PHYS 2386 - Optics - Lecture 3 PDF

LECTURE 3: SUPERPOSITION OF WAVES PHYS2386: Electromagnetism and Optics Review  Recall a mathematical description of a wave  Derive a one-dimensional wave equation  Represent waves using complex numbers  Provide a general description of harmonic waves Plane...

LECTURE 3: SUPERPOSITION OF WAVES PHYS2386: Electromagnetism and Optics Review  Recall a mathematical description of a wave  Derive a one-dimensional wave equation  Represent waves using complex numbers  Provide a general description of harmonic waves Plane Spherical cylindrical Learning Objectives  At the end of this lecture, you should be able to:  Describe the interaction of propagating waves  State the principle of superposition of waves  Wave equations to show destructive and constructive interference  Describe ‘the beat phenomenon’  Describe phase and group velocities Wave Equation Interpretation  The initial amplitude of a wave can be zero or a maximum depending on whether or not we use a sine or cosine function. This parameter can assume any value in principle. In other words, the wave can be shifted to the left or right by any angleWe describe this angle as a phase shift.  Thus, the general form for a propagating wave is given by :  Suppose that when and , then. Thus, the initial phase is : Interaction of Propagating Waves  Please watch the following link and make notes : https://www.youtube.com/watch? v=CAe3lkYNKt8 Superposition Principle  When 2 or more waves travel through the same medium simultaneously, the resultant displacement at any point is the vector sum of the displacement due to the individual waves.  Superposition Principle  Consider two waves travelling simultaneously in opposite directions. We can see images of waveforms at each instant of time. NB : As per the principle of superposition, we can add the  It is observed that the net displacement of overlapped waves algebraically the waveform at a given time is the algebraic to produce a resultant wave. sum of the displacements due to each wave.  Let us say two waves are travelling alone, and the displacements of any element of these two waves can be represented by y1(x, t) and y2(x, t). When these two waves overlap, the resultant displacement can be given as y(x,t).  Mathematically, y (x, t) = y1(x, t) + y2(x, t). Constructive & Destructive Interference  Superposition Superposition Principle  The resultant displacement of wave is the sum of the separate displacements of the constituent waves. Superposition of waves of the same frequency E0 sin( kof ConsiderEawave 0 ) t form: r the k r Set equal to 0 since we wish to examine E E0 sin(t   ) waves at a fixed point. Thus  Superposition  The 2 waves may be expressed E1 E01 sin(t  1 ) The 2 waves possess the E2 E02 sin(t  2 ) same frequency.  Superposition by the phasor method: EO 2 EOR R 2 In this case the EO1 E waves are 1 treated as E R E0 R sin(t   R ) scalar quantities. where 2 EOR EO21  EO2 2  2 EO1 EO 2 cos(2  1 )  Superposition  New wave has the same frequency as original waves but different amplitude and phase. Assumptions  Electric fields are all parallel  Coherence NB : Coherence means that two or more waves in a radiation field are in a fixed and  Beat Phenomenon  When two sound waves of different frequency approach your ear, the alternating constructive and destructive interference causes the sound to be alternatively soft and loud.  This wave disturbance is called ‘beats’  Beat frequency is the difference in frequency for two waves.  e.g., Two instruments playing the same note.  Feedback loop designed to ensure two sources have the same frequencies.  Data Transmission  Any source of waves has a superposition of different frequencies;  Higher freq. – carrier waves  Lower freq. – envelope waves Wave have to undergo frequency modulation (fm) or amplitude modulation (am) to transmit signals. am fm  Phase and Group Velocity analogy  A group of city marathon runners, start from the beginning at the same time.  Initially it would appear that all of them are running at the same speed.  As time passes, group spreads out (disperses) simply because each runner in the group is running with different speed.  If you think of phase velocity to be like the speed of an individual runner,  then the group velocity is the speed of the entire group as a whole.  Phase and Group Velocity  Moving at different frequencies can affect velocity.  The group velocity of the signal is the velocity at which the positions of maximal constructive interference propagate.  The phase velocity of an EM signal is a measure of the velocity of the harmonic waves that make up the signal. Summary  Describe the interaction of propagating waves  State the principle of superposition  Wave equations to show destructive and constructive interference  Describe ‘the beat phenomenon’  Describe phase and group velocities Supplementary Notes  Superposition Principle Constructive Interference Constructive interference occurs when ( Destructive Interference Destructive interference occurs when (  Superposition Principle General Superposition +)) Defining +  Superposition Principle where and For N harmonic waves, we have =+ Beat Phenomenon Consider two waves of comparable amplitudes but different frequencies in a non-dispersive medium i.e., same speed. the super-positioning of these waves is given by; Using the trig. identity ) where Beat Phenomenon Now let , The beat frequency is given by = Beats are caused by the interference of two waves at the same point in space.  Phase and Group Velocity ω  The general relation for velocity: v = k ω  Velocity of higher freq. carrier wave k vp =  Velocity for low freq. Envelope wave dω vg = dk The media in which vg = vp is called the non-dispersive medium. But the media in which vg < vp is called normal dispersion. The media in which vg > vp is called anomalous dispersive media. Where the refractive index will influence the media The relation between phase and group velocity is given by, vg = vp [ 1 − λ dn ] n dλ