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Rone if \ 6. Superposition of Waves 12th Science 12th - Physics Syllabus 1. Introduction 2. Progressive wave 2.1 Properties of progressive waves 3. Reflection of waves 3.1 Reflection of a transverse wave 3.2 Reflection of a longitudinal wave 4. Superposition of waves 5. Stationary waves 5.1 Formation of stationary waves 5.2 Equation of stationary wave on a stretched string 5.3 Properties of stationary waves 5.4 Comparison of progressive wave and stationary waves 6. Free and forced vibrations 7. Harmonics-and overtones 74 End correction 7.2 Vibrations of air column in a pipe close at one end 73 Vibrations of air column in a pipe open at both ends 7.4 Practical determination of end connection 75 Vibration produced in a string 7.6 Laws of a vibrating string 8. Sonometer 9. Beats 9.1 Analytical method to determine beat frequency 9.2 Application of beats 10. Characteristics of sound 10.1 Loudness 10.2 Pitch 10.3 Quality of timbre 11. Musical instruments 11.1 String instruments 11.2 Wind instruments Theory Notes Q. Define wave and give types of waves. Q. Give the types of mechan- ical waves. Q. What is wave motion Q. Define- i) Amplitude ti) Period iti) Frequency iv) Wavelength uv) Speed of Wave Wave i rough the elasti 1) The oscillatory disturbance of the particles throug Ic medium is called as wave. ; int in the form 2) Energy is transferred from one point to another Po of waves. 3) Wave motion is the mode of trans’ medium. There are different types of waves. Classification of waves . Depending upon the necessity of the medium, the waves are classi fied into two types. 1, Mechanical Waves . ‘ The waves which require i ium-for their propagation are | called as mechanical waves. Ex. Sound waves, water waves etc. 2. Electromagnetic Waves . 7 The waves which do_not require material for their propagation are called as electromagnetic waves. Ex. Light waves. fer of energy through the Types of mechanical Waves Mechanical waves are classified into two types - 1. Transverse Waves . : The waves in which particles of the medium. vibrate in a direction perpendicular to the direction of propagation of waves are called as transverse waves. Ex. Waves on rope, water waves etc. 2. Longitudinal Waves The waves in which particles of t! _parallel to the direction of propagati tudinal waves. Ex. Sound waves. he medium vibrate in a.direction on of waves are called as longi- Wave Motion It is the mode of transfer of energy in medium by transfer of oscil- latory disturbance through medium without changes in its form. 1. Amplitude (A) Maximum displacement of particle in a medium from its mean position when wave travels through medium is called as amplitude of wave. 2. Period (7) Time required by particle in medium to complete its one oscillation (vibration) is called period of wave. 3. Frequency (7) Number of oscillations (vibrations) performed by particle in unit time is called frequency of wave. Frequency of wave is equal to number of waves passing through given point in medium in unit time. n = 7 4. Wavelength (A) 1) Distance between two successive crests or troughs of a wave is called wavelength. Or The distance between two consecutive particles of the medium which are in the same phase is called wavelength. Or The distance between two consecutive particles of the medium which differ in phase by 27 radians is called wavelength. Q. Derive an expression for a one dimensional simple harmonic progressive wave traveling in the direction of Positive X-axis. Express it i two different forms, [Oct. 13, Mark 3] ive Waves. : , . , The side which travel through the medium continuously in a given direction are called as progressive waves. Simple Harmonic Progressive Waves , . . 1) The waves which travel through the a normmmaer Air particles of the medium performing simple harm called as Simple Harmonic Progressive waves. ; 2) Consider the wave is traveling along positive x-axis and particles of the medium vibrate along y-axis. wet oO i Q >< Fig. Simple harmonic Progressing wave 3) Here the wave Starts from its origin. Therefore at ¢= 0, the displacement of the particle from origin is zero. 4) Consider that the particle at origin is at mean position. 5) Hence the displacement of the particle at time / is given by - y= Asinwt Where 'A' is the amplitude and wis the angular velocity of S.H.M. 6) Every particle of the medium vibrates with same period and amplitude but at given time, its displacement is different so they are in different phases. 7) Consider a point at a distance x from O on x-axis and lags behind by phase 6 with tespect to origin. Then the displacement of this particle at time 1 is given by- Y= Asin(@t— 8). (1) 8) Two successive 2z (equal to A) 9) Thus a path difference of 'J'c of 2M radians. Hence a path d phase difference of 2ax particle are separated by the distance equal to orresponds to a phase difference ifference of 'x'will correspond toa Put above value in equation (1) y=A sin(or - 2a) But k= on = wave number Y= Asin(at— kx) This is the equation of simple harmonic Progressive wave travelling along positive x-direction. 10) The Simple Harmonic Pro: 1e Sin gressive wave travelling along negative x-direction has displacem ent at any instant- y= Asin(kx + wt) Q. Give the different forms of | Different forms of Simple Harmonic Progressive Wave Simple Harmonic Progressive | Equation. Wave Equation. Q. Explain the reflection of transverse waves from a denser medium and a rarer medium. The equation of displacement of Simple Harmonic Progressive wave travelling in positive x-direction is given by- y= Asinde(h— 4%). uD) = Asin{ 2% - 22) But 5 = wand 74 = k = Propagation constant oy = Asin(@t — Kx) .cc(2) We have Je pan | Therefore equation (1) becomes- = Asin 2re( nt — F) unnen(3) «y= Acin2x AL “y= Asin 2an(t =) Li eSlaeeed (4) But y= nd “.y = Asin 2an(t -).. wu l5) =v But n=] y= Asin(2)(1 - ? oy = Asin2Z (tx) esses (7) Equation no. 1,2,3,4,5,6,7 represent the different forms of Simple Harmonic Progressive wave equation. Reflection of Transverse Wave from Denser Medium 1) Consider a string fixed to a rigid wall. When its free end is moved up and down, a pulse consisting of one crest is propagated by the string. 2) The crest reaches the fixed end and exerts a force in the upward direction. 3) Due to the equal and opposite reaction, the velocity is reversed. incident wave rarer . denser medium medium —— reflected wave Fig. Reflection of transverse wave 1) Suppose that the crest of transverse wave is incident on the denser medium then, crest is reflected as trough and trough is reflected as crest. 2) At the time of reflection the particle velocity is also reversed, this reversal of particle velocity causes phase change by 180° (x radian). ee Q. State and explain the principle of superposition of waves. Q. Derive the equation for the amplitude of the resultant wave produced due to super- position of two waves. What will be the resultant ampli- tude if the two interferring waves have phase difference Oand x Principle of Superposition of Waves ; 1) Wen two or more waves, travelling through the same medium, arrive at a point simultaneously, each wave produces its own displacement at that point independently of the others. The resultant displacement at that point is equal to the vector sum of the displacements due to all the waves. This is called the principle of superposition of waves. Wave 1 Wave 2 Wave 3 Constvciv interference Destructive interference Fig. Superposition of Waves 2) The effect of superposition of waves is called interference. 3) If two waves arrive at a point in phase, then crest due to one wave coincides with the crest due to the other wave or trough due to one wave coincides with the trough due to other wave. The resultant amplitude at that point is maximum and is called constructive interference. 4) If two waves arrive at a point out of phase, then the crest due to one wave coincides with the trough due to other wave or trough coincides with crest. The resultant amplitude at that point is minimum and is called destructive interference. / 5) It is applicable to all types of waves. Amplitude of the resultant wave produced due to superposi- tion of two waves Consider two waves having amplitudes A; and A>. Their frequency is same. Let @ be the phase difference. The displacement of each wave x = 0 is yy =Asinwt ys = Asin(wt +g) According to the principle of superposition of waves, the resultant displacement at x = 0, is yEnty = Asinwt + A.sin(wt+ @) = A\sinwt + A)sinwt cosg + A:cos wf sin e = (A, + Arcos @)sin wt + Asin g cos wt Let Ai + A> cos g = Acos6.........(1) and A:sing =
[email protected]) “. ¥ = Acos@sinawt + AsinOcoswt “y= Asin(wt + 8) This is the equation of the resultant wave. Squaring and adding equation (1) and (2), we get 7 Q. Explain formation of Stationary wave on string. Q. What are stationary waves? Why are they called stationary waves? Q. Explain the formation of stationary waves by analyt- ical method. A’cos’@+ A’sin’@ =(A; + A:cosg) + Aising «A? (cos?6+ sin @) = A? + 2A,A.cos@ + Aicos’g + Aisin’ 9 A} + 2A,A2cos@ + A}(cos’g + sin’) vA’ = Ai + 2A\A.cosp + A? A= JA? + 2A\A;cosg + A This is the amplitude of the resultant wave produced due to super- Position of two waves. Special cases 1) If p= 0 i.e. waves are in phase. ; A= Ai + 2A\A,cos0 + A? = J (Ai + Ar) = Ai + Ad The resultant amplitude is maximum. If A= A,=A “A=2A 2) If @ = 7 i.e, waves are out of phase - A= VAi+t 2A,A cosa + Ai = V(Ar~ Ary =|Ai~ A2l The resultant amplitude is minimum If A= A.=A “A=0 Intensity of the wave is proportional to the square of the amplitude. When g= 0 Imax 0¢ (Anus = (As + Aa)? Trin © (Amin? = (Ay — Aa)? Formation of Stationary Wave on String 1) When two identical progressive waves (transverse or longitudinal) travelling along the same path in opposite directions, interfere with each other, by superposition of waves, the resultant wave obtained in the form of loops is called stationary wave. 2) Consider two simple harmonic progressive waves of equal ampli- tudes 'a' and frequency 'n' propagating along the string in opposite directions. 3) If wave is travelling along positive x-direction, then its displace- ment is given by- yi = asin 2n(ne—*) Similarly in negative x-direction, displacement is given by- ys = asin 2n(mt+ +) Where, 'n' is frequency and '/' is wavelength. 4) Two waves interfere to produce stationary wave. The resultant displacement of stationary wave is given by principle of superposition- yayty “y= asin2a(nt+%)+ asin 2z(nt~ 4) “y= 2a|sin( rar + %*) + sin(am1— We have sinC + sinD = asin( +? \cos( S52) ana cos(—6@) = cos y= 2a sin(2nnt)cos( 72) y= (2a.cos 2%) sin(2ant) y = Asin(2znt) Rut m= Irn + v= Acinmr Q. Write the properties of Stationary waves. Q. State the Characteristics of stationary waves. Q. What are pressure node and pressure antinode? Q. In sound waves, a displacement node is a pres- sure antinode and the vice versa. Explain why? *Q. Distinguish between progressive waves and stationary waves. 5A A 3 4°47 47°" The distance between two success!ve ween two successive nodes or internodes is Ew .... nodes are produced. For x = . 3A nodes is %2~ %1 4. rls ns Thus the distance bet same i.e. half of wavelength ( equally spaced. A)pnis shows that nodes and antinodes are 2 Se Se Properties of Stationary Waves ; 1) Stationary wave is the resultant wave in the form of loops produced due to superposition of two identical edie in the same medium along the same path in oP Ones rections, Stationary wave consists of nodes and ane ais "ace nodes displacement of particle is zero and at antinodes aisplacement of icle is maximum. 4 : Fart fittanoe bebween two successive antinodes or nodes is half of wavelength ie. (A/2).Thus antinodes and nodes are equally ced. the nodes and antinodes are alternate to each other. nN The distance between node and adjacent antinode is ( /4). Particles in stationary wave perform S.H.M. with period as that of component waves but of different amplitude. vibrati 7) All the particles within the loop are in the same phase of vibration, 8) The particles in adjacent loops are vibrating out of phase. _ 9) Stationary wave is periodic in space and time i.e. doubly periodic. 10) Resultant velocity of stationary wave is zero. There is no transfer of energy through medium. In case of longitudinal stationary waves (eg. sound waves), the points at which displacement of particle is minimum but change in pressure is maximum are called pressure antinodes and the points at which displacement of particle is maximum but change in pressure is minimum, are called pressure nodes. 2) 3) 4) 5) 6) 11) Distinction between Progressive waves and Stationary waves Progressive Wave Stationary Wave 1) Itis produced duetodistur-} 1) It is produced due to bance in the medium. interference between two identical progres- sive waves travelling in opposite directions. 2) It travels continuously| 2) It does not travel. until it is dissipated. 3) It transfers energy from] 3) There is no transfer of one point to another. energy. 4) Amplitude of all the] 4) Amplitude of all the particles is same. particles is not same. 5) Phase changes from] 5) Particles in the same loop particle to particle. have same phase but the particles in adjacent loop have different phase. Some particles of the medium do not vibrate. 6) All the particles of the medium vibrate with some amplitude. 6) Q. Explain free vibration with example. Q. Explain forced vibrations with example. Q. What are forced vibrations? (Mar. 15, Mark 1) Free Vibrations , a 1) When a body capable of vibrating is displaced from its equilibrium position and released, then this vibration is called free vibration. The frequency of such vibration is called natural frequency of the body. , - 2) The amplitude of free vibration decreases due to air or frictional resistance and finally comes to rest. 3) The natural frequency of vibration depends on the dimensions, mass, elastic properties and mode of vibration of the vibrating body. The natural frequency of the body remains constant. | 4) The energy of vibrating body goes on decreasing. Examples ; 7 1) When bob of a simple pendulum is displaced from its equilib- rium position and released, it oscillates freely with its natural frequency. Its amplitude decreases due to frictional resistance and finally it comes to rest. equilibrium position Fig. Free vibrations of simple pendulum 2) When a tuning fork is struck on a rubber pad and set into vibrations, its prongs vibrate freely with natural frequency. The amplitude of vibration decreases due to frictional resistance and finally it comes to rest. Forced Vibrations 1) The vibrations of body under the action of external periodic force in which body vibrate with frequency equal to frequency of external periodic force other than its natural frequency are called forced vibrations. 2) The amplitude of forced vibrations depends upon difference between the frequency of external periodic force and natural frequency of vibration of body. It also depends on the amplitude of applied force and damping force. If this difference is small, then amplitude of forced vibration is large and vice versa. 3) External periodic force maintains constant amplitude. The frequency of forced vibration changes. 4) The energy of the vibrating body remains constant. Example When a person swinging in a swing is pushed periodically by another person, the vibrations are forced vibrations. Q. Distinguish between free vibrations and resonance. Q. Distinguish between forced vibrations and resonance, (Oct. 14, Mark 2) (Mar. 13, Mark 2) Q. Explain the terms harmonics and overtones. Distinction between Free vibration and Resonance Free vibrations Resonance 1) When a body is displaced from its equilibrium position and released, free 1) In case of forced vibrations, if frequency of external periodic force is equal to vibrations are produced. the natural frequency then resonance is produced. 2) Energy of the body| 2) Energy of the body remains decreases due to external constant due to external resistance of air periodic force 3) Frequency of the body is} 3) Frequency of the body is called natural frequency resonant frequency. 4) Amplitude of vibration is} 4) Amplitude of vibration is small. large. Distinction between Force vibrations and Resonance Force vibrations Resonance 1) Vibrations of the body under the action of external periodic force are called forced vibrations. 1) In case of forced vibrations, if frequency of external periodic force is equal to the natural frequency then resonance is produced. 2) Frequency of the body] 2) Frequency oftheisresonant is other than natural frequency which is equal to frequency. natural frequency. 3) Amplitude of vibration is] 3) Amplitude of vibration is small large. 4) If external force is removed,} 4) If external force is removed, force vibration stops resonance stops after some immediately time. Harmonics and Overtones 1) The vibrations of a string or air column consist of the funda- mental frequency together with certain higher frequencies. The lowest allowed fundamental frequency is called first harmonic. If the first harmonic is “n then the value of second harmonic is ‘2n’, the value of third harmonic is ‘3n’ and so on. The word harmonics indicate the fundamental frequency and all its integral multiples. They may be present in given sound or not. 2) The higher allowed frequencies are called overtones. The first higher frequency greater than the fundamental frequency is called first overtone, the next higher frequency is called second overtone and so on. 3) Ex. Let the air column in a tube closed at one end is set into vibrations and 'n' be the fundamental frequency. Then first harmonic is 'n', second harmonic is '2n', third harmonic is '3n' and so on. But the sound consists of only odd multiples of funda- mental frequency i.e. n,3n,5n,.........The first higher frequency greater than the fundamental frequency is called first overtone. Therefore first overtone is '3n' (third harmonic), second overtone is 'Sn' (fifth harmonic) and so on. *O. Distinguish between harmonics and overtones, Q. Explain the modes of vibration of an air column in @ pipe closed at one end. Q. Show that only odd harmonics are present in an air column vibrating in a pipe closed at one end. (Mar. 15, Mark 3) Q. Draw neat labelled diagrams for the modes of vibration of an air column in @ pipe when it is a) open at both ends b) closed at one end Hence, derive an expres- sion for the fundamental frequency in each case. (Oct. 14, Mark 4) a a Distinction between Progressive waves Harmonics nd Stationary waves Overtones | 1) Frequencies higher than natural frequency are known as overtones, 1) Natural frequency (lowest allowed frequency) of vibration of a bounded medium along with all frequency j its integral multiples are| 2) w mature st ene is known as harmonics. 1, onor3n, second 2) If natural frequency is n, either 21 ‘nek noe then first harmonic is n, overtone 1S e second harmonic is 2n and so on. a are alway 3) Harmonics may be preset] 3) Overton’ given sown’ or absent in given sound. presen! Pe ee Ans. 1) Consider a cylindrical Pipe, closed at ane eng eee ‘uniform cross-section and length containing . 2) When a vibrating tuning fork is held near the a ond of the pipe, sound waves produced by the fork travel t rough e air column in the pipe and get reflected from the close en 7 3) Reflected waves interfere with the incident waves and stationary waves are formed. , 4) A node is formed at the closed end and antinode at the open end of the pipe. , 5) The air column vibrates in a number of different modes as below- A LA J N N (a) (c) Fundamental mode In this mode, one node is formed at the closed end and one antinode is formed at the open end as shown in fig. (a). Distance between the node and consecutive antinode is 4/4, Let 1 be the wavelength of the wave, Then- Length of pipe, A (b) vA 4L Q. Explain the modes of vibration of an air column in @ pipe open at both ends, Q. With a neat labeled diagram, show that all harmonics are present in an air column contained in a pipe open at both ends. (Mar. 13, Mark 3) Q. Show that even as well as odd (all) harmonics are present as overtones in the case of an air column vibrating in a pipe open at both ends. Q. Show that the funda- mental frequency of vibra- tions of the air column ina tube open at both ends is equal to double the funda- mental frequency in a tube of the same length and closed at one end. rac ri e ends having unifor 3 lindrical pipe, open at both gz " 1) Consider a cyli ; i i umn. "I taining air co od oF , d length |/' con one open end of the ost sete an tuning fork is held neat evel Gea the oe 2 pes ee wad ave produced by the for ipe, sound w ; églumn in the pipe and get reflect ae Reflected waves interfere wit waves are formed. e 4) Antinodes are formed at both the oP 5) The air column vibrates in anu nt waves and stationary 3 ds. ‘different modes as below- (a) (b) (c) Fundamental mode In this mode, two antinodes are formed at two open ends and one node is formed at the centre of the pipe as shown in fig. (a). Distance between two successive antinodes is A/2. Let A be the wavelenth of the wave. Then- A Length of pipe, L = > 2 A= QL voy Frequency of vibration is, n = V7 OL This is the lowest frequency of vibration called fundamental frequency of vibration of air column in a pipe open at both ends. The fundamental frequency of vibration of air column in a pipe closed at one end is 77. Thus fundamental frequency of vibrations of air column in a tube open at both ends is equal to double the funda- mental frequency in a tube of the same length and closed at one end. Second mode In this mode, three atinodes and two nodes are formed as shown in fig. (b). Let A, be the wavelength of the wave. Then- Length of pipe, L = A, Frequency of vibration is, =tuLv mea n= 2n The frequency of second mode is twice the fundamental frequency. It is first overtone or second harmonic. 16 *Q. A tube open at both ends has fundamental frequency ‘n'. If one end of the tube is dipped in water to half of tts length, What would be its fundamental frequency? Q. What is end correction? What is its cause? What are its limitations? Q. Define end correction. (Mar. 13, Mark 1) Third mode In this mode, four antinodes and three nodes are formed as shown in fig. (c). Let 2. be the wavelength of the wave. Then- Length of pipe, L = Me Frequency of vibration is, m= 7 3B Thus the frequency of third mode is three times the fundamental frequency. It. is second overtone or third harmonic. Conclusions 1) Thefrequencies ofmodes of vibration arein the ratio m:1:/ ~ 1:2:3. 2) In the modes of vibration of air column in a pipe open at both the ends, all harmonics are presents as overtones. (P+) = n, 3) In general, the frequency (np) of P" overtone is 7”, where n is fundamental frequency and P = 12,3... Ans. If'L' is the length of the pipe open at both ends and v is the velocity of sound in air, then its fundamental frequency 7”? isn = v/2L.When this pipe is dipped in water to half of its length, then it becomes : = L/2 an pipe closed at one end with an air column of length. L' Vig yl = ee fundamental frequency n’ is 7! = aD ~ 4/2) 2L Thus its fundamental frequency remains constant. End Correction | 1) In different modes of vibration of air column in a pipe closed at one end or open at both ends, antinode is always formed at the open end and node is always formed at the closed end. But the antinode is not formed exactly at the open end, but it is formed a little distance beyond it. The distance between the open end and antinode is called end correction. 2) End correction ‘ce’ is given as- e= 0.3d Where, d is the inner diameter of the pipe (tube) Hence, (ore wnat) -( length of air }+( end of air column column in pipe correction « L=1+03d For a pipe closed at one end, the fundamental mode is- v= 4nL = 4n(1+ 0.3d) Thus velocity of sound in air at room temperature can be deter- mined, if n,/ and d are known. For a pipe open at both ends, an end correction (e = 0.3d) for each open end must be added to the measured length of the pipe. Therefore end correction for a pipe closed at one end is 0.3d while that for a pipe open at both ends is 0.6d. Q. With neat diagram, explain various modes of vibration of a stretched string. Q. Obtain an expression for the fundamental frequency of vibration of stretched string. Q. Show that all harmonics are present ona string stretched between two rigid support. (Oct. 15, Mark 3) Q. Show that even as well as odd harmonics are present as overtones in modes of vibration of string. OQ. Draw neat, labelled diagrams for the modes of vibration of a stretched string in second and third harmonic. (Oct. 14, Mark 2) Modes of Vibration of String wien a string clamped between two rigid supports is plucked at lilferent points, transverse waves are produced, Due to the refiec- tion of transverse waves at the supports stationary waves are formed with a node at cach support. The different ways in which the stationary waves are formed, are called modes of vibration of string. 1) When the string is plucked at the centre, it forms a single loop with one node at each support and one antinode at the centre. Such mode is called as fundamental mode. P Q Fig. 1st harmonic Let 'n' and '\' be the frequency and wavelength of th string. Length of each loop is- e wave on l= oo AS BL » Wa) Nils We have y= nd ay n= ve ai 21 Bp T wn Where T is tension and m is linear density. This is the lowest frequency of the stationary waves and is called fundamental frequency or threshold frequency. It is the frequency of first harmonic. 2) Let the string forms two loops with three nodes and two antinodes. Ih, Fig - 2nd harmonic Let 'm!' and 'A,' be the frequency and wavelength of corresponding wave then- m= 2 vm = 2n This shows that, the frequency of this mode is twice the funda- mental frequency. It is called second harmonic. It is the first higher frequency greater than fundamental frequency, so it is called first overtone. Q. State the laws of vibrating strings. ith four any 3) Let the string forms three loops W! modes and yy antinodes. *e Fig - 3rd harmonic not Let 'm' and ‘A;' be the frequency and wavelength o COFFESPONGing wave then- ; 8 Length of each loop is- As 2 2 3 i 3 NM We have v= mA2 maa Bl 3 T naa! ) “2M = 3n This shows that, the frequency of this mode is thrice the funda. mental frequency. It is called third harmonic and second overtone, General Formula If np is frequency of P” overtone then- PAD [EL ° 21 m The P" overtone is (P + 1)"harmonic. Conclusion i) Frequency of P” overtone = (P+ 1)n. ii) Modes of vibration of string gives frequency n,2n,3n,... so on. iii) Therefore stretched string vibrates with all odd and even harmonics. Laws of Vibrating String The fundamental frequency of vibration of a stretched string is given as- Where, '!' is the vibrating length of string (wire), 'm'is linear density (mass per unit length) and 'T' is the tension in the string. From above equation we have following three laws of vibrating string- i) Law of Length The fundamental frequency of vibrations of a stretched string iS inversely proportional to its vibrating length, if the tension and linear density are kept constant, ; OCH Tone crv ne at constant T and m * mh = mlb = constant *. al = constant
