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= Wave is a disturbance. ; (i) One dimensional wave. Ex: Waves moves along a string Gi) Two dimensional Ex: Ripples on a liquid surface Ex: Sound waves and light waves (i) Three dimensional Sound waves, waves on a string, waves on a water , Non-Mechanical waves: - Does not need medium Ex: bight waves, radio waves. 7 ~ rays. X rays r The longitudinal waves ean travel in solids, liquids and gases. ei Progressive waves: - The waves proceeding in the medium with velocity and never returns to the origin, (i) Frequeney (7) = a ay T (ii) Angular frequency (w J (ii) Time period T = . 2 = =< (iv) Wave function y= f(x = v1) *— along the +ve x- axis +4 along the —ve x — axis (vy) Other forms of wave function y =(x-v)", v= y(x—vi), Ae Hee (vi) Wave function equation /(x,7)= f(axtbt) x - 5, 3 dy (vii) The differential form of wave equation x = ‘ ay a ; 2z (viii) Propagation constant k = — Aa — _ @ a wl Gt | {ix) Wave velocity v =e =nh=— Wwe ‘ U vo gm WG locity V do x) Group velocity V7, = — Pe ar OR (xi) Progressive wave cquation ye asin(cot = kx +) Z. *~ tveX — axis “+ -ve x - axis _ . x (WF x) =asinw(tF— (xii) Other forms of progressive wave equation y= asin v —EEEEE ee (xiii) The displacement of wave function at any time Y= any > The relation between phase difference and path difference of the wave A¢= Phase difference A= wavelength Ax = Path difference > Particle Velocity: V, = Awcos(wr - kx) Maximum particle velocity [= 40) Particle Acceleration: a, =—Aa? sin(dt — kx) Relation between particle velocity (Vp) and wave velocity (V,,) : a ) =(V,) (-slope of the wave) > Energy density of the wave is K.E RE 1 =—— =~ p#'e' cos" (wt — ke) yolume volume eae coe ) Average energy associate with volume E = ; pA eo? xVolume = ; p&a? xSAx T= Tension in the string # = Linear mass density > If ‘A’ is the area of cross section of the wire then = , asin(Ax +!) It y, sasin(ky-or) & 8 Stationary wave equation v= 2acos kysinof Amplitude of the wave As = 2acoskx Stationary wave Ina stretched string: i) Fixed at both ends} (p=t.2 Rs ae length of a string, p = number of loops T=Tension: p= linear density: /= ¢ available All harmonics ¢ \ (p= 1.2.30) ii) Fixed atone end. free at other end 1 Only odd harmonics are available sae 1 ii) maa 7 =9 mh = Ml Also AB = iolog( 2) 4 J oe Open Organ Pipe: (Stationary wave [rave 2 PSII (i) Fundamenta lamental fre: 4 7 equency or I“ harmenic is 7 = I ,=— (ii) 2" harmoni cal narmonic or 1“ overtone n, = a » =—=2n, (iii) 3" harmonic or 2" onic or 2™ overtone n= wv 33 ita nV (iv) General formula, 1 = oy poled deere JEE Marre — | (v) Ratio of the frequencies (vi) All harmonics exish, » Closed Organ Pipe: a A Aa | v, | 7 | | [ if } _ Vv (i) Fundamental frequency or 1 harmonic is ™ =]; - 3aV Gi) 3" harmonic or I’ overtone 1, =" a a SV _ (iii) 5" harmonic or 2" overtone HM; = rial (iv) General formula n, =p) (p= 1.2, 3,0 ) an (v) Ratio of the frequencies = (vi) Only odd harmonics exist. resence of end correction & closed pipes in the p > The fundamental frequency of open . V (i) Open Pipe: == a » 2426) {e=0.6r}, (r — radius of the organ pipe) Vv ' =———>~x 241.2") (ii) Closed Pipe: Vv n= 4( +e) 4 1.=———— A( +0.6r) > Resonance: If 41, 4, hare the I*, 2" and 3" resonating lengths then j+eae 4 2Qnv. 2ny, sam apparent frequency Aan=—y ans souree then change 10 ii) Observer crossing the stationary source then chang Psound; pler effect in case o! of reflection of wal Outs observer is at rest in between source of sound and “ , yes tcflecied ( a eet Sound ° ( 4 of see ibserver wall Image SUM ( ; at -( * } n 7s v-¥, v Fy = n mu [ vy, Ans ~n) An=0 (ii) 1f the source of sound is in between observer and wall ats, ned oh 2 0 Direct Reteeaed Bp ( f ey = (@ ubserver wall Ti of tdi ! v i n = trees vey, n! a) - decd =| —— reflecetd vey, (iii) When both source (s) and observer are both moving towards the wall, then the apparent Beats frequency An = anv, frequency heard by an observer n’ a Y% I v-y, Doppler effect due to rotating source of sound (or) observer. (I) Source of sound is rotating (V, =ra,@ — angular velocity) : ,_onv at=—— (a) Source of sound approaches stationary observ maar nv A scecneaceuh (b) Source of sound recedes stationary observe 7 vay ‘s (I) Observer is rotating: »_ {vr (a) Observer approaches stationary source of sound # -( 5 . }. i ae” (b) Observer recedes stationary source of sound n” -( = tn When velocity of wind is considered, effective velocity of sound v =v+¥,, cosd, 0, is angle between wind and direction of path of sound between source and observer. Doppler effect in light: i) When source and observer approach each other n' = 1, on oft +4) c-u c v= a(1-) [Wavelength decreases- blue shift] c ii) When source and observer move away from each other n'=n, j— = oft -2) ctu c Mz at +4) [Wavelength increases- red shift] C-> Speed of light, u—> relative velocity between source and observer 
