Periodic and Oscillatory motions: + Oscillations ore defined as to and fra motion which repeat itself after regular intervals of time. In oscillations, the frequency of vibrations is comparatively less. For example: The to and fro motion of a pendulum clock + Every oscillatory motion is periodic motion that is every oscillatory motion repeats itself after the ‘fixed interval of time, But every periodic motion is not oscillatory. For e.g.:- Motion of planets cround the sun is periodic but is not oscillatory motion. 4 Ttisdenoted by T. @ sats a + Its ST unit is second. >> Frequency: it is defined as number of cycies per second. + Ttisdenoted by v. « [ts SIunitis sec’, + Special Unit is Hert2(H2) of cycles in 1 second eyele Displacement: Displacement in. ron can be represented by @ function which is periodic which repeats after fixed interval of #Q)eA ‘As cosine ‘50 it can be written as U 2m) Equation (1) Let f where displacement keeps on repeating after (1*T) Acos (wt)? cosw(t+T) = Aeos (wt+ wT) Acosut = A cos (wtwT) Equation (2) From Equation (1) and Equation (2) Tad MwF(t) = Asinwt+ Acoswt LetA=sDcos® = B= D sin’ (1) = Deos® sinwT + Dsin® cos wt =D (cos@ sinwT + sin® cos wt) F(A) =D sin (ut + 3) (Using sinAcos® + sinBeosA = sin (AsB)) Din terms of A and B: =o & a ety Simple Hormonic Motion (HM): I* iso periouic motion the body fro about its mean position, The restoring farce-on the oscillating body is directly displacement and is ‘always directed towords its mean position. a @ +A — Oscillatory motion is soid to be SHM if the di particle from origin varies with time t: (1) = A cos (wt + 2) Az amplitude (It is defined as displacement of the particle from its mean position.) wt + = phase angle (time- Phase: Tt is that quantity that state of motion of the particle. + Tts value is (wt+ (0)= A (cos0=1) 451/4, t= 12, 1237/4, 1-7 ond 1257/4 In the given graph the displacement as a function of time is obtained when » = 0. The curves (1) and (2) are of two different amplitudes A and 8.In the above graph the curves (3) ond (4) are for » = O and Dislocenent -1/4 respectively but the amplitude is same for both, Angular Frequency (yw): It refers to the angular displacement per unit time, Angular frequency is larger ‘than frequency v by a factor of 2m, Consider the oscillatory motion which is varying with time t and displacement x of the “is origin w= 2e/T SHM & Uniform Circular Motion: Uniform Circular motion can be i rasa SHM. Consider any particle moving in. cireular path whose radius is R At Angular position =J@ dt swt +9. Consider the projection of particle on x-axis be P' ~ Displacement = Acos® % =A cos(wts 6) As the particle is moving in the tions are also moving, We can conclude that the ing from left to right and ‘gain from right to left. cle moving in circular path, The velocity at any point P at ony time 1 will be tangential to the paint P. v(t) = - Aw sin(wt + @) Acceleration in SHM: Acceleration in uniform circular mation always directed towards the centre. Tt is known as radial acceleration. a (t) = - wA cos (wt + ¢)Energy in SHM: The Kirstie ond Potential enerses ine SHIM and thelr maxim values, etc energy, potential enerzy andthe tote energy is gfpen of tien the chove graph. Bethknatic energy ond potential energy repeats after tine T +The netic energy (0) of «particle executing eskapu ‘efmed os e Ke bm = =A mats Gt +9) Ke Bk a ait Gat» + The potential energy executing SHIN, uw: # Up K ) +1 system anys rans the some Usk = HAE sie ut 9) +4 HA cot (at og) E> bk Aen Gat = 9) + coat (wt =) ESERAT ——Totlenergy is clays const. we T Qscillations due to spring: Cosi block f its pulled on onesie ed is released, te hen it ‘exesites toons fro motion about a mean postin Fedepe fe Aand sa mann + Kisteown as eping constant — ‘Ae Songs eon eee fens snl pncn cn be in reine ae ‘CLLLLLs a ionWave: « wave is « disturbance that propagates through space and time, usually with transference of ‘energy. ‘Types of Waves: 1. Mechanical waves 2. Electromagnetic waves 3. Matter waves 1) Mechanical waves: In mechanical waves, ‘sonnonenstnoa AG, ‘+ The mechanical waves are governed by all the Newton's lows of motion. For Example: - Water Waves, Sound Waves Electromagnetic waves: An clectromogneric wave, dees not need 0 3 no mass, does carry energy, + Electromagnetic waves are related to electric and magnetic fi Examples: - Sotellite system, mobile phones, radio, music play: Oo" Matter waves: Matter waves are associated with mo ‘otons, neutrons & other fundamental particles etc. + It is on abstract concept. Examples: - pencil, sun, moon, earth, ball, Transverse Waves: 7: se waves which propagates perpendicular to the direction of the disturbance. a + Direction of disturb jon of motion of particles of the medi A sinusoidal ‘the string, A typical element = Warclengh of the string ind down continuously as the wave Transverse Wove pa = In longitudinal waves direction of disturbance or displacement in the medium is ‘along the propagation of the wave. Longtudial ware + Ina Longitudinal wave there are regions where ce particles are very close to each other. These appeal ot I tl Will MI In some regions the particles are far apart. Those regions are known as rarefactians, CCompreniion RaretactionDisplacement _in a progressive wave: Amplitude and phase together describe the complete displacement of the wave + Displacement function is ¢ periodic in space and time. + Incase of transverse wave displacement is given as Wave travelling along +X-axis: y(x. t)= a sin (kx - wt + @). yrasin(kx = wt+ 9) => wia=sin{kx - wt + @) Siny/a) = lout =ahexesin’ (y/a) owt x=(1/kjsin’ ye} (wit?k) 8 Wave travelling along -X-axtis: =(1/k)sin"*(y/a)-(utt/k) a As time t increases the vatue of x increases, This implies the x mo As time t decreases the value of x decrease, This implies the (Dive x-axis Amplitude: Amplitude is the maximum di positions as wave passes through them. Ir is denoted by A. In case of transverse wave, of the particles is known as ipiutude® In case of longitudinal represented by the rarefaction region where the partic! to oscillate to its ‘AtPointP 2evo Displacement Compression tes ‘describes the state of motion as the wave sweeps through an element at a Gc.1 asin(kex - wit + 9) where 0: amplitude, (kx - wt + @)=phase In-phose: Two points are said to be in-phase with each other when these twa points are at the same position and they both are doing the same thing i.e. both the two points are exhibiting the same behaviour. Out-of -phase: Two points are said to be out of phase even though they are at the seme points but they are doing oppasite thing i.e. both the points are exhibiting the different behaviour.Wavelength and wave number— Wavelength? Wavelength is defined as the minimum distance between two consecutive points in the same phase of wave mation. + Tt is denoted by A ck Longudinal Wave Hi warsiengin No VAXY ill iT LL werent Comin Rarataction Movement Teoneverse Wane Wave Number: Wave number describes the number of wavelengths per nig + Denoted by k’ also known as propagation constant. 2 ~e Time Period, and lar ‘Time Period: Time Period of o wave is the time token oscillation. It is denoted by T. Frequency: Frequency of a wave is defined as per unit time.It is denoted by v veVT=u/2n Angular frequency: Angular frequency is Frequency of the wave in terms of a circulor motion. It is denoted by ‘w’. w = 2av = 2a/T. & s are those waves which travel from one medium to another, ive wave. Consigge transverse wove moving along (+) ive x axis. wave speed v = dufdt = w/k By definition of we (2 n/T Also k=(2 m)/A Therefore v=(2 /TV(2 n/a) =W/T venT As wavelength of a wave increases as a result frequency of the wave decreases as a result speed of wave is constantTransverse wave speed determined by! ‘As mass gives rise to Kinetic energy. If no mass then na kinetic energy. Then ‘there will be na velocity, © Trisdenoted by. Tension- Tension is the key factor which makes the disturbance propagates along the string. Because of tension the disturbance travels thraughout the wave. + Ttis denoted by T. K, w= (MVEL]=(ML*] T=Fema =(MJ[LT?] = (MLT?] fii) Dividing equation (i) by (il) =~ (ML"VIMLT?] = [TL 7} =1/ AS Therefore w/T = tv? ve€ E where C=dimensionless constant . aoe icy of the wave, Longitudinal wave speed determined by: 1. Density - Longitudinal wave is + Itis denoted by p 2 Bulle modulus Bui m does the volume of a medium changes when the pressure on it changes. ° Tris olume= [ML *] = - (Change in pressure (AP))/Change in volume(AV/¥)) AV/V is a dimensionless quantity. OP =F/A = ma/A =(MJLT2VEL2] = [ML"T 2) + Dividing o/B= (ML°VML'T?] = (LTE (TLF =? p/B = Lv on v*=B/ p vel £ where C-dimensionless constantThe principle of superposition of waves: It describes how the individual waveforms con be algebraically added te determine the net waveform. + Waveform tells about the overall motion af the wave. It does not tell about individual particles of the wave. Cosel:-Consider 2waves which ore in phase with each other. They have the some amplitude, same angular frequency, and same angular wave number «If wave Lis represented by yi(x, t) i + If wave 11s represented by y.(x, t) + Wave 2 és represented by yz + Therefore by the princi sition af waves y = yt yz zalsin(ko-wt }sin(lox = wste 9) ¥; (ke - ut + (9/2) + Amplit and Phase will be determined by (9/2). 3 When o wave strikes any interface between any two mediums the bouncing back of wave is telgged os reflection of waves. Interface can be categorised into 2 types: * ~Qpen boundary: - When o wave: strikes an interface in case of open boundary it will get reflected as well os refracted. + Closed boundary or a rigid boundary: - When a wave is incident on an interface it will completely get reflected, Example:-Wove striking wall(echo)Reflection at rigid boundary the amplitude at the boundary is phase 9-7. Therefore, y(x, 1) = asin (kx - wt) yu, 1) =a sin (lox ¢wt + 2)=- asin (kx + wt) By superposition principle y= y,* y, =O Conclusion: The reflection at the rigid body will take place with a phase reversal of m or 180. int matali % + amplitude at the boundary is maximum & phase difference ¢=0. Therefore, y(x, 1) = a sin (kx - wt), 5 y-(x, t)= asin (kx = wt), By superposition principle y= y:+ y-= 2a sin (kx - wt) Standing(stationary) Waves: 4 stationary waye is which is'hot moving. ‘amplitude travelling in opposite ‘towards right y. swt) ¥ (x, t)= yx, 1) + ye [kx - wt) +o sin (kx + wt) wheet)= cos wt + The above, 'S the standing wave expression. —_ Wave: travelling towards I VVUVY litude is dependent on the position of the particle. cos ut represents the time dependent variation or the phase of the standing wave, Nodes and Antinodes of Standing Wave ¢ The amplitude of a standing wave doesn't remain the sarne throughout the wave. Nodes:- Nodes represent the positions of zero amplitude. © Incase of the standing wave: amplitude is given as :~ 2asinkx + Zasinkx = 0 => sinkx = Q => sinkx =sinn m => kxenx=(nmVk — ~equation(?) From the definition of k-(2x)/A ... equation(ii) + The position of nodes is represented by: x=(m AV2 From(i) and (ifj,where n=l, 2, 3.. + Note: Holf a wavelength (\/2) separates two consecutive nodes, Antinedes:~ - Antinodes represent the positions of maximum amplitude + Incase of the standing wave amplitude is given as i- Zasinkx > 2asinkx = maximum, This valve is maximum only when sink => sinkx = sin(no{1/2)) => kx = (ne(1/2)) => (€2 m)/ A) x= (eC) ‘The position of nodes is represented by:~ x= (no(1/2)X System closed at both ends means both This implies at boundary amplitude is Standing waves on a string af length WhenxeL, Le(naVe possi value of frequency. + 2 nodes ot the ends ond 1 antinode, ‘Second Harmonie:-For n=2, vo=(2v)/ (2L) =w/L. © = 3 nodes at the ends and 2 antinodes. ‘ThirdHarmonie:-For n=3,vs= (3v)/ (21). + 4 nodes and 3 antinodes. Nodes‘Nodes and Antinodes: system closed at one end ‘+ For asystem whichis clesed at one end, only one node is formed at the closed end. ‘+ AtxcL,antinades will be formed. This means amplitude will be maximum ot this end + Condition for formation of entinodes is x (n+ (1/2)) (M2) b= (mcvan vey 2 A seLymeere2y. ‘This expression shows that the vlues of wavelength is restricted. n=0,1,2.3. 4 1+ Corresponding frequencies wil be v= v/(2L}nel/2) (By using vev/A}re 0.12.3. — Medes of xciations:- @, Fundamental frequency Also known os Firs Herr. Tt velvefor f. That i 0. ty ds +The expression for fundamental frequency ise = w/2U(/; olen ae + Odd Harmonics | fs © mel: weW/QLM/2) # (BYVAL) #30 i 2 med v=WM2L{eoQ/2)> (BEAL \ 9 revs ALXSHt/2)H(7eVEL7 + Forasjstem which ia sed ato another end wil get one fundamental frequency ord al ie. ates snaked Sardgmie nO Beats: Beors isthe phe sound waves of nearly some frequencies and amplitudes ‘rovelng nthe some cand not the position dependent part of the wove. wit: where amplitude and phase of the waves ore same, but the Aisa considering w, > we, + scafcos unt + cos wat] = 20 (costs -wn}/2}t costun + an)/2)t) ‘sau - eis ery smell as wi? ie Let (wr - we) = wm ‘=m + is very large Let (wy + we) = we Therefore, £= 2a 008 unt cos unt ‘+ cosuct wil very rapidly with time and Zocosunt wil charge slowly with te. + Therefore we con say 2ecosunt = constont, As a result Zecosust = amplitude es it has small angular variation Beat Frequency: Seat frequency con be defined as the difference in the frequencies of two waves, © Trisdenoted byw + Altos anv “Therefore tes * vi~¥e Doppler’s Effect: boppier Effect ic the phenomenon cf motion-releted frequency change, oy)