UNIT 6 WAVE MOTION Structure Introduction Objectives Basic Concepts of Wave Motion Types of Waves Propagation of Waves . Graphical Representation of Wave Motion Relation between Phase Velocity. Frequency and Wavelength Mathematical Description of Wave Motion Phase and Phase Difference Energy Transported by Progressive Waves One-Dimensional Progressive Waves : Wave Equation Waves oh a Stretched String Waves in a Fluid Waves in a Uniform Rod Wave Motion and Impedance Impedance offered by Strings : Transverse Waves lmpcdance offered by Gases : Sound Waves Waves in Two and Three Dimensions Summary Terminal Questions Solutions 6.1 INTRODUCTION In Unit 5 you have learnt that when one mass in a system of N coupled masses is disturbed, the disturbance is gradually felt by all other masses. You can think of many other similar situations in which oscillations aJ one place are transmitted to some other place through the intervening medium. When we talk, our vocal cord inside the throat vibrates. [t causes air molecules to vibrate and the effect-speech-is transmitted. When it makes our ear drum to vibrate, it is heard. Do you know what carries the audio information? The information is carried by a (sound) wave which propagates through the medium (air). If you have ever stood at a sea shore, you would need no description of waves. In addition to sound and water waves, other familiar types of waves are : Ultra-sound waves and electro-magnetic waves, which include visible light, radio waves, microwaves and x-rays. Matter waves, shock waves, and seismic waves are other less familiar but important types of wavb. You will note that all our communications depend on transmission of signals through waves. The use of x-rays in medical diagnosis is so very well known. Now-adays we also use ultra-sound waves - sound waves of frequency greater than 20 kIk-to make images of soft tissues in the interior of humans. Sound wava are used in sound ranging, sonars and prospecting for mineral deposits and oil - commadities governing the economy of nations these days. This means that understanding of the physics of wave motiorl is of fundamental importance to us. in this unit we will confine to mechanical wava with padcular reference to sound waves. When a progressive wave reachesjtheboundary of a finite mkium or an interface between two media, waves undergo reflection and/or refraction. These will be discussed in detail in the, next unit. ~rorn'~1ock-1you would redl that our discussion of oscillations .was simplified bcdurc of some basic similarities between different physical system. Once we unde*ltood.Ipe behaviour of a model spring-mass system, we could easily draw analogies for others. Exactly the same simplification occurs in the study of waves. The basic Wption d a wave and the parameters reqpired to quantify this description remain ,$R same when we deal with a one dimensional (I-D) wave' travelling along a strik, a 2. D wave on the surface of a liquid or a 3.0 sound wave: For this reason, j.l1 this unit we shall' fmt consider basic chant&- of wave motion. Then we would calculate the energy bgasporlted by progrbive qraves. The vocabulary, language and i&as developed here will thm be applied to waves on *gs, liquids and gases. Waver Objectives After going through this unit, you should be able to @ define wave motion and state its characteristics @ dislinguish between longitudinal and transverse waves @ represent graphically waves at a fixed position or at a fixed time @ rclate wavelength, frequency and speed of a wave s establish wavc equations for longitudinal and transverse waves @ compute thc encrgy transported by a progressive wave @ dcrivc expressions for velocities of Ibngitudinal and transverse waves 9 derive cxprcssions for'charactcristic irnpcdancc and acoustic impedance e write two and three diniensional wavc equaticlns. You may hltvc enjoyed dropping small pebbles in'still water. It will not take you long to convince yoursclf that water itsclf docs not move with the wave (evidenced as circular discurhancc).. If you place ;I paper boat, a flower or a small piece of wood, you will observe thai it bounccs up and down, without any forward motion. You may like to know : Why the papcr boat bounccs up and down'? I1 bounccs due to the energy imparted by waves. Let us reconsidcr lhc motion of a systcm of N coupled masses (Fig. 6.1). If we disturb the first mass from.its cquilibriu~nposition, individual masses gradually begin to oscillate about their respcctivc cquilihrium positions. That is. neither of the masses (or connecting springs) nor the systcm as ;I wholc tnoves liom its position. What moves instead is a wave which carries encrgy. How can yo11 say that'? It is evidenced by compression and stretching of springs as the wi~vcpropagutcs. Thus thc most'important characteristic of wave motion is :AwavcJ 1rrrtt.spurr.s canilrg)hu~ no1 nrallcpr. Fig. 6.1 : The motion ofa disturbed nlass in the coupled spring-mass system. The disturbance ir eventually communicated to adjacent masses. This results in a wave propagation. You .would note that regions of compression nndclonealion move along ihe system, which is shown here at two different times. A vivid demonstration of the energy carried by water waves is in damage caused in coastal areas by tidal waves in stormy weather. You w~llbe astonished to know that in April 1991 an oceanic (tidal) wave generated in the Bay of Bengal created havoc in Bangladesh. It is estimated that about hundred thousand people lost their lives and more than a million were " rendered homeless. In May 1990 a similar oceanic tide destroyed property over coastal areas in Andhra Pradesh. An earth quake in Chile produced2 tidal wave which carried huge amount of energy across 15,000 km of the Pacific Ocean and caused untold damage in Japan. Do you know that 3m high oceanic wave can lift 30 bags of wheat by about 10 ft.? You may have read that we now plan to harness tidal energy to meet our increasing energy requirements. Another important characteristic of mechanical waves is their velocity of propagation, referred : to as wuve velociry. It is defined as the distance covered Gy a wave in unit time. It is different from the particle velocity,.i.e. the velocity with which the particles of the medium vibrate to- , and-fro about their respective equilibrium positions. Moreover, the wave velocity depends on ! the nature of the medium in which a wave propagates. A wave has a characteristic amplitude, i wavelength and frequency. You must have learnt about these in your earlier classes. We will, , however recapitulate these in sub-section 6.2.4. i I You can see with unaided eyes the actual propagation of a disturbance in water, Can you see a ' sound wave propagating in air? Obviously, you cannot. Then you may like to know as to how ; we detect sound waves. We obServe the motion at the source (like'sitar string or tabla .+ membrane) or at the receiver (microphone membrane). Another question that comes to Our i mind is : Are sound waves and water waves similar? If not, hod are waves classified? Let us 1 now proceed to know the a&er to this question. f i 6.2.1 Types of Waves Wave Motion In your school you must have learnt that waves can be classified as transverse or longitudinal depending upon the direction of vibration of particles relative to the direction of propagation of the wave. In fact, we can classify waves in many other ways. For instance, we have , mechanical and non-mechanical waves depending on whether a wave needs a medium for propagation or not. Sound waves and water waves are mechanical (or elastic) waves whereas light waves are not. Waves can also be classified as one, twe and three-dimensional waves, according to the number of dimensions in which they propagate energy. Waves qh strings or the slinky are onedimensional (1-D). Ripples on water are two-dimensional (2-0). Sound waves and light waves originating from a small source are three-dimensional (3-0). Sometimes we classify waves as plane waves or spherical waves de nkling on the shape of the wavefront. in 2-0, a spherical wave appears circular, as in case o ?=\kaves on the surface of water. Waves set up by a single, isolated disturbance are calledpulses. The dropping of a stone in still water of a pond, the sound produced by clapping ~f hands, a single word of greeting or command shouted from one person to another belong to this category. When an engine joins the compartments, the jerk produces a disturbance which is carried through as a pulse. But continuous and regular oscillations produce periodic waves. This, alongwith wave forms for sound produced by a violin and a piano, is shown in Fig. 6.2. The simplest type of a periodic , wave is a harmonic wave. Violin Fig. 6.2 : Waveronns for (a) hannonk. wave (b) he dinand (c) phm. When the motion of panicla of the medium is perpendicular td the direction in which the ,.f wave propagates, it is alled a tm~persewave. Wava on a stiing under teaion are Wave% : as in a ektara, sarangi, sitar, vina and violin. You can generate transverse waves on a coup14 , spring-mass system of Fig. 6.1 by displacing a Mass at right angles to the spring as own in a Fig. 63a. (Electromagnetic waves are also transverse ia nature. But they-Beqpre~. medium for propagation.) Wave Fig. 6.3 (a) : A transverse wave and (b) a longitudinrl wave on a coupled spring-mass system.