WAVES - Lecture Notes

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 WAVES Wave motion is defined as a form of disturbance transferred from one point to another point involving transfer of energy but no transfer of matter Classification of Waves 1) Mechanical Waves : Waves which require a medium for their propagation. Mechanical wave can be produced and propagated only in those material media which possess elasticity and inertia eg  Waves on water surface, sound waves 2) Non mechanical waves or electromagnetic waves : EM waves do not require any material medium for their production and propagation. Eg.  light waves, radio waves, x-rays 3) Matter waves or de-Broglie waves : Matter waves are associated with the motion of microscopic particles like electrons, protons, neutrons and other fundamental particles Types of Mechanical Waves (i) Transverse Waves : A transverse wave is the one in which the particles of the medium execute oscillations in a direction perpendicular to the direction of propagation of waves. Properties 1) It produce crest and trough in a medium 2) It changes shape of the medium 3) It can produce only in a medium with shear modules 4) It propagates only in solids or on the surface of liquids (due to surface tension) (ii) Longitudinal Waves In a longitudinal waves particles of the medium execute oscillations in a direction of propagation of wave. Properties i) It produce compression and rarefaction in a medium ii) Medium required Bulk modulus iii) It propagate in all material media be it solid, liquid or a gas 1

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023 Path difference and phase difference The mutual separation between two points is called path difference. x  x2  x1 Path difference  = phase difference 2 When a wave travelling through a x distance the phase difference   2 x or   2 t T Equation of plane progressive simple harmonic wave A wave which advances in a medium called a progressive wave. If the particles of the medium vibrate simple harmonically and if the wave front is a plane wavefront, then it is called plane harmonic wave Suppose a simple harmonic wave which starts from the origin O at t = 0 and travels along the positive x direction with a speed v. The displacement of the particle at x = 0 at any instant ‘t’ is given by y0,t  A sin t If  is phase lag of a particle at P w.r.t the particle at the origin O, then the displacement of this particle at P, at the same time ‘t’ can be written as yx,t  A sin t   2

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 At a distance x, the phase change = 2 x  yx,t  A sin  t  2 x     Put 2 k  propagation constant or angular wave number   yx,t  A sin t  kx  If 0 is the initial phase (for a general choice of the origin of time) yx,t  A sin t  kx  0  If the wave travels in the –ve x direction, the equation of the wave yx,t  A sin t  kx  0  –ve sign in between t and kx implies wave travelling along +ve x direction +ve sign in between t and kx implies wave travelling along –ve x direction Terms in wave equation yx,t  The displacement of an element at position x at any instant t A  Amplitude : Which is the measure of magnitude of the maximum displacement of the particle from their equilibrium position. t  kx  0   phase : The argument t  kx  0  is the phase of the wave. It describe the state of motion of the element located at coordinate x at time t. 0  Initial phase (phase constant) : It describe the phase of the element at location x = 0 and at t = 0   Angular frequency   2  2 T k  Angular wave number k  2  3

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023 Yx,t  A sin t  kx  0    2, k  2   yx,t  A sin 2  t  x   0      The equation of the wave depends on the position of the observer which is taken as the origin of the reference frame with respect to which the wave motion is observed. If 0   If 0   2  yx,1  Asin t  kx   2 yx,1  A sin t  kx    yx,tA cos t  kx   A sin t  kx yx,t  A sin kx  t  Difference between A sin t  kx and A sin kx  t  yx,t  A sin t  kx  yx,t  A sin kx  t  put t = 0 put t = 0 y  A sin kx y  A sin kx dy  Ak cos kx dy  Ak cos kx dx dx Slope  –ve slope  +ve 4

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 Wave equation - Longitudinal wave Sx,t  A sin t  kx  0  Sx,t  displacement of particle along the direction of wave propagation Particle Velocity Acceleration of the particle yx,t  A sin t  kx  d2y ap  dt2 dy  A cos t  kx   ap  A2 sin t  kx dt Vp  Acos t  kx  ap  A2 max  Vp  A max Speed of a travelling wave General relation v   v  displacement time v for 1 time period T, displacement =  k v     or v T k Velocity of a mechanical wave depends on elastic and inertial properties of the medium. 3. Relationship between particle velocity and wave velocity y x, t   A sin t  kx dy  Ak cos t  kx   Vp  Acos t  kx  dx  vp  cos t  kx   A    Slope  Ak cos t  kx   Ak  VP A 5

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023 Slope = Vp  Vp  slope  V V Velocity of transverse wave on a stretched string The speed of transverse wave on a string depends on (i) Tension (ii) linear mass density  (mass per unit length) v T   Mass of the string  M  Length of the string L v T A   M  MA A  Area of cross section L LA MA M =   density  V  V NOTE Consider a vertically hanged massive string in which a transverse wave originating from the lower end propagates to the upper part At y tension in the string arises due to weight of hanging mass below it At y, T = mg T  yg m  y then v  T yg   6

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023  v  gy  Velocity of wave when it travels a distance ‘y’ from the lower end. Velocity of transverse wave in solids v  where   modulus of rigidity    density Velocity of Longitudinal Wave The speed of all mechanical waves follows an expression of the general form v elastic property or V  E inertial property  Velocity of longitudinal wave in a thin rod v Y Y  Young’s modulus    density In fluids v B B  bulk modulus ,   density In solids B 4 B  bulk modulus v 3    shear modulus   density Speed of Sound wave in a gas Newton’s formula : He assumed that when a sound wave propagates through a gas, the temperature variations in the layers of compression and rarefaction are negligible. Hence the conditions are isothermal. v p PV = a constant  7

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023 Partially differentiating PV  VP  0 P  P V V B  P The speed of sound in air at this condition should be 280 m/s. but measured value is 332 m/s Laplace’s Correction Laplace suggested that the compression or rarefaction takes place to rapidly and the gas element being compressed or rarified does not get enough time to exchange heat with the surrounding. Thus it is an adiabatic process P  = a constant, p  P V V V  p   CP  CV  M Ideal gas equation PV = nRT V If n = 1 mole, then PV = RT V  PV M V  RT M Vrms  3RT M Vsound   Vrms 3 Factors affecting speed of sound 1) Temperature V  T or V1  T1 V2 T2 8

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 Vt = V0 + 0.61t  For small values of t V0  Velocity at 0oC Vt  Velocity at toC It is found that for every 1oC rise in temperature Velocity of sound increases by 0.61m/s 2) Pressure p If pressure changes density also changes in such a proportion, hence  remains a constat (at constant temperature). Hence pressure has no effect on the speed of sound in gas 3) Humidity With the increase in humidity, the density of air decreases.  Speed of sound will increase Wind As the sound is carried by air, so its speed is affected by the wind velocity Resultant velocity = V  Vw cos  Intensity of a Wave It is defined as the transfer of energy per unit time per unit area, perpendicular to the direction of motion of the wave I  Energy unit  J or I  Power unit  watt Area  time m2s Area m2 The average intensity transmitted through the medium is given by I  1 V2A2   density 2 v  velocity   angular frequency I  A2 A  amplitude A human ear can tolerate and detect sound waves in the intensity range of 1 w to 1012 w m2 . The m2 lowest intensity of sound that can be percieved by the human ear is called threshold of hearing,  I0  1012 w m2 9

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023 Sound Level   10log I I1  1 2  1  10 log  I2  I0  I1    I2  2 Reflection of a wave 1) Reflection from a Rigid boundary As the pulse reaches the wall, it exerts an upward force on the boundary. By Newton’s third law, the wall exerts an equal amount of force on the string in downward direction. Thus an inverted pulse produces which travels in reverse direction. Hence there is a phase reversal of 180o or  radian Eg. Let the incident wave Yx,t  A sin t  kx Reflected wave Yx,t  A sin t  kx   Reflection from a open boundary When the pulse arrives at the ring, the string exerts an upward force on the ring due to which the ring moves up the rod. As the ring moves, it pulls the string upward producing a reflected pulse of same amplitude that travels back without any phase change along the string. When a wave is reflected from a free boundary, it suffers no phase change Eg. Let the incident wave yx,t  A sin t  kx  the reflected wave yx,t  A sin t  kx  10

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 Super Position of Waves When any number of waves meet simultaneously at a point in a medium, the net displacement at a given time is the algebraic sum of the displacement due to each wave at that time yr  y1  y2  y3  ...... Here, we shall discuss the superposition of two individual waves only. There are three types of superposition 1) Interference 2) Beats 3) Standing waves or stationary waves 1) Interference When two or more waves of same frequency same wavelength and a constant phase difference travelling along same direction, superimpose on each other give a new disturbance Let y1  A1 sin kx  t y2  A2 sin kx  t   yx,t  y1 x, t   y2 x, t  Yx,t  Ar sin kx  t    Ar  A12  A22  2A1A2 cos  Initial phase   tan1  A2 sin    A1  A2 cos     I  I1  I2  2 I1I2 cos  Destructive Interference Constructive Interference Ar  min Ar  max cos   1 If cos   1, for this   2n For this   2n 1  where, n  0,1, 2,..... Path Difference where n  1, 2,3...   2 x Path Difference  2n 1   2 x  11

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023 2n  2 x x  2n 1   2 x  n Amax  A1  A2  Amin  A1  A2  2  2 Imax  I1  I2 Imin  I1  I2 Amax  A1  A2 Imax  I1  I2 2   A1  A2 2 Amin A1  A2 Imin   I1  I2   A1  A2     BEATS The periodic variations in the intensity of sound due to the superposition of sound waves of slightly different frequencies are called beats, one rise and one fall of the intensity constitute a beat. Two harmonic sound waves of nearly equal angular freuqency 1 and 2 and fix the location to be x = 0, for convenience, with a suitable choice of phase. 1  2  Let S1  a cos 1t S2  a cos 2t S  S1  S2 S  2a cos  1  2  t cos  1  2  t  2   2  S  2a cos bt cos a t where b  1  2 2 a  1  2 2 The resultant wave is oscillating with the average angular frequency a . However its amplitude is not constant in time unlike a pure harmonic wave. In other words due to the variation of amplitude intensity of the resultant wave waxes and wane. Beat frequency Number of beats per second, n  1  2 1 & 2  frequencies of super imposing waves 12

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 Beat Period Time interval between two successive beats or time interval between two successive maxima or time interval between two successive minima. TB  1 1  2  A maxima and its nearest minima will differ in time by TB  2 1 2 1  2 Stationary waves / Standing waves When two identical waves of same frequency, same velocity travel in opposite directions along the same path super impose each other give rise to a new wave. The resultant wave does not travel in the either direction and therefore is called stationary wave. Let y1  A sin kx  t  y2  A sin kx  t y  y1  y2 y  2A sin kx cos t In the resultant wave particles are oscillating with same angular frequency  , but the amplitude of the particle is varies from point to point. Amplitude of the resultant wave is A'  2Asin kx Amplitude of the particle is determined by its position (x) Node : The points at which amplitude is zero Antinode : The points at which amplitude is the largest Node Antinodes Amplitude = 0 Amplitude  max A  2A sin kx  0 i.e. sin kx  0 A '  2A sin kx  2A max for this sin kx  1 13

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023 For this kx  n kx  2n 1  2 2 x  n 2 x   2n 1  2   x  n 2 (n=0,1,2,...) x  2n 1  4 Nodes occure at, x  0,  2 , , 3 2 ..... Antinode occures at , x =  4 , 3 4 ,5 4 ..... Distance between two adjacent nodes =  2 Distance between two adjacent antinodes =  2 Loop length =  2 Distance betwen a node to nearest antinode =  4 If the amplitude of the standing wave, A '  2A cos kx then the graph becomes Properties of standing waves 1. From node to antinode amplitude gradually increases and decreases from antinode to nearest node. Amplitude varies from 0 to 2A. 2. All the particles in a particular segment between two nodes vibrate in the same phase but the particles in the neighbouring segment vibrate in opposite phase. 3. All the particles (except those at nodes) crosses their mean points simultaneously but with different velocities. 4. Energy becomes alternatively wholly potential and wholly kinetic twice in each cycle. Transverse standing waves in a stretched string Consider a uniform string of length L under tension T lying along the x-axis with its ends fixed x = 0 and x = L. 14

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 The ends x = 0 and x = L are fixed, so they must be nodes. The boundary conditions are x = 0, y = 0 for all time x = L, y = 0 for all time At x = L, y = 0 (position of node) A  2A sin x  0 For this kx  n 2 L  n   L  n 2   2L n The frequency of vibrations of the string in its n th mod e    V    n  nV or 2L  Natural frequency or Harmonics n T n  2L  If n  1 1st Harmonic / fundamental frequency 1  V 2L   2L The lowest possible natural frequency of the system If n  2  2nd harmonic / 1st overtone V 2  2 2L 2  21 2  2L 2 15

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023 If n  3  3rd harmonic / 2nd overtone 3  3 V 2L 3  31   2L 3 Ratio of harmonics = 1 : 2 : 3 : ......... n Difference between any two consecutive harmonics gives 1st harmonic n1  n  1 Longest wavelength   2L 1st harmonic Ratio of wavelengths = 1: 1 : 1 : 1 : ..... 234 No.of segments/loops in nth harmonic = n No.of antinodes in nth harmonic = n No.of nodes in nth harmonic = n + 1 Laws of vibration For a string excited to nth harmonic n T n T n = 2L  2L A ; A  r2 i. Law of length : If T and  are constants, then  1 L ii. Law of tension : If L and  are constants, then   T iii. Law of  : If L and T are constants   1  Also n  n T ie   1 2L r2 Lr 16

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 SONOMETER : Its working is based on transverse standing wave. If the frequency of a tuning fork happens to be equal to one of the natural frequency of the wire, standing waves with large amplitudes are set up on it. The tuning fork is then said to be in resonance. Longitudinal standing waves An organ pipe is a tube of uniform area of cross section in which air (or gas) is trapped to form an air column. Sound waves in air entering from one end gets superimposed with its own reflected wave from the other end to form longitudinal standing waves inside the tube. Open pipe  Both ends of the pipe are open Closed pipe  one end open other end closed In a pipe, the closed end behaves as a rigid boundary and the open end behaves as a free boundary for the displacement wave. Thus at the closed end displacement node or pressure antinode forms and at the open end displacement antinode or pressure node forms. Closed organ pipe at x = 0, displacement node  A '  2A sin kx 17

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023 At x = L (Position of an antinode) A  2A sin kx  Max. For this sin kx  1 kx  2n 1  2 2 L  2n 1  2  L  2n 1  4   4L ;  V  2n 1 n  2n 1 V 4L Natural frequency / Harmonics 2n 1  n  4L  n  0  1st Harmonic / fundamental frequency 0  V ;   4L 4L n  1 3rd harmonic / 1st over tone 1  3 V 4L 1  3V0   4L 3 n  2  5th harmonic / 2nd over tone 2  5 V  50 4L   4L 5 18

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 For a closed pipe only the odd harmonics of the fundamenal are the allowed frequencies. nthovertone  2n  1 th Harmonic Ratio of frequencies = 1 : 3 : 5 : 7 ......... Longest wavelength = 4L Ratio of wavelength = 1 : 1 : 1 : 1 ...... 3 5 7 No.of nodes = No.of antinodes Open organ pipe Both ends are open, each end is an antinode At x = 0 is the position of antinode  amplitude A  2A cos kx At x  L (Antinode) A  2A cos kx  Max For this cos kx  1 kx  n 2 L  n  L  n or   2L 2n n  n V 2L n  n p  Natural frequency or Harmonics 2L  n  1  1st harmonic / fundamental frequency 1  V 2L   2L 19

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023 n  2  2nd harmonic / 1st over tone V 2  2 2L  21   2L 2 n  3  3rd harmonic / 2nd over tone V 2L 3  3 2L  3V1;   3 An open pipe generates all the harmonics Ratio of frequency = 1 : 2 : 3 : 4 : ...... Ratio of wave length = 1: 1 : 1 : 1 : ..... 2 3 4 No. of nodes in the nth harmonic = n No.of antinodes in the nth harmonic = n + 1 End correction Till now we have assumed that antinodes are formed exactly at the open end of the tube. But due to the inertia of vibrating particles antinodes are formed a little above the open end of the tube. This additional distance is called end correction. For a tube of radius r, e = 0.6 r For closed organ pipe, the effective length = L + e For open organ pipe, the effective length = L + 2e Resonance column 20

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 It is used to determine the speed of sound in air with the help of tuning fork of known frequency. It is a closed pipe whose length can be changed by changing level of liquid in the tube. When a vibrating tuning fork is brought over its mouth, its air column vibrates longitudinally. The length of the air column is varies until its natural frequency becomes equal to the frequency of fork, then resonance will occur and loud sound is heared. Ist resonating length L1 e   ...........(1) 4 IInd resonating length L2  e  3 4 .......(2) L2  L1   2   2L2  L1  Velocity V   V  2L2  L1  From the eqns (1) and (2) e  L2  3L1 2 Doppler Effect The apparent change in the frequency of sound due to relative motion between source and observer is called doppler effect. Consider a source of natural frequency , when source is in relative motion with listner, the apparent frequency  measured by the listner is given by    V  VL   V  Vs    VL  Velocity of listner either towards/away from source VS  Velocity of source either towards/away from the listner V  Velocity of sound Sign convention     when source and listner approaches each other     when source and listner moves away from each other 21

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE)-2023     when source and listner at rest ie. VL  VS  0 or source and listner moving with same velocity along same direction When oblique motion is considered, only velocity components in the line joining source and listner are considered. 1    v  vL cos 2   v  vs cos 1        V V0     VS cos        V  V cos    VS    If    2 , then    Note: When a source and listner moving towards a stationary wall Original force  The frequency of sound perceived by the wall (wall listner)    V V    Vs    22

BBrilliant STUDY CENTRE PHYSICS (MED.- ONLINE) -2023 Frequency of the reflected sound perceived by the listner (wall source)     V  VL  V        V  VL    VS   V  V      V  VL   V  VS    Apparent wave length  '  Re lative velocity of sound with respect to source Original frequency  '  v  vs  23