dick Orr Phase difference A phase difference exists between two points on the same wave. Consider the snapshots below of a wave travelling to the right in the positive x-direction. Points O and D have a phase difference of 2π radians. They are separated by one wavelength (λ). Points O and B have a phase difference of π radians. They are separated by λ/2. Notice that points A and B have a phase difference of π/2. Essentially any phase difference can be defined as either a separation [wavelength] or an angle [radians] Since a separation of λ is equal to a phase difference of 2π radians, it is possible to derive a relationship for any phase difference (φ) in terms of separation. phase difference 2 separation 2 x 2x where: is the phase difference (angle) measured in radians x is the separation between two points on a wave measured in metres is the wavelength measured in metres Page 6 Dick Orr Wave equation for a travelling wave It is possible to represent the displacement produced by a travelling wave by using the relationship below x y Asin2(ft ) where the symbols have their usual meanings. This represents a wave travelling in the positive direction. Were the wave travelling in the negative direction it would be represented by the expression x y Asin2(ft ) Applying your knowledge from N5, Higher and unit 1 it is possible to obtain alternative versions of these relationships. Divide through bracket by f x y Asin2(t ) v Since fλ = v Also 2πf = ω, which gives y Asin(t 2x ) You are expected to be able to obtain values for speed, amplitude, wavelength, frequency, angular frequency and period from the wave equation. You may be asked to write an expression for a travelling wave for particular parameters. Page 7 Dick Orr Stationary waves If two waves of equal frequency and amplitude but travelling in opposite directions combine then a stationary wave will form. This is an interference effect. The pattern produced will look like that below. The positions of the nodes and antinodes will not change, hence the term stationary wave. The equation of such a wave is y Asin2ftcos 2x Time dependant part describes how the displacement y varies at any position x along the wave. Position dependent part describes the maximum amplitude at that position. Nodes occur at a separation of along the wave. 2 Page 8 Dick Orr Interference Hopefully you can remember studying the phenomena of interference at Higher. Interference is the test for wave properties. Interference can only occur if coherent waves combine. Waves are coherent if they have a constant phase difference. This can only occur if they have the same frequency, since both positional and angular phase difference must remain constant. Path difference This is the difference in distance travelled from the source of the waves to the point of combination X. X S1 S2 Path difference = S2X –S1X Constructive interference will occur if path difference is an integer number of wavelengths path diff = mλ : m = 0, 1, 2….. Destructive interference when path diff = (m + ½) λ : m = 0, 1, 2….. However if one path travels through a different medium from the other then we have to take this into account. The reason for this is that the wavelength will change in the new medium, so the phase difference is no longer simply related to the distance the wave has travelled. The path length through the medium is known as the optical path length. Page 9 Dick Orr Optical path difference Consider the path of two rays of light as shown below. The left hand shaded box, P, represents a medium of refractive index n. The right hand box, Q, represents a path through air refractive index = 1. A P Q X B Consider the time taken for the ray to travel distance X. X c v c c n air so vmed vmed vmed n X nX tP c c n Effective distance travelled: tQ X Q c X c nX P c nX c optical path length is n geometric path length We must always consider the optical path length when deciding if constructive or destructive interference takes place. This allows for one path to be in a medium and the other in air. Page 10 Dick Orr Interference by division of wavelength This is when a single source produces a wave which is split and follows different paths to a point. Since a single wave front is divided the light passing through the slits must be coherent. Most commonly, a double slit [Young’s slits] or a diffraction grating is used. When light is shone through the double slit an interference pattern is formed on the screen. screen light source Δx d D The slit separation is d metres The slit to screen distance is D metres The fringe separation is Δx metres These quantities are related to the wavelength of light used, λ, by the relationship D d This set up can be used to determine the wavelength of laser light. x Increasing the distance D will reduce the uncertainty in the measured value of D. The separation of maxima will also increase reducing the uncertainty in the measured value of Δx. Page 11 Dick Orr Interference by division of amplitude This is when a ray is sent by more than one path through a medium. The fringes exist when they are observed using a lens, camera or eye. They cannot be projected onto a screen. Wedge fringes: Two glass slides formed into a wedge. Length l and height d d t l Consider light incident at near normal incidence on an air wedge as shown above. A ray splits at the lower boundary of the first glass slide, one ray reflects from that surface, the other passes through and reflects from the top of the second glass slide. The path difference for the light rays as shown above will be 2×t. The angles are exaggerated in the diagram. When a wave reflects at a boundary between a less optically dense material to a more optically dense material [air to glass] there is a phase change of π. This means that instead of a maximum when path difference = mλ, this set up will give a minimum when path difference = mλ. Page 12 Dick Orr If we consider the horizontal distance between two successive observed minima then we can construct the diagram below. min(n + 1) minn θ t Δx The horizontal distance Δx between minima will result in a vertical height difference of . One wavelength increase in path difference will result in 2 the next minimum. tan 2 x The triangle formed by the glass slides (p 12) will be a ‘similar triangle' to the one above. tan d l d equating gives 2 x l so x Page 13 l 2d Dick Orr Thin film interference If light is incident on a thin film, thickness t and refractive index n, then interference can take place. Consider the diagram below. A phase change of π will take place at the upper surface. No phase change will take place at the lower surface. Air t film The condition for a minimum will again be, path difference = mλ. However since the path difference is due to the path through the film we need to take the optical path difference which will be 2nt in this case. Equating gives the condition for destructive interference 2nt = mλ If the film is able to change thickness, such as a soap film then the varying thickness will result in varying interference for different wavelength. This explains the rainbow effect seen on soap bubbles. Page 14 Dick Orr Non- Reflecting coating. Some lenses are coated with a film that reduces reflection. The coating is chosen to have a refractive index between that of air and the glass it covers. This means that there will be a phase change of π at both reflections. Air t Coating Glass The minimum thickness of coating will give rise to the first minimum, no reflection if destructive interference takes place. optical path difference = (m + ½ )λ 2nt = (m + ½ )λ for first min, m = 0 2nt = 2 t= 4n The non – reflection will only occur 100% at one wavelength. This wavelength is normally chosen to be in the middle of the visible spectrum (green). This means that red and blue are reflected giving the lenses a purplish hue. Page 15 Dick Orr Polarisation Light from the sun and other sources is unpolarised and consists of ‘vibrations’ in every plane perpendicular to the direction of travel. This can be illustrated using the diagram below. The arrows represent the plane of vibration of the electric field vector E. In all electromagnetic waves the electric field E and the magnetic field B vary in size and are at right angles to each other and to the direction of travel, hence ElectroMagnetic waves. https://www.ndeed.org/EducationResources/CommunityCollege/RadiationSafety/Graphics/elec_mag_field.gif If the plane of the electric field does not change, that is it always vibrates in one plane, it is said to be linearly polarised. Illustrated by the diagram below. Page 16 Dick Orr Polarisation By Reflection Plane polarised light can be produced naturally when light is reflected from any electrical insulator (e.g. glass, water). Polarising angle At one particular angle of incidence, known as Brewster’s angle, the reflected light is completely Linearly Polarised. Hence the statement can be made that… At an angle of incidence equal to Brewster’s Angle the reflected light is linearly polarised. When light is incident at a boundary between air and an electrical insulator, the polarising angle ip is the incident angle in air which causes the reflected light to be linearly polarised. air iP iP insulator r Page 17 Dick Orr The refractive index of the insulator is taken as being n. siniP sinr but iP + r + 90 = 180 n so r (90 iP ) but sin(90 iP ) cosi siniP so n taniP cosiP iP is known as Brewster’s angle. How do we know if light is in fact polarised? We need to use a polarising filter. The filter is rotated and if the light passing through it is linearly polarised there will be no transmission at a particular angle of alignment. If the incident light is unpolarised, then rotating the filter will result in no change in the transmitted light level. Page 18 Dick Orr