Domasi College of Education Physics Section PHY 112 PDF

Domasi College of Education Faculty of Sciences Physical Science Department Physics Section PHY 112: Oscillation and Waves Topic 2: Waves and wave motion Prepared by Andrew E...

Domasi College of Education Faculty of Sciences Physical Science Department Physics Section PHY 112: Oscillation and Waves Topic 2: Waves and wave motion Prepared by Andrew E.P.Phaundi-Shonga 2023 Version Shonga  Waves and wave motion  Areas Covered a. b. c. d. e. f.  OSCILLATIONS and WAVES  Oscillations are periodic to and fro movement about fixed/rest/equilibrium position.  Waves are disturbances in medium that transfers energy from one point to another without particles  Characteristics on Waves  Amplitude (A) is maximum displacement from equilibrium position.  Period (T) is time taken for complete cycle. (T = 1/f)  Frequency (f) is the number of complete cycles per second. (f = 1/T)  Wavelength (λ) is the distance between successive points in consecutive waves e.g. crests or troughs.  Wave speed (v) is distance covered by a wave per unit time. (v = fλ). This is called wave equation.  In-phase is when two or more particles in a wave have the same direction.  Out of phase is when two or more particles have the different directions.  Activity 1 Displacement  In the figure, draw and label the following: I Amplitude ii. Period iii. Wavelength  Activity 1 Cont’d B D G A C F H  In the figure, identify (a) in-phase and (b) out of phase.  Basic Properties of Waves  Reflection  Refraction  Interference  Diffraction  Properties of Waves: Interference  Interference is the effect of superposition of two or more waves travelling in a medium.  Superposition principle states that when two or more waves meet at the same point, the resultant displacement of the medium at any point is algebraic sum of the displacements of individual waves at that point. In-phase Out of phase Constructive Destructive Interference Interference  Activity  Use letters in the figure above to identify positions of where (a) constructive interference and  (b) destructive interference would occur.  Properties of Waves: Diffraction  Diffraction is the spreading out of waves as they pass through an aperture or around objects.  Diffraction depends on size of wavelength of the incident wave and aperture.  For example, complete diffraction occurs when the size of the aperture or obstacle is equal to size the wavelength of the incident wave. For very small aperture sizes, the vast majority of the wave is blocked. For large apertures the wave passes by or through the obstacle without any significant diffraction.  When aperture width is smaller than the wavelength, the wave transmitted through the aperture spreads all the way round and behaves like a point source of waves.  Basic Types of Waves  Direction of scillation of particles Direction of travel Direction of scillation of particles  Transverse Wave is wave whose particles move perpendicular to direction of wave travel  Longitudinal Wave is wave whose particles move parallel to direction of wave travel  Forms of Waves  There are two forms of waves, namely: a). Progressive / Travelling Waves and b). Standing / Stationary Wave  Progressive / Travelling Waves  Travelling waves moves in space at a velocity given by v = fλ  Properties of material or medium through which wave moves determine speed of the wave. For example, speed of a transverse wave on a string is determine by tension (F) in the string and linear density of the string (𝜇 = m/L) while speed of a longitudinal wave in a medium (e.g liquid) is determined by bulk modulus and density of the material. 𝐹  Speed of a transverse wave in a string is given by 𝑣= 𝜇 𝛽  Speed of a longitudinal wave in medium (e.g liquid / air) is given by 𝑣= 𝜌 where 𝛽 = Bulk Modulus and 𝜌 = Density = m/V 𝐸  Speed of a longitudinal wave in medium (e.g solid bar) is given by 𝑣 = where 𝜌 𝐸 = Elastic Modulus and 𝜌 = Density = m/V  Travelling waves transfer energy from one point to another. Activity  Types of Progressive / Travelling Waves  There are three types of progressive waves. These are Mechanical Waves, Electromagnetic Waves and Matter Waves.  i). Mechanical waves require material medium (e.g. air, liquid, solid) for their propagation.  ii). Electromagnetic waves travels through a medium and vacuum.  iii). Matter waves are associated with electrons, protons and even atoms or molecules.  Production and propagation of mechanical waves requires three things i). The medium has to be elastic so that it can be disturbed. ii). Some physical connection between adjacent particles or medium which can influence each other. iii). An energy source to provide disturbance to the medium. Mathematical Description of Travelling Waves  Equation of travelling wave in positive direction is given by wave function, 𝑦(𝑥, 𝑡) = 𝐴𝑠𝑖𝑛 𝑘𝑥 − 𝜔𝑡 where A is amplitude,  k is wave number and 𝜔 is angular velocity.  ∴ 𝑦 𝑥, 0 = 𝐴𝑠𝑖𝑛 𝑘𝑥  Displacement, y is the same after every wavelength, λ. i.e 𝑥 = 𝑥1 and 𝑥 = (𝑥1 +λ)  𝑦 𝑥1 , 0 = 𝐴𝑠𝑖𝑛 𝑘𝑥1 = 𝐴𝑠𝑖𝑛 𝑘(𝑥1 +λ) = 𝐴𝑠𝑖𝑛 (𝑘𝑥1 +kλ)  Sine function repeats itself when then the angle is 2𝜋 𝑟𝑎𝑑𝑠 2𝜋 2𝜋  ∴ 𝑘= and 𝜔= = 2𝜋𝑓 λ 𝑇  Activity 𝜔 1. Show that 𝑣= 𝑘 2.  The figure above shows sinusoidal wave travelling to the right direction. Its wave function can be written as 𝑦(𝑥, 𝑡) = 𝐴𝑐𝑜𝑠 𝑘𝑥 − 𝜔𝑡.  Calculate: (i) Wave number, k. (ii) Angular frequency, 𝜔. (iii) Speed of the wave, v  (iv) Displacement, y at x = 20 cm and t = 13 s  Electromagnetic Waves Understanding in Electromagnetic Waves helps to answer a number of questions like:  How do televisions, radios, microwaves, mobile phone, bluetooth speakers, remote controls and infrared thermometers work. Exposition: Electromagnetic Waves  Electromagnetic Waves Cont’d  Electromagnetic (EM) waves are progressive transverse waves.  They consist of travelling the electric and magnetic fields which are at right angles to each other and to the direction of travel of the wave. Exposition: Electromagnetic Spectrum Activity 2.1: Electromagnetic spectrum Discuss your understanding of the term electromagnetic spectrum Exposition: Electromagnetic Spectrum Cont’d  Electromagnetic spectrum is a range of all types of electromagnetic radiation.  The electromagnetic spectrum is a range of frequencies, wavelengths and photon energies.  Note: 1. waves with a very short wavelength have high frequency and high energy 2. waves with a very long wavelength have low frequency and low energy Exposition: Electromagnetic Spectrum Cont’d  The Electromagnetic Spectrum can remembered using the phrase is: Roman Men Invented Very Unusual X-ray Guns  Thus, radio waves, microwaves, infrared waves, visible light, ultra-violet, X-rays and gamma rays Exposition: Properties of electromagnetic waves 1. Electromagnetic waves are transverse in nature since oscillating electric and magnetic fields are perpendicular to the direction of propagation of the wave 2. They exhibit interference, diffraction and polarization, which suggests that they have a transverse wave nature  Exposition: Properties of electromagnetic waves Cont’d 3. All types of electromagnetic waves travel through a vacuum (space) with a speed of light (3 x 108 m/s) 4. Electromagnetic waves are non-mechanical waves in nature since they do not require any material medium for propagation.. Exposition: Properties of Electromagnetic Waves Cont’d 5. They obey the wave equation v = fλ where v is speed of light in metres per second, f is the frequency in Hertz and λ is the wavelength in metres. The product of wavelength and frequency is equal to a constant c, the speed of light which is equal to 3 x 108 m/s. From this relationship between wavelength, frequency and speed of light, we can understand that as wavelength decreases with increase in frequency. Exposition: Properties of electromagnetic waves Cont’d 7. They carry energy from one place to another and can be absorbed by matter to cause heating and other effects. The energy is carried in photons and is given by E = fh where E is energy in joules, f is the frequency in Hertz and h is Plank’s constant (6.6 × 10−34 ) in Joules second. In this relationship, it is clear that energy increases with increase in the frequency. Thus, overall relationship between frequency, wavelength and energy in electromagnetic waves is: the higher the frequency, the smaller the wavelength, and the greater the energy.  Some methods of detecting EM waves There are many devices that can detect electromagnetic waves depending on the frequency.  Ourbodies detect infrared radiation (IR) by its heating effect on the skin. Infraredthermometers are used to measure body temperature. Some methods of detecting EM waves Cont’d Infrared sensors are used on satellites and aircraft for weather forecasting, monitoring of land use, assessing heat loss from buildings, intruder alarms and locating victims of earthquakes. Uses of electromagnetic waves Different groups in electromagnetic spectrum have different uses in our daily life. Uses of electromagnetic waves: Radio waves A radio basically captures radio waves that are transmitted by radio stations. Radio waves are mainly used for TV/mobile communication/satellites.  Radio waves are transmitted easily through air.  Radiowaves can be produced by oscillations in electrical circuits.  When radio waves are absorbed by a conductor, they create an alternating current. Uses of electromagnetic waves: Microwaves  Microwaves are used for cooking at home/office.  Highfrequency microwaves have frequencies which are easily absorbed by molecules in food. The internal energy of the molecules increases when they absorb microwaves, which causes heating in the food.  They are also used by astronomers to determine and understand the structure of nearby galaxies and stars (satellite communications). Uses of electromagnetic waves: Infrared (IR)  Itis used widely in night vision goggles. These devices can read and capture the infrared light emitted by our skin and objects with heat.  In space, infrared light helps to map the interstellar dust. This finds much use in military, at night military targets are identified using infrared. Also when filming animals at night, infrared is used. Uses of electromagnetic waves: Infrared (IR) Cont’d  Infrared (IR) light is used by electrical heaters, cookers for cooking food, short-range communications like remote controls, optical fibers, security systems and thermal imaging cameras which detect people in the dark. The heating effect of IR can cause burns to the skin. Uses of electromagnetic waves: Infrared (IR) Cont’d  Infrared light has frequencies which are absorbed by some chemical bonds. The internal energy of the bonds increases when they absorb infrared light, which causes heating. This makes infrared light useful for electrical heaters and for cooking food. All objects emit infrared light. The human eye cannot see this light, but infrared cameras can detect it. This ‘thermal imaging’ is useful for detecting people in the dark. Uses of electromagnetic waves: X-ray  X-raysis used in hospitals to take an image of bone or teeth.  Itused at airport by security personnel to see through and check bags. Uses of electromagnetic waves: Gamma rays  They have a wide application in the medical field. They are used to see inside our bodies.  They are used for sterilising food and medical instruments.  They are used in the treatment and detection of cancer.  Interestingly, the universe is the biggest gamma-ray generator of all. Uses of electromagnetic waves: Ultraviolet (UV)  Sun is the main source of ultraviolet radiation. Also hot materials that are in space also emit UV radiations.  We cannot see ultraviolet (UV) light but it can have hazardous effects on the human body. Ultraviolet light in sunlight can cause the skin to tan or burn.  The hazardous properties of UV kills bacteria and because of that property, UV is used for disinfecting water. Uses of electromagnetic waves: Ultraviolet (UV) Cont’d  Fluorescent substances are used in energy- efficient lamps - they absorb ultraviolet light produced inside the lamp, and re-emit the energy as visible light. Similar substances are used on bank notes to detect forgeries. Uses of electromagnetic waves: Visible light  Visible light can be detected by our eyes.  Light bulbs, stars, etc. emit visible light.  Visible light is used in photography and illumination.  Itis also used in fiber optic communications, where coded pulses of light travel through glass fiber from a source to a receiver.  Standing / Stationary Waves  Standing / Stationary Wave is result of superposition of two waves moving in opposite directions within the medium, each having the same amplitude and frequency / wavelength  The peak amplitude of standing waves does not move along through a medium as a traveling wave.  A standing wave consists of nodes and antinodes  Nodes are fixed points on a wave that do not vibrate. Antinodes are maximum amplitude points where the wave is oscillating vertically. Antinodes form the maxima and minima of the wave  Standing waves do not transfer energy. The energy of the traveling waves that formed the standing wave is stored within the standing wave.  Mathematical Description of Standing Waves  Equation of travelling wave moving in positive direction is given by wave function,  𝑦1 = 𝐴𝑠𝑖𝑛 𝑘𝑥 − 𝜔𝑡  Equation of travelling wave moving in negative direction is given by wave function,  𝑦2 = 𝐴𝑠𝑖𝑛 𝑘𝑥 + 𝜔𝑡  Therefore equation of standing wave, 𝑦 = 𝑦1 + 𝑦2 = 𝐴𝑠𝑖𝑛(𝑘𝑥 +  Nodes and Antinodes Positions  Find angles at 5 positions for each of the following:  (i) Node  (ii) Antinodes  Node and Antinode Positions Cont’d  Node Positions  The amplitude has minimum value when 2A sin 𝑘𝑥 = 0 i.e sin kx = 0  ∴ kx = 0, 𝜋, 2𝜋, 3𝜋, … … … … = 𝑛𝜋 where n = 0, 1,2,3,4……. 2𝜋 λ 3λ 5λ 𝑛λ  Since 𝑘= then 𝑥 = 0, , λ, , …..= where n = 0,1,2,3, λ 2 2 2 2 4……..  Antinode Positions  The amplitude has maximum value when 2A sin 𝑘𝑥 = 1 i.e sin kx = 1 𝜋 3𝜋 5𝜋 1  ∴ kx = , , , ….= 𝑛 + 𝜋 where n = 0, 1,2,3,4……. 2 2 2 2 𝑛𝜋  Or kx = where n = 1, 3, 5……… 2 λ 3λ 5λ 𝑛λ  Since 𝑘 = 2𝜋 then 𝑥= , , …..= where n = 1,3, 5….. λ 4 4 4 4  Activity  Imagine one of the equation of travelling wave moving in positive direction is given by wave function, 𝑦1 = 𝐴𝑠𝑖𝑛 𝜔𝑡 − 𝑘𝑥  Determine:  (a). Equation of standing wave that can be formed.  (b). First 5 horizontal positions of antinodes in terms of wavelength.  Standing Waves and Resonance  Describe relationship between frequency and amplitude displayed in the graph.  Standing Waves and Resonance Cont’d  When external driving frequency is near natural frequency, 𝑓 ≈ 𝑓0 , the amplitude becomes large.  This effect of attaining maximum amplitude is called resonance.  For example, pushing a child on a swing. The swing has natural frequency. If you push the swing at random frequency, the swing bounces round and reaches no great amplitude. But if you push it with a frequency equal to natural frequency of the swing, the amplitude greatly increases. At resonance, relatively little effort is require to obtain and maintain large amplitude.  Natural frequency is the frequency at which a system tends to oscillate in the absence of any driving force.  Standing waves are only set up at resonance frequency. Standing Waves and Resonance Cont’d L  Determine relationship between length of the string and wavelength for each of the above diagrams. Standing Waves and Resonance Cont’d λ1 3λ3  (a) L = (b) L = λ2 (c) L = 2 2 𝑛λ𝑛  ∴ L= where n = 1, 2, 3, 4……… 2 2𝐿  λ𝑛 = where n = 1, 2, 3, 4……… 𝑛 Standing Waves and Resonance Cont’d  λ1 = 2L and λ2 = L  V = fλ.  But velocity is constant in any medium regardless of changes in frequency and wavelength.  ∴ 𝑉 = 𝑓1 λ1 = 𝑓2 λ2 = 𝑓3 λ3 and 2𝐿  = 𝑓1 2L = 𝑓2 𝐿 = 𝑓3 3  = 2𝑓1 = 𝑓2  Determine 𝑓3 in terms of 𝑓1  ∴ 𝑓𝑛 = 𝑛𝑓0 where f0 is fundamental frequency  Standing Waves and Resonance Cont’d  Standingwaves can form at numerous natural frequencies.  Theset of all possible standing waves are known as the harmonics of a system.  Thesimplest of the harmonics is called the fundamental or first harmonic. It has one antinode and two nodes.  Subsequentstanding waves are called the second harmonic, third harmonic, etc.  Activity  One of the harmonic frequencies for a particular string under tension is 325 Hz. The next higher harmonic frequency is 390 Hz. Calculate harmonic frequency next higher after harmonic frequency of 195 Hz.