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Chap-12 - Waves Displacement relation for a progressive wave. Principle of superposition of waves, reflection of waves, Standing waves in strings and organ pipes, fundamental mode and harmonics, Beats, Doppler effect in sound TYPES OF WAVES A wave is disturbance that propagates in space, transports energy and momentum from one point to another without the transport of matter. The ripples on a water surface, the sound we hear, visible light, radio and TV signals are a few examples of waves. There are two types of wave. Mechanical Waves : Require material medium (elasticity and inertia) for their propagation. These waves are also called elastic waves, water waves and sound waves are example of mechanical waves. They are of two types : Transverse and longitudital. Comparison between the two is given there : Transverse Longitudinal Particles of the medium vibrate at right angles to the direction of wave motion Particles of the medium vibrate in the direction of wave motion Particle velocity is always perpendicular to wave velocity Particle velocity is parallel or antiparallel to wave velocity Waves on strings are always transverse Can not be produced on stretched strings These can be polarised Can not be polarised Do not exist in gases as they do not possess shear modulus or modulus of rigidity Can exist in a solid, liquid or gas Electromagnetic or non-mechanical Waves : Do not require any material medium for their propagation, such as light and TV signals. Important points regarding these waves are : Elasticity or inertia do not affect their propagation. They are always transverse in nature. CHAPTER COVERS : Types of waves Wave function Velocity of wave Sound waves Superposition of waves Standing waves Free oscillation Forced oscillation Beats Interference Doppler’s effect WAVE FUNCTION Various functions and their implications : If y = f (x + vt), then wave is moving in negative x-direction with velocity v. If y = f(x – vt), then wave is moving in positive x-direction with velocity v. y = f (x ± vt)2, y = f ( x ± vt ) or f (x ± vt)3 are valid wave equation. y = f ( ± t ), y = f (x2 ± v2t) or f (x3 ± v3t) are not wave equation. Here y, x, v, t stand for displacement, position, speed of wave, time respectively. Differential Equation of Travelling Wave d 2y = dt 2 d 2y v dx 2 Speed of Wave Motion Speed of non-mechanical i.e., electromagnetic wave in vaccum is c = 1 where μ0 = absolute permeability and ε0 = absolute permittivity Speed of mechanical waves : Transverse wave in a stretched string T = tension in the string μ = mass per unit length D = diameter of pipe ρ = density v = = = Transverse wave in a solid In a long bar v = where Y = Young’s modulus, ρ = Density of material In an extended solid v = where η = modulus of rigidity, ρ = Density of material Longitudinal Waves In liquid v = In gases v = Case - I : K = bulk modulus of elasticity ρ = density . For gases, K depends upon the process. [Suggested by Newton] Taking isothermal process K = P ⇒ v = Put P = 1 atm, ρ = 1.23 kg/m3 ⇒ v = 280 m/s (more than 15% error) Case - II : For Adiabatic k = γP [Corrected by Laplace] ⇒ v = . Taking γ = 1.4, we get, v = 330 m/s Factors affecting speed of sound : If temperature is kept constant. (1) v is independent of pressure (2) v ∝ 1 or v ∝ 1 Velocity of a wave depends on medium. v ∝ Velocity of sound in humid air is more because its density is very less. Velocity of sound in humid hydrogen is less than in dry hydrogen due to similar reason. Reflection of Waves If a wave travelling in a medium of high velocity gets reflected from the surface of a medium of low velocity, it suffers a phase change π. At the interface of a rarer and denser medium. Wave is moving from rarer to denser medium. Incident wave ⇒ v1 Reflected wave v2 Transmitted wave v2 < v1 Wave is moving from denser to rarer medium ⇒ v1 Transmitted wave Incident wave v2 2 Reflected wave < v1 The transmitted wave is always in phase with incident wave. Reflection from fixed end : ⇒ Reflection from free end : frictionless ⇒ ring Refraction of waves Media can be classified as Rarer Medium : A medium in which speed of wave is greater. Denser Medium : A medium in which speed of wave is smaller. For example, in case of light, air is rare medium and water is denser medium as speed of light is more in air than in water. But in case of sound air is denser medium and water is rarer medium as speed of sound in air is less than in water. Harmonic Wave and Various Terms If the source of the wave is a simple harmonic oscillator, then the function f (x ± vt) is sinusoidal and it represents a harmonic wave. This function, in general, can be written as, y = A sin[k(x ± vt) + φ] or y = A sin(kx ± ωt + φ) Various terms used to describe wave are : Amplitude (A) : It is the maximum displacement of a particle in the medium from its equilibrium position. Wavelength (λ) : It is the distance between the two successive points with the same phase. Propagation constant or Angular wave number (k) : k = 2π λ Wave velocity (v) : v = λ T = λf = ω k Phase : kx ± ωt + φ Initial phase : φ Wave number : 1 λ Relation between phase difference and path difference Δφ = 2x Δx λ Δφ = Phase difference Δx = Phase difference Analysis of harmonic waves For a transverse wave y = A sin(ωt – kx) at t = 0, photograph of the wave is For the particle at x = 0 y v y D R S A C E x P Q B t = 0 t y = A sin(–kx) y = A sinωt Particles at A, P, S, E are moving upwards This particle is moving upward at t = 0 Particles at Q, C, R are moving downwards Particles at B and D are at rest For wave y = A sin(kx – ωt) at t = 0, photograph of the wave is For the particle at x = 0 y v y B P Q C E A x t = 0 t R S D y = Asin kx y = Asin(–ωt) Particles at A, P, S, E are moving downwards This particle is moving downward at t = 0 Particles at Q, C, R are moving upwards Particles at B and D are at rest SOUND WAVES They are mechanical and longitudinal waves. They propagate in form of compressions and rarefactions. Particle displacements can be represented by wave function S = A sin(ωt – kx) As particles oscillate, pressure variation takes place according to the wave function. ΔP = ΔP0 cos(ωt – kx), ΔP0 = maximum pressure variation Characteristic of Sound Loudness Sensation of sound produced in human ear due to amplitude. It depends upon intensity, density of medium, presence of surrounding bodies, Intensity of Wave I = 2π2f 2A2ρv I ∝ f 2 and I ∝ A2 I = P 4πr 2 I ∝ 1 r 2 P = Power of point source (for a point source) I ∝ 1 r (for a line source) Intensity Level or (Sound Level) (β) ⎛ I ⎞ ⎡I0 = minimum intensity of audible sound = 10−12 W/m2 ⎤ β = 10 log10 ⎜ I ⎟ (dB) ⎢ ⎥ ⎝ 0 ⎠ ⎢⎣I = measured intensity ⎥⎦ Sound level range for audible sound [0 dB to 120 dB] ⎛ I2 ⎞ β2 – β1 = 10 log10 ⎜ I ⎟ , Unit of sound level β is decibel (dB) ⎝ 1 ⎠ Quality : Sensation produced in human ear due to shape of wave. Quality is that characteristic of sound by which we can differentiate between the sound of same pitch and loudness coming from different sources. Pitch : Sensation produced in human ear due to frequency. Pitch is the characteristics of sound that depends on frequency. Smaller the frequency smaller the pitch, higher the frequency higher the pitch. Humming of mosquito has high pitch (high frequency) but low intensity (low loudness) while the roar of a lion has high intensity (loudness) but low pitch. Classification of Waves based on Frequency : Infrasonic Wave : Longitudinal waves having frequencies below 20 Hz are called infrasonic waves. They cannot be heard by human beings. They are produced during earth quakes. It can be heard by snakes. Audible Waves : Longitudinal waves having frequencies lying between 20-20,000 Hz are called audible waves. Ultrasonic Waves : Longitudinal waves having frequencies above 20,000 Hz are called ultrasonic waves. They are produced and heard by bats. They have a large energy content. Shock Waves : A body moving with speed greater than speed of sound (supersonic speed) produces a conical disturbance called a shock waves. SUPERPOSITION OF WAVES If number of waves are travelling through a medium then resultant displacement of a particle of medium is sum of individual displacements produced by individual waves in the absence of other. Wave (I + II) y = y1 + y2 Standing Waves When two waves identical in all respects, but travelling in opposite direction along a straight line, superimpose on each other, standing waves are produced. Let y1 = A sin(ωt – kx) and y2 = A sin(ωt + kx) + ⇒ y = y1 + y2 = 2A coskx sinωt 2A coskx represents the amplitude of particle located at ‘x’. Some Important Points : For x = 0, x = λ , λ and so on, amplitude is maximum i.e., 2A. These points are called antinodes. 2 For x = λ , 3λ , and so on amplitude is minimum i.e., O. These points are called nodes. 4 4 λ Distance between consecutive nodes = distance between consecutive antinodes = 2 . λ Distance between adjacent node and antinodes = 4 . All the particles in same loop i.e., between two adjacent nodes vibrate in same phase. Particles on the opposite side of a node vibrate in opposite phase. Sonometer : In this case, transverse stationary waves are formed. T = Mg λ (tension in wire) l The wire vibrates in n loops, then l = nλ 2 or λ = 2l n velocity v = T where ‘μ’ is mass per unit length of wire. μ ⇒ νn = v = nv = λ 2l If the wire vibrates in simplest mode, ν1 = For nth harmonic, νn = [Fundamental mode, Ist harmonic] [(n – 1)th overtone, nth harmonic] Case : A wire is to be divided in three parts whose fundamental frequencies are f1, f2 and f3. l1 + l2 + l3 = l …(1) l : l : l : : 1 : 1 : 1 …(2) 1 2 3 f1 f2 f3 From (1) & (2), we get, f1 f2 f3 l1 = f1f2 f2f3 + f2f3 l f1f3 l2 = f1f3l , l Σf1f2 = f1f2 l Σf1f2 Organ Pipe : In this case, longitudinal stationary waves are formed Open organ Pipe : l l Displacement Node Pressure antinode l = λ 2 or λ = 2l l = λ l = 3λ 2 ν = V = V 1 λ 2l ν = V l = 2ν1 λ = 2l 3 1st harmonic or 2nd harmonic ν = 3V 0 2l = 3ν1 Fundamental mode Ist overtone 3rd harmonic 2nd overtone Closed organ pipe : l l = λ ⇒ λ = 4l l = 3λ ⇒ λ = 4l l = 5λ λ = 4l 4 ν = V 4l 4 3 ν = 3V 4l 4 5 ν = 5V 4l Fundamental mode Ist overtone 2nd overtone Ist harmonic 3rd harmonic 5th harmonic Pipe length ‘’l Fundamental Mode Ist Overtone (n – 1)th overtone Open ν = V 2l I stHarmonic ν = V l 2ndHarmonic ν = n V 1 : 2 : 3 : 4 2l nth Harmonic Closed ν = V 4l IstHarmonic ν = 3V 4l 3rd Harmonic ν = (2n − 1) V 1 : 3 : 5 : 7 l (2n –1)th Harmonic End correction : As the antinodes are formed slightly out side the open end. e = 0.6r = end correction. Thus, we have, For closed organ pipe For open organ pipe e e l + e l + 2e ν = (2n − 1)V 4(l + e) Resonance Tube: If resonance is obtained first at length l1. then at length l2, then λ = 2(l2 – l1) ν = nV 2(l + 2e) l1 e ‘ν’ l2 ⇒ distance between two successive lengths is λ 2 BEATS It is the phenomenon of periodic variation in intensity at a particular position on account of superposition of wave of nearly equal frequencies. When two waves of same amplitude and nearly equal frequencies ν1 and ν2 superimpose on each other. The amplitude at a given position varies with frequency ν1 − ν2 2 . The intensity at a given position varies with frequency |ν1 – ν2|. This frequency of variation of intensity is called beat frequency. Frequency of the resulting wave is (v1 + v2)/2. Interference Consider two waves of same frequency and wavelength, y1 = a1 sin (ωt – kx), I1 = Ca 2 y2 = a2 sin (ωt – kx + φ), I2 = Ca 2 Equation of resultant wave is, −1⎛ a2 sinφ ⎞ y = y1 + y2 = A sin (ωt – kx + θ), where A = and θ = tan ⎜ a + a ⎟ cosφ Resultant Intensity is given by I = I1 + I2 + 2 ⎝ 1 2 ⎠ Max. Int. : I max = ( + )2 where phase difference φ = 2nπ, path difference = 2n λ 2 Min. Int. : I min = ( − )2 where phase difference φ = (2n + 1)π, path difference = (2n + 1) λ 2 ⎛ I ⎞2 ⎛ a ⎞2 ⎛ ⎞2 ⎜ 1 + 1⎟ 2 ⎜ 1 + 1⎟ Imax ⎜ I1 + ⎟ ⎜ I2 ⎟ ⎛ a1 + a2 ⎞ ⎜ a2 ⎟ I = ⎜ I − ⎟ = ⎜ I = ⎜ ⎟ a − a = ⎜ a ⎟ min ⎝ 1 ⎠ ⎜ 1 − 1⎟ ⎝ 1 2 ⎠ ⎜ 1 − 1⎟ ⎜ I2 ⎟ ⎝ a2 ⎠ For equal intensity I = 4I , I = 0, I = 4I cos2 φ . max DOPPLER’S EFFECT 0 min 0 2 If a wave source and a receiver are moving relative to each other, the frequency observed by the receiver (f) is different from the actual source frequency (f0) given by, ⎛ v ± v0 ⎞ f = f0 ⎜ v μ v ⎟ Various cases : Where v = speed of sound, v0 = speed of observer, vs = speed of source Source at rest, observer moves Observer moves away from source, f = f ⎛ v − v0 ⎞ 0 ⎜ v ⎟ Observer moves towards source, f = f ⎛ v + v0 ⎞ Observer at rest, source moves 0 ⎜ v ⎟ ⎛ v ⎞ Source moves towards observer, f = f0 ⎜ v − v ⎟ Source moves away from observer, ⎛ v ⎞ Both move Both approaching each other f = f0 ⎜ v + v ⎟ ⎛ v + v0 ⎞ S v v O f = f0 ⎜ v − v ⎟ S 0 S ⎠ Source following the observer ⎛ v − v 0 ⎞ S v O v f = f0 ⎜ v − v ⎟ S 0 S ⎠ Observer following the source ⎛ v + v0 ⎞ v S f = ⎜ ⎝ v + v ⎟f0 S ⎠ S v0 Both moving away from each other ⎛ v − v0 ⎞ v S v f = ⎜ ⎝ v + v ⎟f0 S O 0 S ⎠ Application of beats and Doppler Effect : Case - I : f = frequency of source vs = velocity of source f ′ = frequency of direct sound v = velocity of sound f ″ = frequency of reflected sound S ⎛ v ⎞ O vs f ′ = ⎜ ⎟ f ⎝ v + vs ⎠ (source moving away) f ′ = ⎛ v ⎞ f ⎝ v − vs ⎠ (source moving towards) Cliff/Wall at rest ⎛ v v ⎞ ⎛ v × 2vs ⎞ Beat frequency = f ′ − f ′ = ⎜ − ⎟ f = ⎜ ⎟ Case - II : ⎝ v − vs v + vs ⎠ ⎜ − vs ⎟ A source of frequency ‘f ’ is revolving in a circle of radius R with speed vs. An observer is standing at a distance x from the centre. O At B and D, observed frequency is ‘f ’ . At ‘P’ frequency is maximum as OP ⊥ PR, i.e., ⎛ v ⎞ f ′ = ⎜ ⎟ f ⎝ v − vs ⎠ OP = OR 2 − PR 2 = x 2 − R 2 ⎛ v ⎞ At Q frequency is minimum as OQ ⊥ QR, i.e., f ′′ = ⎜ ⎟f ⎝ v + vs ⎠ Case - III : v1 v0 S1 O (f1) ⎛ v − v0 ⎞ S2 (rest) v = velocity of sound (f2) f1′ = ⎜ v − v ⎟ f1 1 ⎠ f ′ = ⎛ v + v 0 ⎞ f 2 ⎜ v ⎟ 2 beat frequency = f1′ – f2′ Case - IV : Direct frequency = f ⎛ v + v ⎞ Reflected frequency = f ⎝ ω ⎠ ‘f ’ (rest) vω v = velocity of sound (movable reflecting wall) ⎛ v + v ω ⎞ ⎛ 2v ω ⎞ Beat frequency = − f ⎜ v − v − 1⎟ = f ⎜ ⎟ v − v ⎝ ω ⎠ ⎝ ω ⎠   