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 Wptiond 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) hedin and (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. Thc broad arrow indicates the direction of wave propagation. When the motion of particles of the medium is along the direction in which wave propagates. the wave is cglled a longi~udinalwave. Sound waves inhairare the most familiar example or longitudinal waves. You can generate a longitudinal wa've on a coupled spring-mass system of Fig. 6.1 by displacing a mass along the length of the spring (Fig. 6.3b). In your school you may have been told that water waves, produced by winds or otherwise. arc transverse and the motion is confined to the surface. But this is not correct. Strictly speaking, the motion gradually extends with diminishing amplitude io deeper layers. Moreover, oscillations have longitudinal as well as transverse components. That is, water waves are .composite; partly transverse and partly longitudinal. This is illustrated .in Fig. 6.4. similar waves can occur at the surfaces of elastic solids. Such waves are called Rayleigh waves.
Fig, 6.4 : Water waves ate composite. The arrows at different points indicate instantaneous velocity of water whereas dotted circles are paths tmed out by parcels of water as the wave passes. The direction of wave pmpagation is showri by the broad arrow. In general, only longitudinal waves can propagate in gases and liquids but in solids both transverse and longitudinal waves can propagate. In the previous block, you have read abouttorsional oscillations, When such a disturbance propagates in a medium, we have a torsional wave. In the following sections, ypu will'learn about.mechanical waves in general and sound waves in particular. But before that you should answer the following SAQ.
SAQ 1 (i) The frequency of ultrasound wave is more than ...... (ii) Water waves are ...... wv~ves (iii) Waves transfer ...... not ......
(iv) Light waves require ...... medium I
I (v) Waves on sitar strings are ...... 6.2.2 Propagation of Waves , To see how waves propagate in n medium, you can pertorm the following activity : Take a long elastic string and-fix its one end to a distant wall. Hold the other end tightly. Move your hand up and down. What do you observe? A disturbance travels along the string. This disturbance is due to the up and down motion of the particles of the string about their 1 respective mean positions. When the motion of the arm [hand) is periodic, the disturbance on ' the string is a wave with a sinusoidal profile. The shape of a portion of the string at inte~al$of. i T/8 is shown in Fig. 6.5. Yoc will observe that the waveform moves to the right, as shown by 1 the broad arrow. You may ask : Why is the whole string not displaced simultaneousiy? The 8 time lag between diffe;ent parts is due tdgradual transfer of disturbance between successive . ' '1 , particles. , I I Wave Motion
Fig. 6.5 : A periodic motlon of the hand generates waves with a sinusoidal profilc. In parts (b)-(j) we have depicted wave progle on the string at Intervals of 778.
1 In this connection, it is important to distinguish between'the motion of the waveform and the 1 motion of a particle of the string. While lhc wavcform moves wit11 a constant speed, the ,, particles of the string executc SHM. To illustrate this difference clearly, let us mark nine ' equidistant marks on the initial portion of the string. We assume thal this string oscillates with ' a period T. Let us tie one end of this string (at mark 1) to a vertielly oscillating spring-mass system as:shown in Fig. 6.6. As the mass m on [he spring moves up and down, the particles at
Atziuku.
FIR. 6,6 :A stdng fastened to on oAlllnting spring rnnw system : Illumtntion of the dlflerence between mown of Uaa Wavefonn ma the mdlon of partkles the marked positions'begin to oscillate one after the other. In time 1; the distuibance initiated at the first particle will reach the ninth particle, This means that in the interval Z18,'the disturbance propagates from the particle at mark 1 to the particle at mar6 2. Similarly, in the next T/8 interval, the disturbance travels from the particle at mark 2 to the particle at mark 3 and so on. In parts (a)-(i) of Fig. 6.7 we have shown the instantanea'us positions of particles at all nine marked positions at intervals of T/8.(The arrows indicate the directions of motion along wirh which particla at various marks are abut tp move.) You will observe that . Fip. 6.7 : Inslnntsneous profiles at Intervals of T/8 when a transverse wnve is generated on n string.
(i) At f = 0, all the particlei are at their respective mean positions. I I (ii) At f = T, the first, tifth and ninth particles are at their respective mean positions. The first and ninth particles are about to move upward whereas the fifth particle is about to move downward. The third and seventh particles anat position of maximum displacement but on opposite sides of the horizontal axis. The envelop joining the instantaneous positiaa~of all the particles at marked positions in Fig. 6.7 (i) similar to those in Fig. 6.5 and represents a transverse wave. The positions of third and seventh particles denote a trough and a crest, respectively. The impqrtant point to note here is that while the wave moves along the string all particles of lhe string are oscillating up and down about thei~respective equilibrium positions with the same period (T)and amplitude (a).This wave remains progressive till it reaches the fixed end. 1 I I We would now like you to know to represent a wave graphically as well as mathematically. This forms the subject of our discussion in the following sections, 1 6.2.3 Representation o[ Wave Motion I Fro the abpve activity you would recall that when a wave moves along a skng/spring, three parat eters are involved : particle displacement, its position and time. In a 2-JJ graph, you can either plot displacement against time (at a given positidn) as shown in Fig. 6.8a or displacement against position (at a given time) as shown in Fig. 6.8b. You can easily identi& ' that both plots are sinusoidal and have amplitude a We can represent these as 10 y (x).= a sin 2n (x/A) t y (1) = a sin 2n and The argumenr of the sine function ensura that the: function reneats itself regularlv. Wave Matian
I (a) (''1 Fig. 6.8 : (a) The profile of vibrations at a given position in the path of a wave, and (b) the profile of a wave at a particulrlr instant. It is snapshot of the wave travelling along the string. We can draw an analogy between the wavelength and the period. The wavelength is sepgration in space between successive in-phase points on the wave. On the other hand, period is separation in time between equivalent instants in successive cycles of vibration. This means that the wavelength and the period are respectively the spalial and the temporal properties of a wave.
It is important to note that the scales for y(x) and y(l) are different. For sound waves, the displacemhnt amplitudes are a small fraction of a mm whereas x extends to several metres. Human ears can hear sound of 1000 Hz quite clearly. The amplitude of the wave corresponding to the faintest sound that a normal human ear can hear is approximately 10-"m. This is smaller than the radius of the atom (= 10'"'m). Yet our ears respond to such a small displacement! Another point about graphical representation is that it can be used for both,transverie and longitudinal waves.
Fig, 6.9 : Graphical reprewnlntion of r longitudirinl wave.
In longitudinal waves, the displacemcnt of particles is along the direction of wave propagution, In Fig. 6.9, the hollow circles represent the mean positions of equidistant particles in a medium. The arrows show their (rather magnified) longitudinal displacements at a given time: You will observe that the arrows are heithcr equal in lengtl~nor in the same direction. This is clear from the positions of solid circles, which describe'instantancous positions of thc particles corresponding to (he heads of the arrows. The displacements to the right arc shown in the graph towards t y axis and displacements to the left towards the - y axis. For every arrow directed to the right, we draw a proportionate line upward. Similarly, for every arrow directed to the left, a proportionate line is drawn downward. On drawing a smooth curve through the heads of these lines, we find that the graph resembles the displacement-time curve for a transverse wave. If we look at the solid circles, we note that around the positions A and B, the panicles have crowded together while around the position C, they have separated farther. These represent regions of compression and rarefacrion. That is, there are alternate regions where density (pressure) are higher and lower than average. A .sound wave propagating in air is very similar to the longitudinal waves that you can generate on your stringlspring. This similarity is clearly illustrated in Fig. 6.10. A sound wave may be *i(puarsu! lsolule sn saq~a~y~ea aql Jeau JOuo S~JFU!~!JOIR~J lqa!l Lyfi s! J~J,.l~s~ ,,()I x 6 jo paads E q1!~ss~olu jqa!l PUB IsnJs Jalno s,qi~eau! , sw ,01 x g jo Japm aqt JIY pad$ e qj!m show sahem 3!us!as aqL ', slu z.0 inoq~jo paads e ql!m ahow puod a loa3trJins ayl ua saldd!a (.J!V aqj qsno~qlslaAeJ1 I! se u!e8e pue q3e~ipeo~l!w aq~ qBnoiqa slanuj punos aq$ rstr )sl!j-a3!ml pJsaq aq Leu u!w] 8u!q~eo~ddeut! jo 311s!q~aql Aqfi su!aldxa s!q~),~la~flsadaar 'I sw. 001 pue , SKI 00s I ', sw ~[fam laajs pue lajefi 'r!e u! saafi punosjo ~@WCISaya 'd~sIV .saAem leu!pn~!%uolse llaM se '31s ssel%'J~I~M 'J!E aq!l tr!pw 13qIO 111! 10J Sppq I! 1Rg . '~u!J]sI? U! aABM aSJaASUeJl E 01 33UaJajaJ q]!M (~'9)'b3'pa~!~ap 3Aey ah%tE[ll asu ll!M ffa
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Waves You will recognize that*the wave number is the spatial analog of the angular frequency. In terms of cull and k, we can also write the wave equation as . y(x I-) = a sin (w,, - kx) (6.6d)
The simple way in which k and w;, enter this qescription of'the wave explains why these quantities are so often used. . .
On comparing Eqs. (6.6a) and (6.6d) you will observe that v and k are connected through the relation
Eqs. (6.6a - d) provide us equivalent description of a travelling wave moving in the + x direction. But the choice of the particular form to be used for a specific problem is a matter of convenience. How would you describe a wave propagating in the negative x-direction? You can easily convince.yourself~thatto describe a wave moving in the negative x-direction, we should replace x by - x in Eq. (6.6). SAQ 4 . A sinusoidal water wave having a maximum height of 7.4 cm above the equilibri~lmwater level is.propagating In the - x direction with a speed of 93 cm s-'. The distance between twa
successive crests is 55 cm. Write the wave equation in terms of angular frequency and wave ' number. Also calculate the particle velocity.
.6.3.1 Phase and Phase Difference In a periodic motion, particle displacement, velocity, and acceleration repeatedly undergo a cycle of changes. The different stages in a cycle may be described in terms of phase angle. The argument of the sine function is called the phase angle or simply phase. We will denote it by the symbol 0. Thus, the phase at x at time f in a wave represented by Eq. (6.6d) is given by
0 = wui - k..- ' (6.8) You will note that the phase changes both with time and the space coordinate. With time, it changes according Jo
A 0 = 0<,1I 5 2~rvAf for fixed .T (6.9a) and with position according to
A,+ = - k3x for fixed 1 ' (6.9b) The minus sign in this equation signifies that in a wave moving along + x direction, the forward points lag in phase. That is, ihey reach the successive stages of vibration later.
6.3.2 Phase Velocity From our experience we know that water waves travel with constant velocity as long as the ( properties of the medium remain constant. For harmonic progressive waves, this velocity is 1 . called the phase velociry, i,. To show this, let us follow a given wave crest or through as the wave propagates. In order to keep the phase @(x,t) defined by Eq. (6.8) constant, we must lookfor different xas t changes. Thus by taking the differential of $ (x, t) and setting the result equal to zero, you can find the relation between x and i for a point of constant phase. The 1 differential of 4 (x, i) IS given by \
It will behezero provided dx and dt are related by
On comparing Eqs. (6.7) and (6.10). you will observe that the expression deduced earlier is aclually the phasc~ rrloci!~~ 'fi 6.3.3 Energy Transported by Progressive,Waves Wc nou know that the most spec~acularcharacteristic of progressive waves is that they 6, transport energ). We will now calcula~eLhe energy carried by a wave. To do so, we should f FI ;now both the kinetic energy and the potential energy. If the instantaneous displacement of a Wave Mollon )article isy(x, t), then the equation of a wave moving along C x direction is ~(x,t) = asin (w,,t - kx)
xt US consider a thin layer of thickness Ax and cross-sectional area A at a distance x from the ource. If p is the density of themedium, the mass of the layer is pAx A. Therefore, kinetic nergy of the layer
1 = -p&Aol,~tos2(w,,t - kx) 2 = 2 .rr2vip~dua' cos2(o1,t- k~) 1 .his expression implies that kinetic energy oscillates between zero and T pAx A ol: a'. This is
because the value of the function cosZ (w,,t - kx) varies between 0 and I
1 her one full cycle, the average value of cos28 iS -. So, the average kinetic energy over a 2 irne period is
Yhat about the potential energy? From Unit 1 of Block-1 you would recall that in SHM,the* verage kinetic energy and average potential energy are equal.' Is the same true for a harmonic lave as well? Physically, we except so. Let us now compute potential energy analytically. 'he layer under consideration will be subjtct to a force
Vc know that the work done by this force, when the layer of interes>is displaced through y urn its equilibrium position, is stored in the layer as its potential energy. So we can write
A 4 T' v,?,p ky'dy'
= - 2a' v; p-~x~y2 = - 2~'v,pA AX sin2(w,,t- kx). (6.13)
'hc minus sign tells us that the work is done on the layer. (This is of no consequencc when wc glculate total energy of the wave.) The time-averaged potential energy of the wave is : u>=~v~~A~'A~=
hn combining Eqs. (6.1 1) and (6.13), we find that the total energy of the wave is
' .- K.E. + U =- 27?'a2i;PA ax =
= 2a2a'v,?, pvA (6.1 68)
his shows that average rate of energy flow, or what we call power, is proportional lo *avc ecd and the square of the amplitude.
3.4 Intensity and the Inve~Square Law om our common experience we know that chirping of birds, the shout of a person; vchicuiar biw. sound of crackers or light from a ;amp fade out beyond a certain distance. If it wcrc not IS Waves true; noise pollution would have made life hell in our planet. To understand the principle go~erningsuch situations, we note that amplitude of an outward spreading wave decreased as the disfance from the source increases. This means that the average rate of energy flow associated with a wave decreases as it spreads out. It is therefore not very useful to talk about the total energy of progressive waves, In general, it makes more sense to describe'the strength of a wave by specifying its intensity. It is defined as the energy carried by a wave in unir rime across a unit area normal to the direcrion of morion. Using this definition, Eq. (6.16) gives
where ck = 2wv,a The S1 units of intensity are J m-'s ' or W m-'. From Unit I of Block I, you will recall that total energy is proportional to square of the amplitude of oscillation. In the same way, the intensity of a wave at a given position is proportional to the square of amplitude at that position. For a second wave, we can write '
wherep,, is the maximum change in pressure over normal pressure. Note that when we express intensity in terms ofp,, the frequency does not appear explicitly in the expression. This means that 100 Hz sound wave has the same intensity as a 10 kHz sound wave; both have the same amplitude. How does the intensity of a wave at a point vary with distance from the source? We know that the area crossed by a wave increases as it spreads out. If it originates from a point source or the distance from the source is much greater than the size of the source, the area will be almost spherical (a?). Then principle of conservation of energy demands that E = 4x19 be constant. So as r increases, intensity decreases as 1 /? :
On combining Eqs. (6.18 a) and (6.18 .b), we find that 1 pa a - r ., Since a a pa, this relation implies that amplitude of a wave 1s inversely proportional to the ' distance from the source. This explains why we can be heard upto a certain distance. (Beyond this the amplitude becomes too small to affect our ear.drums.) You must note that these results will hold if the wave is not absorbed or obstructed. In Table 6.1 we have lisred intensities of waves generated from different sources.
Table 6.1 Wave Intensities Source/ Wave Intensity (Wm-') Sound . . Threshold of hearing 10-l2 Rustle of leaves lo-"' Whisper, intensity at eardrum lo-"' Ordinary conversation 3.2 X Street traffic 1o-~ Bunting cracker, at lm , 8 X lo-' Jet Taking off, at 30m 5 I IZbtronlfagnetic Waves Radio home TV signal, 5.0 km from 50 kW trammitter 1.6 X 10-1' Sunliaht intensity it.earth's orbit 1368 , Im from typicalcamera flash rUlOO ~ i Inside microwave oven SqoO 'i Target of lwt fiuion experiment 1018 4 Sew uaw F I 16 5 Lm hrn Richter 7.0 quake 4 x lo4 t .-
I SAQ 5 Wave Motion At a distance of Irn from a bursting cracker, the intensity of sound is 8 X 10~' WIEI-~.The thrashold of human hearing is about Wm-'. Bf sound waves spread out evenly in dl dirwtions, ho~tfar from the source could such a sound be heard?
Our car is sensitive to an extremely large range of intensities. So we andefim a lo;garitRmk intensity saLe. The intensity level of a sound wave i. defined by the equation
p -1 10 ln (//I,,,). wherel,, (= 10"2 Wrn ') denotes threshold of hearing. lntensity levels are expressed in decibels. abbreviated as db. Our ear can tolerate intensity of upto 120 db.
6.4 ONE DlNENSIQNAL PROGRESSIVE WAVES : WAVE EQUATION
As a child, the music of ektara must have attracted yqu. The performances of sitar-genius. Pt.lRaLi Shankar, shahnrti maestro Ustad Bismilla Khan, or music queen Lata Mangeshkar must have delighted you. Do you know how this nlusic reaches you? What detkrmines whether or not waves can propagate in a medium and when they move, how fast they do so? Experimental investigations show that speed of waves does not depend on the wavelength or period. Th'is means that adswer to these questions should lie in the physical properties of the medium. To discover this, now we consider particular physical systems. For simplicity, we first study waves on a stretched string.
6.4.1 Waves on a Stretched String Consider a uniform string, having mnss pcr unit lcngth m, stretched by a force F. Let us chaose x-axis along the length of the string in its cquilibriurn stale. Suppose that the string is plucked so that a part of it is normal to the length of thc string, i,e, along y-axis (Fig. 6.13). Whnt happens when the string is released? It results in wnve motion. We wish to know the speed of this wave, We expect that interplay of inertia and elusticity of the mcdiurn will determine the wave speed. For a stretched string, the elasticity is rneusurcd by thc tension in the string and inertia is measured by m. Before proceeding furthbr, we would like you to carry aut the following exercise.
Fig. 6.13 : A magnified view ala small elcmcrll of the slrelchetl string, 'fhc ner rorco acllng on il is nun-zero. llle vertical disturtiun k rxaggeraled (h~clarity 1, SAQ 6 Using dirnensionai analysis, show that v=lu&% where K is some dimensionless constant.
We will now analyse the problqm mechanically by c~osiderioga small element along the string. Suppose that the string is distorted slightly so that the magni~udcaf tension an the siring mentially remains unchanged. Fig, 4,13 shaws (magnified view af) a ernall slcrncnt of thc iistoad string. You will obfirve that the direction nf.tke (tension) fnrcr; uarim along the :lement under consideratican. Why? This is hnw Iks #bring ia curved, This moans that the .ens@ forces pulling at apposite ends of the clrarnenl, alrhough of thc Rsms magniiude, da ant :xactly hncel out. Ta calculate the net force along x and &axis, we rmlvs Fin rcctnngular Waves components. The difference in the x and y components of tension between the right and the left ends of the e!ement is respectively given by
and F, = Fsin02 - Fsin01 Where el and O2 are angles which the tangents drawn at the ends make with the horizontal. For small oscillations (6 5 4')
This means that there is no net force in the X-direction,Fx = 0, and the string will be very nearly horizontal. This implies that the sines of the angles are very nearly the same as their tangents, i.e.
sin 0 I = tan 0 I and sin82 = tan 02 But the tangents of the two angles are just the sIopes of the string-or the derivadves dy/dx at the ends of the element under consideration. Then, the y-component of force on the element is approximately
F, = P(tan 02 - tan 0 I)
From the previous unit you may recall that the quantity in parentheses is just the change in the first derivative from one end of the interval AX to the other. Dividing that change by drgiv~, Thc Taylor series expansion of a in the limit Sx - 0, the rate of change of the first derivative. But we know that the function Ax + ;Lr) about the pint x displacement of the string is a function of position as well as time. If either of these variables is given by changes, the displacement also changes, You will recognise that Eq. (6.20) is valid for the configuration of the string at a particular instant of time. Therefore, the derivative in this equation is to be taken with time fixed. We call a derivative taken with respect to one'variable while other(s) is (are) kept constant partial derivative. We denote partial derivatives with the symbol a in place of the usual symbol 'd', Then, Eq. (6.20) can be rewritten as
This equation gives the net force on the segment Ax. By Newton's second law of motion, we can equate this force to the product of mass and acceleration of the segment. The mass of the segment of length Ax is mAx Then, we can write
Cancelling 5 on both sides, we obtain
You can check that Flm has dimensions of velocity square. Eq. (6.21) has been obtained by applying Newton's second law to a small element of a stretched string. Since there is nothing special about this particular element on the string, this equation applies to the whole of it. I Let us pause for a minute and ask : What goals we set for ourselves and how Eq. (6,21) helps us in attaining them? We wished to know what determines the speed of a wave, To know this let us assume that a harmonic wave described by 1 I 1 y(x, 1) = a sin (o,f- kx) 4 ' moves on the string. If this mathematical form is consistent with Newton's law, then you . be sure that such waves caa move on the string. To sea this, you should calculetc the ~~nd 18 partial derivatives of the particle dbplacement : I I -
Wave Motion
and a5 - = whsin(co,t - kx). a? Substituting these derivatives in Eq. (6.21). we get, on simplification
What is implied by this equality? We know that it has followed from Newton's lnw of motion applied to a stretched string when a harmonic wave is travelling along it. So, the above relation tells us that only those waves can propagate on the string for which wave properties wo and k are related to Fand m through the relation uh- - fl k m
But w,/k is just the wave speed (Eq. 6.7),so that
I'his relation tells us that velocity of a (transverse) wave on a stretched string depends on tension as well as mass per unit length of the string, not on wavelength or time period. This means that v is not a property of the material of the string. It involves an external factor- tension-which can be adjusted for fine tuning. This explair~swhy musicians are seen adjusting tension in their stringed instruments. However, no such thing is done in case of a flute or a harmonium. Using Eq. (6.22), we can write Eq. (6.21) in the form
rhis result is identical to Eq. (5.41) and expresses one dimensional wave equation. It holds so ong as the oscillations of the string have small amplitude. You may now ask : Will Eq. (6.23) $old for large amplitude disturbances as well? The answer to this question is that large amplitude oscillations result in a more complicated equation and the wave speed tend to jepend on wavelength as well. Before you proceed further, you may like to solve the following 3AQ.
9AQ 7 A 1 rn long string having mass 1 g is stretched with a force of 10 N. Corlculnte the speed of transverse waves. , We now know that the speed of a wave is determined by the interplay of elasticity and inertia 3f the medium. Elasticity gives rise to the restoring force and inertia tells us how the medium responds to them. Since a fluid (gas or liquid) lacks rigidity, transverse waves can propagate 3nly in solids. However, longitudinal waves can propagate in all phases of matter-plasmas, Sases, liquids and solids-in the form of condensations and'rarefactions. We will now consider wave propagation in a fluid.
6.4.2 Waves in a Fluid Let us consider a fixed mass of a fluid of density p contained in a long tube of cross sectional area A and under pressure p,. As for a string, we shall consider a small element (column) of :he flaid; Let us assumcthat the column is at rest and is contained, is the region PQ RSP :xtending between planes at x and x + Aq (Fig. 6.14a). Then the mass of the column PQ RSP is &A. How can you generate longitudipal waves in the fluid? You can do so by placing a vibrating tuning fork at its one end or displacing the fluid to the right using a piston. 4s the wave passes through the column under mnsidention, its pressure, density and volume \ Waves change. Let us assume that in time t planes PC? and SR move to P'Q' and S'R', respectively, as Shown in Fig. 6.14b. If the planes PQ and SR are displaced through $(x) and $(I + ax) the chance in thickness, Al, is
In writing this expression we have used Taylor series expansion for $(x + Ax) about +(x). This means that the change in volume AVis
P P' SS I ...... st.. . I :;:;f;,.):;;;, ...... ;.l.::I.,:(;;; :.. I ,...... ::::..: :,:;;;; I :,;.;:;:;: I~I(~)-2.>:~.:;;;~j$~;~~ -p(x + Ax) I ...... ,:...... :.:,:.,,.p: ..:.:., ' .:;;..,.; I .:. :::..> .y:*.):;,..... ;: .... ,..:. ., , ...... ,. . ,I':::.. Q Q' R R'
Fig. 6.14 : (a) Equilibrium slate of the column PQRS of a fluid contained in a long tube of cross xctional area A. (b) Displaccd position of column under pressure dillerence.
The volume srrain, which is defined as the change in volume per unit volume, is given by
The mirius sign signifies that the column is compressed. This happens because the pressure on its two sides docs not balance. Lel the difference of pressures be p(x + Ax) - p(x). The net force adng on the column is, therefore Ab(x + Ax) - P(x)l, Using Taylor series and retaining only Erst order term in Ax, we can write
where p,, is equilibrium pressure and Ap is the excess pressure due to the wave. Hence, the equation of motion for the column under consideration, according to Newton's secand law, is
To express this re'sult in a familiar form, we note that' Ap and E, the bulk modulu.~ofelaslicily, , are connected by the relation
Stress - Ap 1 E= - -- Volume Strain A V/V I ?he negative sign is included to account for the fact that when pressure increases, volume ' t decreases. This ensures that E is positive.
I a We can rewrite it as 1 on substituting for AV/V from Eq. (6.24)) we get Wave Moth
Using this result in Eq. (6.25) we find that
If we identify v=m as the speed of longitudinal waves, Eq. (6.26) becomes identical to Eq. (5.41).
You will note that the wave speed is determined only by F, and p; two properties of the medium through which the wave is propagating. Let us now consider propagation of sound waves in a gas. a. Sound Waves in a Gas A process is said to be isothermal if For a gaseous medium, such as air, the volume elasticity depends upon the thermodynamic temperature remains constant during, changes arising in the medium, when a longitudinal wave is propagation. These changes can the process. In an adiabatic proms, be kolhennal or adiabalic. For sound waves, Newton assumed that changes in the medium are the total energy of the system isothermal. For an isothermal change, you can check using Boyle's law that the volume remains constant. elasticity equals equilibrium pressure. Then, we can write
For an isothermal prm,Boyle's This is known as Newtun's formula for the velocity of sound. law states that pV = wnstRnt Any change in p and/or Vis relatcd For air at STP, p = 1 :29 kgrn-' and ,p,, = 1.O1 X 10' ~rn-~.Then the velocity of sound in B~VPP-~~~Y=Q air using Newton's formula comes out to be
But experiments show that velocity of sound in air at STP is 332 ms-'. This gives rise to an interesting question : How could Newton come so close to the correct answer and yet miss it by about 15W? It means that something is wrong with his derivation. You may now ask : How to explain the discrepancy? The problem lies with the use of Boyle's law, which holds only at constant temperature. The discrepancy was solved when Laplace pointed out that when sound waves move in a medium, the particles oscillated very rapidly. In the process, regions of compression are heated up while the regions of rarefaction get cooled. That is, local changes in temperature do occur when sound propagates in air. This effect produces a larger phase velocity. However, the total energy of the system is conserved. This means that adiabatic zhanges occur in air when sound propagates. For an adiabatic change, E is y times the pressure, where y is the ratio of the specific heats at For on adiabatic chunse, the mnstant pressure and at constant volume, i.e. E = ypo Then Eq. (6.27) becomes equation of stale upb= constant I The change in p and V are connected through the retation P Ap + py V.'Av = 0 & :or air, y = 1.4. So the velocity of sound in air at STP works out to ms-',which an . Or - -= E = vp. be 331 is A V/Y :xcellent agreement with the measured value. This shows that Laplace's agreement is correct. ita given temperature, PO/ p is constant for a gas. So Eq. (6.28) shows that velocity of a mgitudinal wave is independent of pressure. lou will now like to know why the heat does not have time to flow from a compression to a arefaction and equalize the temperature everywhere. To attain this condition, heat hiis to flow 2 1 Waves . a distance of one-half wavelengh ~n a time much shorter than one-half of the period of oscillation.'This means that we would need vheat> >vsouod (6.29) Since heat- flow is mostly due to conduction, the speed of air molecules is given by
where M is mass of air molecules and T is absolute temperature. We can similarly write 7
Thus, even if air molecules travel a distance of A/2, they will not be able to transfer heal in time, In practice they move randomly in zig-zag paths of the order of 10-'cm and as Iong as A > 10-' em, the adiabatic flow is a very good approximalion. The shortest wavelength for audible sound (l.6km) corresponds to 20 kHz. The ability to measure the s@ of sound has been .put to many uses in defence. During World War I, a technique called sound ranging was developed to locate the position of enemy guns by using the sound of cannon in action.
b. Sound Waves in a Liquid Liquids are, in general, almost incompressible. For water, E = 2,22 X 10' Nrn ' and p = 10' kg m". This gives a wave velocity of about 1500 ms-'. Compare it with the speed of sound jn air at STP. Though air is about lo-' times less dense than water, sound propagates faster in water than air. This means that we can send messages from one ship to another faster via water than air. This has led to the development of a Sonar. High frcquency sound waves are used in a Sonar which can measure the depth of sea bed, detect submarines and enemy torpedoes.
6.4.3 Waves in a Uniform Rod For a solid elastic rod, changes take place,only in length; thc vnlu~iieremaining almost constant. The bulk modulus is replaced by Young's modulus defined as . Y= Stress -=-AP Longitudinal Strain AM Then Eq. (6.26) modifies to
which is the equation bf wave motion on the rod. The velocily of longitudinal witves'isgivell by 'Iv
This shows thatv is independent of the cross-sectional area of the rod.
SAQ 8 For a stcel rod, l' = 2 X 10" ~rn-'and p -. 7800 kg rn '. Compute the speed of sound.
On working out this SAQ you will find that v- 5 X 10' ms-',which shows that longitudinal wavqtravtl faster in a solid than in a gas or a liquid. This mans that you can know about corning train by putting your ear on the rails. However, you are advised never to do sol
6.5 WAVE MOTION AND IMPEDANCE
When a wave travels through a medium, the medium oppost's its motion. This resistance Lo wave motion is called the wave impedance. You should no1 confuse it with the electric impedance in the case of AC circuits where it arises due to resistance offered to the flow of current. The impedance offered to the transverse waves. travelling on strings is called the CharacferhficImpedance:Usually, the impedance offered to the lo~lgitudinal(sound) waves in air is called the 'acoustic impedance'. You may now ask : Why impedance arises and what .22 factors determine it? To discover answer to this question, ne recall that wiien a wave, . , propagates in a medium, each particle vibrates about its mean position. Moreover, each Wave Motion particle in motion,attempts to make the succeeding panicle vibrate by transferring energy. Likewise, each particle at rest tends to slow down the neighbouring particle. That is, a vibratiig particle experiences a dragging force, which is similar to the viscous force. According to Newton's third law of motion, it will be equal to the driving force F. From Unit 3 of Block 1 of this course you would recall that when oscillations of the particles are small, we can model the viscous force on the basis of stokes' law and write F =Zv The constant of proportionality Z is called wave impedance. From this equation it is clear that impedance is numerically equal to the driving force which imparts unit velocity to a particle. We will now consider some specific examples.
6.5.1 Impedance Offered by Strings : Transverse Waves Let us consider a wave ravelling on a stretched string. Let us choose x-axis along the Iength of the string (see Fig. 6.15). The transverse waves are generated by applying a harmonic force F = F, cos w,,l at the end x = 0 of the string. The displacement of the particles of the string at position x and at time t is given by Eq. (6.6d).
Fig. 6.15 :A string vibrating under the harmonic Force F = Fo cos wd Let T be the tension in the string. The vertical component of tension (galong the negative y- direction is equal to the applied transverse force (to give zero resultant force at the x = 0 end of the string) : F, cos w,,t = - T sin 0 For small values of 0(< So), sin O,-- tan B so that we can write F,) cos w,,t = - T tah 0 (6.33) The tangent (or slope) is-definedat x = 0 end of the string. Using Eq. (6.6d) we can relate (dy/dx)to (dy/dt): -d'---- - k dv dx w,, dt Inserting this result in Eq. (6.33), we get kT dy F,, cos w,,t = - ull ( ),=" dy Since idr) = a w.. cos wort this, equality becomes r=l) Tk . Fu cos w,,t =- a w,, cos war , w o
where v = odk.
Writing acu,, =to as veloci@ amplitude of the wave, the above equation reduces to Tio EL,COS Wilt = -cos out L' Waves
This result specifies the ratio of the amplitude of the applied force to the amplitude of particle velocity for transverse waves in terms of the tension in the string and particle velocity. This result can be used to get an expression for the characteristic impedance (2)of the string which is defined as : Amplitude of the transverse applied force (F,) Z= Transverse velocity amplitude of the wave (Vo)
Using Eq. (6.34), we find that
This result shows that the characteristic impedance has units of ~m"sbut its dimension are MT- I. - I From Eq. (6.26a) we recall that v = where m is mass per unit length of the string. 11 Then Eq, (6.36) can be written as : , z=-=fiT Y Alternatively, if we eliminate T we can write 2z-zv2m m v v
From kq. (6.36a) we find that the characteristic impedance is goveined by mass per unit \ length of the string and tension in it. This means that a sonometer wire will offer different 1 impedance when it is loaded by different weights. Eq. (6.36b) tells uS that since Z is related to \I the velocity of the wave, it depends on inertia as well as elasticity of the medium.
SAP9 ~mhkthe characteristic impedance offered by a thin wire of steel stretched by a force of 80 N. It weights 2 g per metre.
6.5.2 Impedance Offered by Gases : Sound Waves For sound waves propagating in a gas, the role played by exc& pressure due to the wave is analogous to that of applied force in case of a transverse wave. So'we define the acoustic impedance as : Excess pressure due to a sound wave =-& z= particle velocity
It means that dimensionally Z is the ratio of force per unit area lo velocity. The excess pressure Ap experienced by the medium when a longitudinal wave propagates through it is given by
where E is the Bulk modulus of elasticity of tho medium. This means th- 2, we must a+ mmpute pndax To do so, we recall that the particle displacement for a longitudinal a wave travelling in the fve xdirection is written a 2n ~(rI) = a sin [y(vl - xl]
Differentiating it with respect to x and 1, we get and
On combining Eqs. (6.38) and (6.39a), we find that
On ~ubstitutingfor Ap and a$/al from Eqs. (6.40) and (6.39b) in Eq. (6.37) we find that the amustic impedance is given by
where v is the wave velocity. This result shows that the units of acoustic impedance are ~rn-'s and the dimensions are ML-~T-" (YOUshould verify these before proceeding further;)
From Eq. (6.26a) we recall that the wave velocity is given by 7
where p is the density of the medium. Hence the acoustic impedance Z can also be expressed
This result shows that the acoustic impedance Z is given by the product of the density of the medium and the wave velocity. This means that denser the medium, greater will be the impedance offered. Yet we know that sound moves faster in solids than gases. In the next unit, you will apply these results to compute reflection and transmission amplitud~ and energy coefficients for a wave incident on a boundary separating two media. .- SAQ 10
Calculate the ricoustic impedance af air at standard tenlpcrature and pressure. Usc p =? 1.26 kg rn-I and v = 332 ms-'. Will this value he more for air or water? ~ustify'youranswer.
6.6 WAVES IN TWO AND THREE DIMENSIONS
So far we have confined ourselves to waves propagating along 1-D,as in a stretched string. The waves are constrained to move along the itring whereas particles vibrate in perpendicular direction. But all musical instruments are not stringed. Have you enjoyed tabla performance of Ustad Zakir Hussain or Allah Rakha Khan? You may have played on Sistrum. What happens when a tabla or drum membrane is suddenly disturbed in a direction normal to the plarie of the membrane? Particles of the membrane vibrate along the direction of applied force. But tension in the membrane makes the disturbance to spread over the surface. That is, waves on stretched membranes are two-dimensional (2-0). Similarly, surface waves or ripples on water, caused by dropping a pebble into a quite pond, are 2-0.In such cases, the displacement is a function of x, y and I, i.e. Jr = $(x y, I). YOU may now ask : What is the equation of a 2-0 wave? Will the preceding analysis as such apply in this case? - We will not go into mathematical details to answer these questions. However, from physical consideration extension of Eq. (6.23) for 2-0 wave is a straight forward exercise. Since forces along x and y axes act independently, each one will contribute analogous term to the wave equation so that Eq. (6.23) modifies to
hesolution of this equation is Waves wherek.r = (kr+ kj).(zfJ$) = k.~+ kd and k: + k;,
Let us pause for a minute and ask ; Do sound and light waves emanate radially from a small twodimensional source? How can we describe seismic waves or a wave propagating in an elastic solid? These are three-dimensional waves. To analyse 3-0waves we have to extend the preceding arguments. The result is a 3-0 wave euuation.
6.7 SUMMARY
* Medhanical (elastic) waves can be transverse as well as longitudinal. In a transverse wave, particles of the medium vibrate normal to the direction in which a wave moves. In a longitudinal wave vibrations of the 'particles of the medium are along the direction of wave propagation. The wave velocity, frequency and wavelength are connecteQ by the relation : v = vA. We wo can also express v as ratio of the angular frequency and wave number : v = -k * A harmonic wave in 1-D is described by the equation
= a sin (w,,t - kx)
where T is the time period. The phase of a wave, + = w,t - kx varies both with time and space., Waves carry energy. The total energy carried by a wave is half kinetic and half potential :
1 * The average rate of energy flow or average power is proportional to the wave speed and to the square of the wave amplitude: L 1 < P > = 25ra'~:~VA * For waves propagating in space, intensity is a more useful measure of energy carried by waves. The intensity of a plane wave remains constant as the wave'propgates. But for spherical waves, the intensity decreases as inversequare of the distance mom the source. * A wave propagating along a string (1-Dwave) is described by the equation
a2w, 0 - 2 dl+(x, l) -Y - at a2 where Jl(x t) is the displacerpent and v is the wave speed.
4. * The speed of a wave on a stretched string & given by v = ,/zi .. where F is tension in the'stiing and m is mass per unit length. For a longitudinal wave ! v=m ! where E t elasticity and p is density of the medium. For sound waves in air, , v=m where y is the ratio of specific hats at constant pressure to that. at constant volume. For sound waves in tali& u=m I * When a wavt travels through a medium, the medium opposes its'motion. This resistance to , wave motion is referred to as the wave impedance. In case of transverse waves, the
cbaraacristic impedance is given 'by I, I
1 I I
For sound waves in air, the acoustic impedance is givsa by
* A wave propagating in 2-D is dekn'bed by the equation
The solution of this equation in given by $(r, r) = asin (ad- k.r) where k.r = kxx kxvmnd k2 = ki + ki - NAL QUESTIONS - i 1. A transverse harmonic wave of amplitude 1 cm is generated'dt one end (x = 0)of a tong string by a tuning fork of frequency 500 Hz. At one instant of time, the displacements of the particles at x = 1O'crn is - 0.5 cm and at x = 20 cm is 0.5 cm. Calculate the velocity and the wavelength of the wave. If the wave is travelling along the positive x. direction and the end x = 0 is at the equilibrium position at r = 0, write the displ in te& of wave velocity. 2. In normal conversation, the intensity (energy flux/area) of sound is 5 X ~rn-~.The frequency of normal human voice is about 1000 Hz. Compute the amplitude of sound waves, given that the density of air at STP is 1.29 kgm-'. Take the velocity of sound at STP &3 332 ms-'. 3. The wavelength of a note of sound of frequency 500 Hz is found to be 0.70 m at room temperature (lS°C). Given that the density of air at STP is 1.29 kgm-3,calculate y. 4. A longitudinal distutbance generated by an earth quake travels 10' km in 2.5 minutes. If the average density .of the rock is taken to be 2.7 X 10' kg m-', calculate the bulk ' modulus of elasticity af the rock.
1 6.9 SOLUTIONS
.f1 i! SAQS $.1. i) 20 kHz ii) composite iii) energy, matter iv) no v) transverse ,/ 2. From Eq. (6.4), we have %\ v=- C 5: A For A = 4000 = 4 X 10-~rn,the frequency of visible light in ,< ! 3 X 10"s-' '1 v = :I 4 X lod7m
;I S-I. . :.; = 7.5 x 10" .:ti] ' j j Similarly for A = 7200 A, we have :, $ ... j,.. .. > 3 X 10' Ips-' 1: . . :: Y= ,. 8 . ; >. 7.2 X iO" m 'I! : $!! .;$. X s-'. : y! = 4.1 10" . .@Y 1 3.. The wave equation is y(x f) = (2.5 X 10-'m) sin [WOs (f - m)X ] The wavelength is given by The velocity of the meditim particles is
and acceleration is a5 X o =- = - (2.5 X lo-' m) (6007~s-I)' sin [ 6OChr (t - x)] a t2 4. A wave moving along -xdirection is described by the equation y(x, 1) = a sirr (ad - kx) The angular frequency is connected to v and A by the relation 2n v Wo = 7 Here v = 93 cm s-' and A = 55 cm. Hence
2a X 93cm s ' lo" = 55 cm 372 -, --- S 55 Simjlarly the wave number is related to wavelength.as
Hence,
Y(X t) = (7.4'cm) sin 35 = (7.4 cm) sir1 (1 0.6t t 0, f x) The particle velocity is given,by
The intensity falls off as the inverse square of the distance. if r denotes the distance at which this sound could just be heard, then
8X l~-~~m-~--- ? lo-'' ~rn-' (lrn12 r = 9 X 103m = 9 km
But this is not observed in practice. it is because of absorption of energy by the medium.
On comparing the powers of T, we get -1=-Za. or 0 = 1/2 On comparing the powers of L, we haw. a-b=i or b=a-.I . "-1/2 "=kgHence 7. We know that wave velocity on a stretched string is given by . Wave Motion v= @k On substituting the given values, we get
kg m-' = 100 m s-I From Eq. (6.32b) v=Jy/p On inserting the given values, you will get,.
9. m = 2.0 g m-I = 2.0 X low3kg m-' and I T=80N 1 From Eq. (6.36b) we recall that impedance offered by a string,is given by I I I 1 z=JTm On substituting the given data, you will get z = J (80 N) x (2.0 x lo-' kg m-') = 0.4 ~m-'s 10. From Eq. (6.41), we have z= pv Here p = 1.29 kg rn-j and v = 332 ms-' :. :. Z = (1.29 kg m-3) X (332 ms-I) = (4.28 X 10') kg m-' s-' = 4.28 X 10' Nm-'s 1 The impedance offered by water will be more €ban that offered by air. This is because the density of water is much greater than that for'.air at STP. Also, the velocity of sound in water is almost five times the velocity of sound in air, 1 11. In 3-D, the wave equation has the form
TQs 1. We know that a harnionic wave in 1-D is described by 27T ~(x,t) = o sin ( v, -- X)I where a = 1 cm (a) At x = 10 cm, 444 0 = - 0.5 cm
2n 'rr sin (T(vt- 10)) = -112 = sin (n + -g-l
This equality implies that 2n 7r (vt- 10) = i h -6 . or 7 (vt - lo=-h 12 0) (b)Atx= 20cm.~l(z1) = + 0,sm Waves :. 0.5 cm = (I cm) sin
so that
From (i) and (ii), we get
or A = 20 cm' = 0.2m We know that 14 = VA :. :. v = (500 Hz) X (0.2 m) = 100 ms-I Hence,
+(x, t)= 10.01 cm) sin = (0.01 m) sin [ 10a (100 t - x) ]
2. The expression for the intensity is I = 27T2p v;,#:t so that 7
Here v = 332 ms'l. p = 1.29 kg m ', v = 1000 HZ, I = 5 X ~m".Hence,
3. Here we use the expressions 7 I
.and the gas equation
The gas equation can be rewritten in terms of p and p, as pop= T, PPo T. since, p = m/Kand po = rn/K Since v, = YA, we find that v, = 500 Hz X 0.70 rn = 350 ms- ' Hence,
sb ihat
Hence, y, the ratio of specific heats it constant pressure to that at constant volume, .k given by The speed of the seismic waves is Wave Motbn
Since - we can write E = vZp. On substituting the given data, we get