Waves & Oscillations PDF

Types of Motion: We very often come across the following types of motions 1. Rectilinear motion: Motion of a particle in a straight line Examples: Motion of ball in straight line, a body that falls freely in vertical direction under the influence of gravity etc. Periodic Motion A motion that repe...

Types of Motion: We very often come across the following types of motions 1. Rectilinear motion: Motion of a particle in a straight line Examples: Motion of ball in straight line, a body that falls freely in vertical direction under the influence of gravity etc. Periodic Motion A motion that repeats itself at regular intervals of time is called periodic motion. Swinging Pendulum and clock In both the cases, the motions repeat after certain interval of time. Such a motion that repeats after certain interval of time is known as periodic motion. The body is displaced from a fixed point and it is given a small displacement, a force comes into action that tries to bring it to its equilibrium point, giving rise to oscillation or vibration. Every oscillatory motion is a periodic, but every periodic motion need not be oscillatory. Oscillations Oscillatory motion is a to and fro motion about a mean position and periodic motion repeats at regular intervals of time. All oscillatory motions are periodic but all periodic motions are not oscillatory. Oscillations or Vibrations There is no significant difference between oscillations and vibrations. Low frequency periodic motions are called as oscillation (like the oscillation of a branch of a tree), High frequency Periodic motions are called as vibration (like the vibration of a string of a musical instrument). Periodic Motions which can be represented by Sinusoidal waveform (Sine or Cosine wave) are known as harmonic motion Simple Harmonic Motion Simple Harmonic Motion (SHM) is a specialized form of periodic motion Periodic vibration around an equilibrium position Restoring force must be proportional to displacement from equilibrium in the direction of equilibrium There are two types of SHM that will be discussed. Mass-Spring System Pendulum Simple harmonic motion (SHM) is a special kind of periodic motion occurs in mechanical system where net force acting on an object is proportional to the displacement of the object from its equilibrium position and the force is always directed towards the equilibrium position. In order for mechanical oscillation to occur, a system must posses two quantities: elasticity and inertia.  When the system is displaced from its equilibrium position, the elasticity provides a restoring force such that the system tries to return to equilibrium. The inertia property causes the system to overshoot equilibrium. This constant play between the elastic and inertia properties is what allows oscillatory motion to occur. The natural frequency of the oscillation is related to the elastic and inertia properties An oscillating system is a mass connected to a rigid foundation with a spring. Example: Spring mass system An oscillating system is a mass connected to a rigid foundation with a spring. The spring constant k provides the elastic restoring force, and the inertia of the mass m provides the overshoot. By applying Newton's second law F=ma to the mass, one can obtain the equation of motion for the system: d 2x d 2x k d 2x F   kx  m   kx   x  0   o2 x  0 dt 2 dt 2 m dt 2 k o  is the natural frequency m The solution of the wave equation x(t )  xm cos(ot   ) where xm is the amplitude of the oscillation, and φ is the phase constant of the oscillation. Displacement x(t)  Acos(t   ) dx Velocity : v(t)   ωAsin(ωt   ) dt d2x Acceleration : a(t)  2  ω 2 Acos(t   ) dt Circular Motion Uniform Circular motion projected in one dimension is SHM The ball mounted on the turntable moves in uniform circular motion, and its shadow, projected on a moving strip of film, executes simple harmonic motion. Simple Pendulum A pendulum consists of an object hanging from the end of a string or rigid rod pivoted about the point. The object is displaced (a small displacement; about 5-10⁰) to one side and allowed to oscillate. If the object has negligible size and the string or rod is massless, then the pendulum is called a simple pendulum. Waves with different amplitudes Waves with different phase Waves with different time period The period of the oscillatory motion is defined as the time required for the system to start one position, complete a cycle of motion and return to the starting position. 2 m T  2 o k v(t )  o xm sin(ot   ) a(t )  o2 xm cos(ot   ) Energy in Simple Harmonic Motion Kinetic energy (K) of the particle executing SHM 1 K mv 2 2 1 K  mω 2 A 2sin 2 ( ωt  φ) 2 1 K  kA 2sin 2 ( ωt  φ) 2 associated potential energy 1 U  kx 2 2 1 U  kA2 cos 2 ( ωt  φ) 2 Total energy, E, of the system is, 1 2 E U  K  kA 2 Throughout oscillation, KE continually being transformed into PE and vice versa, but TOTAL ENERGY remains constant As the system oscillates, the total mechanical energy in the system trades back and forth between potential and kinetic energies. 1 2 1 2 1 2 PE  kx  kxm cos 2 (o t   ) PE  KE  kxm 2 2 2 1 1 1 1 KE  mv 2  mo2 xm2 cos 2 (o t   )  mo2 xm2  mvm2 2 2 2 2 The total energy in the system, however, remains constant, and depends only on the spring constant and the maximum displacement (or mass and maximum velocity vm=ωxm)  Simple Harmonic Motion and Uniform Circular Motion Displacement of oscillating object = projection on x-axis of object undergoing circular motion x(t) = Acos  For rotational motion with angular frequency , displacement at time t: x(t) = Acos (t + ) = angular displacement at t=0 (phase constant) A = amplitude of oscillation (= radius of circle) Velocity and Acceleration in Simple Harmonic Motion Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion takes place. Restoring force F   mg sin  x F   mg   mg for small angles L F  x SHM motion Damped Harmonic motion: Real oscillatory system Have you ever thought why a simple pendulum or spring mass system comes to rest when they are kept in oscillatory motion. Ideally, the oscillatory motion should continue forever. It is because of the resistance created by air, i.e. air works as damping medium which always apposes the motion or in other words we can say that the damping force is always in opposite direction to the restoring force. In general, it is found that the damping force is proportional to the velocity of the oscillatory body. Damped Oscillations For damped oscillations, simplest case is when the damping force is proportional to the velocity of the oscillating object Equation of motion: d 2x dx m 2   kx  b dt dt Forced or Driven oscillation The natural frequency is the frequency at which it will oscillate if there is no driving and damping forces. What is a wave? A disturbance or variation that transfers energy progressively from point to point in a medium and that may take the form of an elastic deformation or of a variation of pressure, electric potential, temperature or more. The Human Wave The human wave is the disturbance (people jumping up and sitting back down), and it travels around the stadium. However, none of the individual people in the stadium are carried around with the wave as it travels - they all remain at their seats. Waves in Everyday Life: Examples Disturbance produced in pond by throwing a stone creates ripples which move outward. Sound: Type of wave that moves through matter and then vibrates our eardrums so we can hear. Visible Light: Kind of wave that is made up of photons. Radio and TV Signals etc. TYPES OF WAVES (a) Mechanical waves Requires medium for propagation Governed by Newton’s laws Example: Water waves, sound waves, seismic waves, etc. (b) Electromagnetic waves Do not require any medium for their propagation Example: Visible and ultraviolet light, Radio waves, Microwaves, X-rays etc. (c) Matter waves  wave associated with the motion of a particle of atomic or subatomic size (electrons, protons, neutrons, other fundamental particles, and even atoms and molecules) Waves differ from one another in the manner the particles of medium oscillate (or vibrate) with reference to the direction of propagation. Transverse Wave A wave in which the particles of the medium vibrate at right angles to the direction of propagation of wave, is called a transverse wave. Longitudinal waves A wave in which the particles of the medium vibrate in the same direction in which wave is propagating, is called a longitudinal wave. Wave Parameters The amplitude A, is half the height difference between a peak and a trough. The wavelength λ, is the distance between successive peaks (or troughs). The period T, is the time between successive peaks (or troughs). The wave speed c, is the speed at which peaks (or troughs) move. The frequency ν, (Greek letter "nu") measures the number of peaks (or troughs) that pass per second. A wave is a disturbance that travels from one location to another, and is described by a wave function that is a function of both space and time. If the wave function was sine function then the wave would be expressed by y  A sin(t  kx ) The negative sign is used for a wave traveling in the positive x direction and the positive sign is used for a wave traveling in the negative x direction.  2 k    2 Wave Speed The speed of a wave depends on the medium through which the wave moves. The speed, wavelength, and frequency are related.   v   λν k T  Speed of a Transverse Wave on Stretched String T v  Where μ is linear mass density of a string, is the mass m of the string divided by its length l.  Speed of a Longitudinal Wave Speed of Sound B v  where B is bulk modulus and ρ is density of the medium.  Speed of a longitudinal wave in an ideal gas Y v  where Y is the Young’s modulus of the material of the bar.  Speed of a longitudinal wave in an ideal gas P v  where P is the Pressure of the ideal gas. Above eqn. is known as Newton relation. Laplace correction For adiabatic processes the ideal gas satisfies the relation, PV   constant where γ is the ratio of two specific heats, γ =C p/Cv P v  This modification of Newton’s formula is referred to as the Laplace correction. Waves meet matter Waves do not travel through the same medium. Waves can be reflected, refracted or absorbed. Sound wave traveling though a long corridor is reflected back as echo. The speed of waves depend on the elastic properties of the medium through which it travels. When a wave encounters different medium where the wave speed is different, the wave will change directions. Sound wave travelling through different medium undergoes change in speed. Refraction enables sound to travel faster along the ground at night than during the say. When waves approach a shore with a gradual depth profile, e.g. a beach, most of the wave energy is absorbed by the breaking of waves. When waves approach a hard vertical surface, e.g. a concrete wall or a dock, most of the energy is converted into waves moving in the opposite direction, a reflection. Of course the reflected waves are superimposed onto the original waves, Superposition of Waves The principle of superposition may be applied to waves whenever two (or more) waves travelling through the same medium at the same time. The waves pass through each other without being disturbed. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave displacements y (x, t) = y1(x, t) + y2(x, t) Reflection of Waves When a wave encounters a boundary, it will reflect back. The way in which it reflects will vary depending on whether it encounters a fixed or free boundary. Fixed Boundary A fixed boundary is when a wave encounters a fixed surface. This would occur for a rope attached to a wall. Free Boundary A free boundary occurs when, for example, a rope is attached to a post and is free to move up and down at the end. Reflection and Refraction Reflection Waves bounce back off of a surface that they encounter. A travelling wave, at a rigid boundary or a closed end, is reflected with a phase reversal but the reflection at an open boundary takes place without any phase change. Refraction Waves bend and pass through a surface that they encounter. Standing Waves and Normal Modes Standing wave oscillates with time but appears to be fixed in its location wave pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere Node A point in a standing wave that always undergoes complete destructive interference and therefore is stationary Antinode A point in a standing wave, halfway between two nodes, at which the largest amplitude occurs For Nodes kx=nπ where n= 0,1,2,3,… distance of λ/2 or half a wavelength separates two consecutive nodes. For Antinodes kx=(n+1/2) π where n= 0,1,2,3,… distance of λ/2 or half a wavelength separates two consecutive antinodes. For a stretched string of length L, fixed at both ends, the two ends of the string have to be nodes λ = 2L/n, for n = 1, 2, 3, … etc The corresponding frequencies can be represented as ν = n v/2L, for n = 1, 2, 3, … etc. For a system closed at one end, with the other end being free the closed end will be node while open end will be antinode λ = 2L/(n+1/2), for n = 1, 2, 3, … etc The corresponding frequencies can be represented as ν = (n+1/2) v/2L, for n = 1, 2, 3, … etc. The oscillation mode with that lowest frequency is called the fundamental mode or the first harmonic The second harmonic is the oscillation mode with n = 2. The third harmonic corresponds to n = 3 and so on. The collection of all possible modes is called the harmonic series and n is called the harmonic number. Beats The phenomenon of regular rise and fall in the intensity of sound, when two waves of nearly equal frequencies travelling along the same line and in the same direction superimpose each other is called beats. One rise and one fall in the intensity of sound constitutes one beat and the number of beats per second is called beat frequency. It is given as: νb = (ν1 – ν2) where ν1 and ν2 are the frequencies of the two interfering waves; ν1 being greater than ν2 Resonance Resonance occurs in an oscillating system when the driving frequency happens to equal the natural frequency. For this special case the amplitude of the motion becomes a maximum. An example is trying to push someone on a swing so that the swing gets higher and higher. If the frequency of the push equals the natural frequency of the swing, the motion gets bigger and bigger. Doppler's Effect The phenomena of apparent change in frequency of source due to a relative motion between the source and observer is called Doppler's effect. When the source and observer are moving toward each other, the frequency heard by the observer is higher than the frequency of the source. When the source and observer are moving away from each other, the frequency heard by the observer is lower than the frequency of the source.