Superposition Interference Waves ADD: Constructive Interference. Waves SUBTRACT: Destructive Interference. In Phase Out of Phase Superposition Traveling waves move through each other, interfere, and keep on moving! Superposition Waves ADD in space. Any complex wave can be built from simple sine waves. Simply add them point by point. Simple Sine Wave Simple Sine Wave Complex Wave Fourier Synthesis of a Square Wave Any periodic function can be represented as a series of sine and cosine terms in a Fourier series: yt()=+∑ ( Annnn sin2ππ ƒ t B cos2 ƒ t ) n Superposition of Sinusoidal Waves • Case 1: Identical, same direction, with phase difference (Interference) • Case 2: Identical, opposite direction (standing waves) • Case 3: Slightly different frequencies (Beats) Superposition of Sinusoidal Waves • Assume two waves are traveling in the same direction, with the same frequency, wavelength and amplitude • The waves differ in phase • y1 = A sin (kx - ωt) • y2 = A sin (kx - ωt + φ) • y = y1+y2 = 2A cos (φ/2) sin (kx - ωt + φ/2) Resultant Amplitude Depends on phase: Spatial Interference Term Sinusoidal Waves with Constructive Interference y = y1+y2 = 2A cos (φ/2) sin (kx - wt + φ /2) • When φ = 0, then cos (φ/2) = 1 • The amplitude of the resultant wave is 2A – The crests of one wave coincide with the crests of the other wave • The waves are everywhere in phase • The waves interfere constructively Sinusoidal Waves with Destructive Interference y = y1+y2 = 2A cos (φ/2) sin (kx - wt + φ /2) • When φ = π, then cos (φ/2) = 0 – Also any even multiple of π • The amplitude of the resultant wave is 0 – Crests of one wave coincide with troughs of the other wave • The waves interfere destructively Sinusoidal Waves Interference y = y1+y2 = 2A cos (φ/2) sin (kx - wt + φ /2) •When φ is other than 0 or an even multiple of π, the amplitude of the resultant is between 0 and 2A • The wave functions still add Superposition of Sinusoidal Waves y = y1+y2 = 2A cos (φ/2) sin (kx - ωt + φ/2) • The resultant wave function, y, is also sinusoidal • The resultant wave has the same frequency and wavelength as the original waves • The amplitude of the resultant wave is 2A cos (φ/2) • The phase of the resultant wave is φ/2 Constructive Destructive Interference φ = 0 φ Wave Interference y = y1+y2 = 2A cos (φ/2) sin (kx - ωt + φ/2) ⎛⎞Δφ Resultant Amplitude: 2A cos⎜⎟ ⎝⎠2 Constructive Interference: Δ=φπ2nn , =0,1,2,3... Destructive Interference: Δ=φπ(2nn + 1) , =0,1,2,3... Ch 18 HO Problem #1 y = y1+y2 = 2A cos (φ/2) sin (kx - ωt + φ/2) ⎛⎞Δφ Resultant Amplitude: 2A cos⎜⎟ ⎝⎠2 Constructive Interference: Δ=φπ2nn , =0,1,2,3... Destructive Interference: Δ=φπ(2nn + 1) , =0,1,2,3... 1-D Sound Wave Interference 2-D Wave Interference? P Interference: Two Spherical Sources 2-D Phase Difference Different than 1-D You have to consider the Path Difference! v 22π π Δφω = Δ=tftt22 π Δ= π Δ= () vtr Δ = Δ λ λλ 2-D Phase Difference at P: Δφ is different from the phase difference φ between the two source waves! P 2π Phase Difference at P: Δφ =Δr λ λ Path Difference at P: Δr =Δφ 2π Spherically Symmetric Waves Intensity Quiet Min Loud Max Quiet Min Loud Max Constructive or Destructive? (Identical in phase sources) P 2π Phase Difference at P: Δφ =Δr λ 2π Δ=φ (1λπ ) = 2 λ Constructive! ⎛⎞Δφ Resultant Amplitude: 2A cos⎜⎟ ⎝⎠2 Constructive Interference: Δ=rnλφ, Δ =2 n π , n =0,1,2,3... λ Destructive Interference: Δ=rn(2 + 1) , Δφπ =(2 n + 1) , n =0,1,2,3... 2 Constructive or Destructive? (Source out of Phase by 180 degrees) P 2π Phase Difference at P: Δφ =Δ+Δr φ λ 0 2π Δ=φ (1λπ ) + = 3 π λ Destructive! ⎛⎞Δφ Resultant Amplitude: 2A cos⎜⎟ ⎝⎠2 Constructive Interference: Δ=rnλφ, Δ =2 n π , n =0,1,2,3... λ Destructive Interference: Δ=rn(2 + 1) , Δφπ =(2 n + 1) , n =0,1,2,3... 2 2-D Phase Difference Different than 1-D You have to consider the Path Difference! v 22π π Δφω = Δ=tftt22 π Δ= π Δ= () vtr Δ = Δ λ λλ 2-D Phase Difference at P: Δφ is different from the phase difference φ between the two source waves! P 2π Phase Difference at P: Δφ =Δr λ λ Path Difference at P: Δr =Δφ 2π In Phase or Out of Phase? A B Ch 18 HO Problem #2 2π Phase Difference at P: Δφ =Δr λ λ Path Difference at P: Δr =Δφ 2π Intensity Quiet Min Loud Max Quiet Min Loud Max Interference of 2 Light Sources Contour Map of Interference Pattern of Two Sources Ch 18 HO Problem #3 You Try Standing Waves: Boundary Conditions Reflected PULSE: Free End Bound End Transverse Standing Wave Produced by the superposition of two identical waves moving in opposite directions. Standing Wave Standing Wave: Standing Waves Superposition of two identical waves moving in opposite directions. y12 == Akxt sin ( - ω ) y Akxt sin ( + ω ) y = (2Akxt sin )cosω •There is no kx – wt term, and therefore it is not a traveling wave •Every element in the medium oscillates in simple harmonic motion with the same frequency, w: coswt •The amplitude of the simple harmonic motion depends on the location of the element within the medium: (2Asinkx) Note on Amplitudes yAkxt= (2 sin )cosω There are three types of amplitudes used in describing waves – The amplitude of the individual waves, A – The amplitude of the simple harmonic motion of the elements in the medium,2A sin kx – The amplitude of the standing wave, 2A • A given element in a standing wave vibrates within the constraints of the envelope function 2Asin kx, where x is the position of the element in the medium Node & Antinodes •A node occurs at a point of zero amplitude nλ xn==0,1,K 2 •An antinode occurs at a point of maximum displacement, 2A nλ xn==1, 3,K 4 Standing Waves on a String Harmonics Standing Waves on a String Harmonics Standing Waves on a String λ1 = 2L λ2 = L 2L λ = 3 3 Standing Waves on a String 2L λ = n n fvnn= / λ v fn= n 2L Standing Wave on a String • One end of the string is attached to a vibrating blade • The other end passes over a pulley with a hanging mass attached to the end – This produces the tension in the string • The string is vibrating in its second harmonic Superposition Sound Waves Longitudinal Standing Wave Standing Waves in a Tube Open at both ends. v fnn = 2L λ = 2L Resonant Frequencies: fn = nf1 λ = L Same as a string fixed at both ends. Standing Waves in a Tube What is the length of a tube open at both ends that has a fundamental frequency of 176Hz and a first overtone of 352 Hz if the speed of sound is 343m/s? v v fnn = Ln= 2L 2 fn 343ms / =1 2(176Hz ) = .974m Standing Waves in a Tube Open at one end. 4L λ = v n fnnodd= nodd 4L What is the difference between Noise and Music? Regular Repeating Patterns Multiple Harmonics can be present at the same time. Which harmonics (modes) are present on the string? The Fundamental and third harmonic. The amount that each harmonic is present determines the quality or timbre of the sound for each instrument. Any complex wave can be built from simple sine waves. Standing Waves Standing waves form in certain MODES based on the length of the string or tube or the shape of drum or wire. Not all frequencies are permitted! Strings & Atoms are Quantized The possible frequency and energy states of an electron in an atomic orbit or of a wave on a string are quantized. Enhf= , n= 0,1,2,3,... v n f = n −34 2l hxJs= 6.626 10 Interference Interference: Beats beats frequency = difference in frequencies Interference: Beats fffB =−21 ff+ f = 21 ave 2 Interference: Beats Particles & Waves •Spread Out in Space: NONLOCAL •Superposition: Waves add in space and show interference. •Do not have mass or Momentum •Waves transmit energy. •Bound waves have discreet energy states – they are quantized. •Localized in Space: LOCAL •Have Mass & Momentum •No Superposition: Two particles cannot occupy the same space at the same time! •Particles have energy. •Particles can have any energy. http://www.kettering.edu/~drussell/Demos.html.