Formula Sheet for Physics Waves

F or m u l ae Ph y si cs | pg. 4 Waves 4. 1st overtone/2nd harmonics: ν1 = 2.1: Waves Motion 5. 2nd overtone/3rd harmonics: ν2 = General equation of wave: 2 ∂ y ∂x2 = 2 1 ∂ y . v 2 ∂t2 2 2L 3 2L 6. All harmonics are present. q q T µ T µ Notation: Amplitude A, Frequency ν, Wavelength λ, Period T , Angular Frequency ω , Wave Number k , 1 2π T = = , ν ω 2π k= λ v = νλ, L String fixed at one end: N A y = f (t − x/v), +x; y = f (t + x/v), 1. Boundary conditions: y = 0 at x = 0 −x y A x λ λ 2 Progressive sine wave: y = A sin(kx − ωt) = A sin(2π (x/λ − t/T )) 2.2: Waves on a String 2. Allowed Freq.: L = (2n + 1) 4λ , ν = 2n+1 4L 0 , 1, 2 , . . . . q T 1 3. Fundamental/1st harmonics: ν0 = 4L µ q T 3 st rd 4. 1 overtone/3 harmonics: ν1 = 4L µ q T 5 5. 2nd overtone/5th harmonics: ν2 = 4L µ q T , µ n = 6. Only odd harmonics are present. Speed of waves on a string with mass per unit length µ p and tension T : v = T/µ 2 2 2 Transmitted power: Pav = 2π µvA ν Interference: y1 = A1 sin(kx − ωt), A N λ/2 Progressive wave travelling with speed v : Sonometer: ν ∝ L1 , ν ∝ √ T, ν ∝ 1 √ µ. ν= n 2L q T µ 2.3: Sound Waves y2 = A2 sin(kx − ωt + δ) y = y1 + y2 = A sin(kx − ωt + ) q A = A1 2 + A2 2 + 2A1 A2 cos δ Standing Waves: A N Displacement wave: s = s0 sin ω(t − x/v) Pressure wave: p = p0 cos ω(t − x/v), p0 = (Bω/v)s0 Speed of sound waves: s s Y B , vsolid = , vliquid = ρ ρ A2 sin δ tan  = A1 + A2 cos δ  2nπ, constructive; δ= (2n + 1)π, destructive. 2A cos kx 2 S h e e t f or Intensity: I = A N A 2π 2 B 2 2 s0 ν v = p0 2 v 2B = vgas = s γP ρ p0 2 2ρv x Standing longitudinal waves: λ/4 y1 = A1 sin(kx − ωt), p1 = p0 sin ω(t − x/v), p2 = p0 sin ω(t + x/v) p = p1 + p2 = 2p0 cos kx sin ωt y2 = A2 sin(kx + ωt) y = y1 + y2 = (2A cos kx) sin ωt    n + 12 λ2 , nodes; n = 0, 1, 2, . . . x= λ antinodes. n = 0, 1, 2, . . . n 2, String fixed at both ends: N A N L Closed organ pipe: L A λ/2 1. Boundary conditions: y = 0 at x = 0 and at x = L q T n 2. Allowed Freq.: L = n λ2 , ν = 2L , n = 1, 2, 3, . . .. µ q T 1 3. Fundamental/1st harmonics: ν0 = 2L µ N 1. Boundary condition: y = 0 at x = 0 v, n = 2. Allowed freq.: L = (2n + 1) λ4 , ν = (2n + 1)4L 0, 1, 2, . . . 3. Fundamental/1st harmonics: ν0 = v 4L 3v 4. 1st overtone/3rd harmonics: ν1 = 3ν0 = 4L F or m u l ae Ph y si cs S h e e t f or | pg. 5 5v 4L 5. 2nd overtone/5th harmonics: ν2 = 5ν0 = Path difference: ∆x = 6. Only odd harmonics are present. N Open organ pipe: L A N A 1. Boundary condition: y = 0 at x = 0 v , n = 1, 2, . . . Allowed freq.: L = n λ2 , ν = n 4L 2. Fundamental/1 harmonics: ν0 = 4. 2 overtone/3 rd Phase difference: δ = ∆x = Intensity: 2v 2L D 2π ∆x λ  nλ, 1  n + 2 λ, constructive; destructive p I = I1 + I2 + 2 I1 I2 cos δ, p p p 2 p 2 I1 + I2 , Imin = I1 − I2 Imax = 3v 2L harmonics: ν2 = 3ν0 = P y θ Interference Conditions: for integer n,  2nπ, constructive; δ= (2n + 1)π, destructive, v 2L 3. 1st overtone/2nd harmonics: ν1 = 2ν0 = nd S d1 S2 A st dy D 5. All harmonics are present. l2 + d l1 + d I1 = I2 : I = 4I0 cos2 2δ, Imax = 4I0 , Imin = 0 Resonance column: Fringe width: w = λD d 0 Optical path: ∆x = µ∆x Interference of waves transmitted through thin film: l1 + d = λ , 2 l2 + d = 3λ , 4 v = 2(l2 − l1 )ν Beats: two waves of almost equal frequencies ω1 ≈ ω2 p1 = p0 sin ω1 (t − x/v), p2 = p0 sin ω2 (t − x/v) p = p1 + p2 = 2p0 cos ∆ω (t − x/v) sin ω (t − x/v) ω = (ω1 + ω2 )/2, ∆ω = ω1 − ω2 nλ, 1 n + 2 λ, constructive; destructive. Diffraction from a single slit: y θ b y D For Minima: nλ = b sin θ ≈ b(y/D) Resolution: sin θ = v + uo ν v − us 0 1.22λ b Law of Malus: I = I0 cos2 θ where, v is the speed of sound in the medium, u0 is the speed of the observer w.r.t. the medium, considered positive when it moves towards the source and negative when it moves away from the source, and us is the speed of the source w.r.t. the medium, considered positive when it moves towards the observer and negative when it moves away from the observer. 2.4: Light Waves Plane Wave: E = E0 sin ω(t − vx ), I = I0 Spherical Wave: E =  (beats freq.) Doppler Effect: ν= ∆x = 2µd = aE0 r sin ω(t − vr ), I = Young’s double slit experiment I0 r2 θ I0 I