AAKASH SG HIGHWAY CH 14 ‘Waves Section A © Write the answer of the following questions. [Each carries 2 Marks] io 1. Describe classification of waves in detail with examples. 2. Write definition of wave speed and derive v = 2 Section B @ Write the answer of the following questions. [Each carries 3 Marks] p71 3. Explain the types of waves on the basis of direction of oscillations of particles of medium. 4. (a) Give reason why transverse waves can be propagated in solids, (b) Give reason why longitudinal waves can be propagated through all the three types of media ~ solids, liquids and gases. (©) Describe in brief about two types of waves that can be propagated on the free surface of water. 5. Give definition, equation and unit for wavelength and wave number for wave. 6. Give definition, SI unit and dimensional formula of time period, angular frequency and frequency of wave. 7. Obtain the equation of speed of transverse wave on tensed (stretched) string 8. Obtain the equation of speed of sound wave in air and give the error in this equations. 9. Write the equation of speed of sound wave of Newton and explain the correction by Laplace. 10, Explain the reflection of wave at rigid support. 11, Explain the reflection of wave at free support. Section C © Write the answer of the following questions. (Each carries 5 Marks) (20) 12. What are stationary waves ? Obtain its equation, 13, Write the equation of stationary wave and obtain the equations of nodes and anti-nodes by defining them. 14. Give explanation of stationary waves produced in closed pipe and obtain equations of natural frequency (normal modes) 15. Obtain the equation of frequency of stationary wave produced in open pipe and show that all harmonics are possible in it © scenesuthoKenscamer (© scanned wit onENSconerAAKASH SG HIGHWAY CH 14 Waves [Section A © Write the answer of the following questions. (Each carries 2 Marks] 4] 1. Deseribe classification of waves in detail with examples. All types of waves can be classified into following three categories. (1) Mechanical Waves : "The waves which require an elastic medium for their propagation, are called mechanical waves. “For example : String waves, spring waves, water waves, sound waves, seismic waves (propagating on the ground during earthquake) are all mechanical waves. “Here waves propagate because of elasticity property of medium. (2) Electromagnetic Waves OR Non-mechanical Waves : "The waves which do not require any medium for their propagation, are called non-mechanical waves. They are also called electromagnetic waves (because during their propagation, electric and magnetic fields oscillate periodically in mutually perpendicular directions in a plane perpendicular to direction of propagation of wave)". For example, radiowaves, infrared light waves, visible light waves, ultraviolet light waves, X-ray and gamma rays ate the examples of electromagnetic or non-mechanical waves. They can propagate in vacuum also. The light emitted from stars very far from us (hundreds of light years away from us) reaches to us after passing through vacuum in the inter - stellar space. s+ All types of electromagnetic waves have same speed in vacuum with a value c= 2.99 792458 x 10® m/s (Which is a well known physical constant known as velocity of light in vacuum). (3) Matter Waves : It has been proved experimentally that, "When a fundamental particle of mass ‘m, moves with a velocity , its motion can be described very well by a wave of wavelength 2 h * mw (Where h = Planck's constant = 6,625 x 10-¥ Js).” Such waves are called “Matter Waves”, (These waves were predicted by Louis De Broglie and so they are also known as De-Broglie waves). “+ Matter waves represent the motion of atom (smallest part of an element), molecule (smallest part of a compound), fundamental particles like electron, proton and neutron (which take part in the constitution of any matter). They interact and show their effect, quite similar to those shown by mechanical or non-mechanical waves. They have been employed in many devices, from basic to modern technology, for example stream of fast moving electrons is employe electron microscope (which gives an image with magnification and clarity, greater than that produced by any light). 2. Write definition of wave speed and derive v : "Wave speed : “The distance covered by wave in unit time is called wave speed’. Us unit is "and dimensional formula is IM? LIT“, "= 1s the distance covered by wave in time T. distance covered(A) s+ Wave speed » = SSSne= oor ved oT © scanned wih one Scam © scanned wh oxen scanerbut i v frequency 2mvxt 2e . baredel 1 wie an” k, [Section B Write the answer of the following questions. [Each carries 3 Marks} e7 Explain the types of waves on the basis of direction of oscillations of particles of medium. Depending upon the direction of oscillations of particles of medium, mechanical waves are classified into following two categories : (1) Transverse waves (2) Longitudinal waves. (1) Transverse Waves : “During the propagation of given mechanical waves, if particles of medium oscillate perpendicular to the direction of propagation of wave then they are called transverse waves.” Figure a} = Consider a horizontal string tied at one end with the fixed rigid support which is stretched and kept under proper tension as shown in fig (a). Now give a little single jerk at the free end of this string on upper side of X-axis such that first particle of string goes up slightly from origin and then comes back to the origin. (‘e. first particle undergoes half oscillation.) By doing so, we notice that a little crest is formed (called pulse) which starts moving away from first particle, causing up and down motion of neighbouring particles. Here we notice that as the single crest produced initially at the free end moves away, its size (i.e. height at the centre of crest) goes on. decreasing. This is called damping of a pulse. If length of string is very large as compared to ini damped out before it reaches opposite end and s ial size of pulse then it will get completely reflection from that end may be ignored. “+ But if the string is short and pulses are produced continuously at the free end then they get reflected from the opposite end. "= Here if initial pulse is in the from of a trough then it travels in the form of a trough. — Figure (b) "Now, consider fig. (b) in which one full oscillation is given to the first particle at the free end. At the end of this oscillation, we find that a sinusoidal wave shape is produced which contains one trough and one crest. Now as this wave shape moves ahead, it gets damped. Here also if string is very long as compared to height of crest or depth of trough, then its reflection can be ignored from the opposite end. (© scare on Scam © scanned with oxen scaner4 But if the string is short and oscillations are produced continuously at the free end then wave shapes are also produced continuously one after another which propagate away from free end. Such oscillations give rise to a “Wavetrain.” In both the cases of fig. (a) and fig. (b) given above, particles of string oscillate perpendicular to the direction of propagation of wave. Hence according to definition, these string waves are transverse waves. (2) Longitudinal Waves : “During the propagation of given mechanical waves, if particles of medium. oscillate parallel (or antiparallel) to the direction of propagation of wave then they are called longitudinal waves.” Figure (c) © Asshownin the figure, consider along pipe filled with airhaving piston atone end. Now whenwe push the piston forward and then pull itbackward suddenly, condensations and rarefactions are generated inthe aircolumn. Ifwe push & pull the piston continuously and sinusoidally the disturbance produced in the air column atthe top end moves away from it, parallel tothe length of the pipe, which producessound. Hence the ‘waves propagating in air column are called sound waves. = Here the air particles inside the pipe oscillate parallel (or antiparallel) to the direction of propagation of sound waves and hence according to definition, sound waves are longitudinal waves. (a) Give reason why transverse waves can be propagated in solids. (b) Give reason why longitudinal waves can be propagated through all the three types of media ~ solids, liquids and gases. (©) Describe in brief about two types of waves that can be propagated on the free surface of water. (a) In order to have propagation of transverse waves, we need to produce crests and troughs. For this we need to produce shearing strain whereby we can change the shape of a medium. For this to happen, medium selected for the propagation of transverse waves, must be capable enough to sustain shearing stress i.e. They must have very high value of modulus of rigidity. Now we know that solids have very high value of modulus of rigidity. Hence transverse waves can be propagatedin solids. (b) Longitudinal waves propagate in the form of condensations and rarefactions, causing changes in density and pressure. Accordingly, compressive and expansive stresses are produced. All the three media - solids, liquids and gases can sustain compressive and expansive stresses andso longitudinal waves can be propagated in all ofthese media. (©) Two types of waves which can propagate on free surface of water (i.e. on the uppermost layer of water which is in contact with atmosphere) are capillary waves and gravity waves. (Capillary waves : They are transverse waves of fairly short wavelengths (few centimeter only). Here necessary restoring force is produced due to surface tension property of water. i) Gravity waves : They are transverse as well as longitudinal in nature. They have large wave lengths ranging from several meters to several hundred meters. Here necessary restoring force is produced by © scanned wih one Scam © scanned wh oxen scanerthe pull of gravity, which compels water surface to remain at lowest possible level. During the propagation of these waveson the free surface of water all the particles of water medium, from surface tothe bottom oscillate with gradually decreasing amplitude Here particles of watermedium oscillate upand downas wellas back and forth. The wavesin an ocean are the combination (superposition) of transverse and longitudinal waves. ‘Transverse and longitudinal waves are found to be travelling with differentspeeds even in the same solid medium (because fora given solid medium, its shear modulus and bulk modulusare different.) Give definition, equation and unit for wavelength and wave number for wave, Wavelength : The linear distance between any two points or particles having phase difference of 2x rad is called the wavelength (1) of the wave. Its SI unit is metre (m), OR The distance between consecutive crests/trough is called a wavelength of a wave. yx, 0) = asinkx [vf 09 =0) From this equation, sin function repeats its value after interval of 2x phase difference solution of function, z Sinkx = sin(kx+2nn) = sof (s+ 22] 1 23, ; i 2ne 1 means displacement at point x and x+77™ are equal where, n = For minimum displacement between two points of same displacement, n = 1 have to be taken, 2: x+5E—x = wavelength 2m _ wavelength % 2x eo ork = > where kis called angular wave no. or wave vector. Unit of k is radian- meter but as rad is unitless, its ST unit is mm Its dimensional formula is [M° L“! T°), Give definition, SI unit and dimensional formula of time period, angular frequency and frequency of wave, Time period : “The time taken by any string element to move through one complete oscillation is called Time period (T)". In equation of displacement of wave, Yer 0) = asin(ke - ot + 9) if > 0, 9 = asin(- of (0, = —asinwt which is shown in below figure. The element moves up and down in SHM with amplitude @ and time period T. 6, then if we observe motion of element at x 2 ~asinot= ~asinlo(t + T)) = ~asin(ot + wT) (© scarab OnE Scam © scanned with oxen scanerbut as sine function is repeating at 2x interval, @T = 2x 2 @= = where « is called angular frequency of wave. ‘Angular frequency of oscillations of particles of medium in wave is called angular frequency of wave. Its symbol is @ and SI unit is rad s“! and dimensional formula is [M°L°T-1), Frequency : The number of oscillations performed by the particle of medium in one second is known as the frequency of oscillation of particle. Its symbol is ‘v' or ‘f. Sometimes it is also written as n. If 1 wave is obtained in T time, then in 1 sec. (f) waves are obtained, :. SI unit of frequency is s or ‘Hz’ and its dimensional formula is (M°L°T~! Obtain the equation of speed of transverse wave on tensed (stretched) string. ‘The speed of transverse waves on a string is determined by two factors, (i) the linear mass density or mass per unit length 1 and (ii) the tension T. The linear mass density, 4 of a string is the mass m of the string divided by its length 1. Therefore its dimension is (M! L-!]. The tension T has the dimension of force - namely (ML! T72). We have to combine pt and T in such a way as to generate v [dimension (LT-!)), It can be seen that the ratio ; has the dimension (L?T. [er] ty) _ [Mer?] Fi [Mi] mn ‘Therefore, if v depends only on T and pi, the relation between them must be, v=Cj— ie Here C is a dimensionless constant and constant C is indeed equal to unity. ‘The speed of transverse waves on a stretched string is therefore given by, Tr res " The speed of a wave along a stretched ideal string depends only on the tension and the linear mass density of the string and does not depend on the frequency of the wave. Other Method : ‘ Obtain the expression for the wave speed propagating in the string kept under tension. ‘The speed of transverse waves in a medium like a string kept under the tension, depends on (2) linear mass density (y) and (2) tension T in the string. Suppose, wave speed ve pe ve T? where a,b, R = ayer a where k= dimensionless constant Dimensions on both the sides (© Saronic © scanned wih oxen camer(ML) = (MIL 4|¢ (MILL = (MALT) (MPLPT-2h) (M°LIT-H) = (Mash Lae p2hy Comparing powers of both the sides, 1 and -2b a+b=0,-a+b 1 sand b= > Substituting the value of a and b in equations (1) ky 4-74, (c= 1 Experimental value) This eqn. shows that wave speed is independent of frequency of a wave and amplitude of a wave. 4 Propagation of a pulse on a rope : ‘You can easily see the motion of a pulse on a rope. You can also see its reflection from a rigid boundary and measure its velocizy of travel. You will need a rope of diameter 1 to 3.cm, two hooks and some weights. You can perform this experiment in your classroom or laboratory. ‘Take a long rope or thick string of diameter 1 to 3 em and tie it to hooks on opposite wall in a hall or laboratory. Let one end pass on a hook and hang some weight (about 1 to 5 kg) t ‘The walls may be about 3 to 5m apart. Take a stick or a rod and strike the rope hard at a point near one end. This creates a pulse on the rope which now travels on it. You can see it reaching the end and reflecting back from it. You can check the phase relation between the incident pulse and reflected pulse. You can easily watch two or three reflections before the pulse dies out. You can take a stopwatch and find the time for the pulse to travel the distance between the walls and thus, measure its velocity. Compare it with that obtained from ir ie ‘This is also what happens with a thin metallic string of a musical instrument. The major difference is that the velocity on a string is fairly high because of low mass per unit length as compared to that on a thick rope. The low velocity on a rope allows us to watch the motion and make measurements beautifully. Obtain the equation of speed of sound wave in air and give the error in this equations. Equation of sound in gas (air), E F5 oO) ‘Newton suggested that the propagation of sound is an isothermal process. In PV = NkgT, NkgT = constant, 2 © scanned wih one Scam © scanned wh oKENscamerFor an isothermal change in equation (2), VAP + PAV = 0 AP. % Therefore, from equation (1) the speed of a longitudinal wave in an ideal gas is given by, B ° It is known as Newton's formula. ya fP . fiorxa0® p 1.29 v= 280 ms Thus, value of speed of sound is 280 ms“! according to Newton's formula but its experimental value is 331 ms. Hence, value from Newton's formula is 15% less. So, assumption of Newton is not correct. Write the equation of speed of sound wave of Newton and explain the correction by Laplace. Speed of sound in ideal gas according to Newton, v= fF a 5 5 It was pointed out by Laplace that the pressure variations in the propagation of sound waves are so fast that there is little time for the heat flow to maintain constant temperature. These variations therefore are adiabatic and not isothermal. For adiabatic processes the ideal gas satisfies the relation, Pv" = constant + ARV) = 0 PIT ay + Vlap = 0 YAP = -VAP AP aw v 2 P=B ~ Q) where 7 is the ratio of two specific heats, £2 v (By using equation (2) in equation (1), v £ @ Pp 1 modification of Newton's formula is referred to as the Laplace correction. For air, (© Saronicz 4 iia ’ 5 Now using equation (3), estimate the speed of sound in air at STR we get a value 331.3 ms“, which agrees with the measured speed. 10. Explain the reflection of wave at rigid support. - “= Apulse (wave) moving in +x-direction and reflecting wave from fixed support are shown in figure. “* _Ifwesuppose that the energyis not absorbed at end, then the shape of the reflected pulsewill besameas incident but the phase will be changed by 180° (n). = The reason behind it is the end is fixed. So that the displacement of pulse should be zero. “Suppose, the incident progressive wave displacement at ‘’ time is yx, ) = asin(kx ~ w). "© Suppose, the displacement of reflected wave is y,- According to superposition principle, VE, 0 = yl, 0 + yl, But yox, 1) = 0 displacement of fixed end is zero) 2 O0= yf 0+ yl% 0 # Yel = Yl asin(kx - of) ysl, 0 = asintke ~ ot + x) Thus, when the pulse (wave) reaches to end the string applies force on fixed end. = According to third law of Newton the end will also apply the same force in opposite direction, Hence, there will be difference in phase of x in reflected wave. It will be progressive = x-direction, Thus, the plane of reflected wave will be increased by x. Hence, the shape of wave will be reversed, It means the crest will be trough and trough will be crest. 11, Explain the reflection of wave at free support. "> Asshown in figure, the string is fastened to a ring which slides without friction on a rod. In this case, when the pulse arrives at the left end, the ring moves up the rod. (© sarah OE Scam © scanned with oxen scaner‘ eae 2. Rod {As the ring moves, it pulls on the string stretching the string and producing a reflected pulse with the same sign and amplitude as the incident pulse. Thus, in such a reflection, the incident and reflected pulses reinforce each other, creating the maximum displacement at the end of the string; the maximum displacement of the ring is twice the amplitude of either of the pulses. Thus, the reflection is without any additional phase shift. Here, the reflection takes place without any phase change. Suppose the progressive wave is, y,(x, ) = asin(kx ~ et) The reflected wave from open end is, y,(x, ) = asin(kx + of) If incident wave is moving in + X-direction, then reflected wave would be moving in ~ X-direction. Note : If progressive wave is moving in + Y-direction then the reflected wave of 24.) = asiniky - of is x, 0 = -asin(ky - on) [Section Write the answer of the following questions. (Each carries 5 Marks] (20) What are stationary waves ? Obtain its equation, Stationary waves : “When two waves of same amplitude, same time period, same frequency, same wavelength are travelling in opposite directions on the same line superposed, then resultant wave formed is called stationary waves. oR When two waves having the same amplitude and frequency (ie. wavelength) and travelling in mutually opposite directions are superposed, the resultant wave formed loses the property of propagation and a stationary pattern is created in the medium. Such a wave is called a stationary wave. These resultant waves cannot move in any direction. Hence, they cannot conduct energy. Suppose, a resultant wave, moving in negative x-direction obtained by reflection of wave moving in positive x-direction of same wavelength and amplitude. Suppose, initial phase @ = 0 which is shown in figure, © scanned wih one Scam © scanned wih oxen scani N N N N N _N ‘Two sinusoidal waves of same amplitude travel along the string in opposite directions. The displacement at positions ‘marked as Nis zero at all times. sin(kx ~ cot) (Wave travelling in the positive direction of x-axis) asin(kx + of) (Wave travelling in the negative direction of x-axis) 2 Alt alt f The principle of superpo: WOE, 1 = le, D+ Yol, 0 = asintkx - of) + asin(kx + wf) ion gives, for the combined wave, fase at +00) coo =a] 2asin kx cos(- ot) «yO, 0 = 2asinkx cost... (1) [+> €0s(-0) = cos6} [-: sinA + sinB = sin(A + B)cos(A - B)) Equation (1) is general equation of stationary wave. Amplitude of stationary wave is 2a sinkx and different points are oscillating with different amplitude. kx and of are different terms but not combined. Hence, stationary wave is not function of fikx—oot) Hence, they are not progressive waves. Wave pattern is not moving on left or right hand so, that it is called stationary wave. Write the equation of stationary wave and obtain the equations of nodes and anti-nodes by defining them. Equation of stationary wave is, YO 1) = 2asinkxcoseat a coset represent that each point of string performs SHM and its amplitude is 2a sinkx. It means, for different values of x on string, amplitudes are different. The points where the amplitude is zero are called nodes. The points where the amplitude is maximum are called antinodes. Nodes : In equation (1), 2a sinkx = 0 :. sinkx = 0 oe = nm where m= 0, 1, 2, 3; oa (© sare OE Same © scanned wih oxen camerHence, first node is obtained at x = 0 and second at x = *, third at x=... Hence, distance a between two consecutive node is >. = Antinades : In equation (1) sinkx| ke = (n+4}x (rs nde fe 2 = (nal) _ xe (n+3}8 where m= 0, 1, 2, 3; Aah 5A This represent that from end of string at distances antinodes are obtained respectively. w(2n+% frst, second, third .. i ive anti sé a = Distance between two consecutive antinodes is > and distance between two consecutive node and antinode is * 4 44, Give explanation of stationary waves produced in closed pipe and obtain equations of natural frequency (normal modes). Aircolumn (glass tube) partially filled with water is example of closed pipe. The end in contanct with water is node and open end is antinode, = At modes, pressure changes are maximum but displacement is minimun (zero). = Atantinodes, pressure changes are minimum but displacement is maximum. > If'we take x = 0 at node and x = L at antinode, for ends, then |sinkx| = 1 +. [sinkL| = 1 u 1 ok < (o$} 2 = (ns) “7h 2 at (nage 2J2 eb = Qnsn% where m= 0,1, 2,00 and possible wavelengths, (© Saronicely Ba : "Normal modes (Natural frequencies of system), a where 1 = 2,3, «. lv » vy = (ne3)e = ensne where m= 0, 1, 2, 3, "= By taking 1m = 0 for fundemental frequency, vt ac and odd number harmonic gives higher frequencies. vs For example, 3, . n+ ae are obtained. Thus, only odd harmonic are obtained in closed pipe. (a) o) CO First Third Fifth harmonic harmonic harmonic (@) @. o Seventh Ninth Eleventh harmonic harmonic harmonic "= The above six figures shows the first 6 harmonics of air column in closed pipe. open pipe and show that all 15. Obtain the equation of frequency of stationary wave produced harmonics are possible in it. 3h, ME where, m= 1, 2,3, a ae 2 In open pipe, antinodes are produced at both ends and consecutive antinodes are obtained at © scanned iho Scam © scanned wh oxen scanerIf wavelength is A for open pipe of length L, then stationary waves will only be produced if L aL du n Frequency of stationary waves in open pipe, 2. [rorwaa=2] ¥ vn Va z .- (1) where v is wave speed. First 4 harmonics are shown for stationary waves in open pipe in figure, Fundamental or ‘Second harmonic First harmonic If we place n = 1 in equation (1), then fundamental or first harmonic is obtained. + aim 2 7 Second harmonic is obtained by putting n = 2 in equation (1), we » ave Bat on{Z)-m Accordingly by putting n = 3, 4, ... nin equation (1), third, fourth, and corresponding (n ~ 1) overtones are obtained. (© Saronic