Interference: Two Spherical Sources Superposition
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Interference: Two Spherical Sources 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! Pulsed Interference 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: y() t ( An sin2ƒ n t B n cos2ƒ) n t n Superposition of Sinusoidal Waves • Case 1: Identical, same direction, with phase difference (Interference) Both 1-D and 2-D waves. • 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 - wt) • y2 = A sin (kx - wt + f) • y = y1+y2 = 2A cos (f/2) sin (kx - wt + f/2) Resultant Amplitude Depends on phase: Spatial Interference Term Sinusoidal Waves with Constructive Interference y = y1+y2 = 2A cos (f/2) sin (kx - wt + f /2) • When f = 0, then cos (f/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 (f/2) sin (kx - wt + f /2) • When f = , then cos (f/2) = 0 – Also any even multiple of • 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 (f/2) sin (kx - wt + f /2) • When f is other than 0 or an even multiple of , the amplitude of the resultant is between 0 and 2A • The wave functions still add Superposition of Sinusoidal Waves y = y1+y2 = 2A cos (f/2) sin (kx - wt + f/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 (f/2) • The phase of the resultant wave is f/2 Constructive Destructive Interference Wave Interference y = y1+y2 = 2A cos (f/2) sin (kx - wt + f/2) f Resultant Amplitude: 2A cos 2 Constructive Interference: f 2nn , 0,1,2,3... Destructive Interference: f (2nn 1) , 0,1,2,3... Ch 18 HO Problem #1 y = y1+y2 = 2A cos (f/2) sin (kx - wt + f/2) 1-D Sound Wave Interference Superposition Sound Waves 2-D Wave Interference? P These two loudspeakers are in phase. They emit equal-amplitude sound waves with a wavelength of 1.0 m. At the point indicated, is the interference maximum constructive, perfect destructive or something in between? A. perfect destructive B. maximum constructive C. something in between These two loudspeakers are in phase. They emit equal-amplitude sound waves with a wavelength of 1.0 m. At the point indicated, is the interference maximum constructive, perfect destructive or something in between? A. perfect destructive B. maximum constructive C. something in between 2-D Phase Difference Different than 1-D You have to consider the Path Difference! v 22 f wt2 f t 2 t ( v t ) r 2-D Phase Difference at P: f is different from the phase difference f between the two source waves! f f21 f P 2 Phase Difference at P: f r Path Difference at P: r f 2 Spherically Symmetric Waves Intensity Quiet Min Loud Max Quiet Min Loud Max Contour Map of Interference Pattern of Two Sources Constructive or Destructive? (Identical in phase sources) P 2 Phase Difference at P: ff r 0 2 f (1 ) 2 Constructive! f Resultant Amplitude: 2A cos 2 Constructive Interference: r n, f 2 n , n 0,1,2,3... Destructive Interference: r (2 n 1) , f (2 n 1) , n 0,1,2,3... 2 Constructive or Destructive? (Source out of Phase by 180 degrees) P 2 Phase Difference at P: ff r 0 2 f (1 ) 3 Destructive! f Resultant Amplitude: 2A cos 2 Constructive Interference: r n, f 2 n , n 0,1,2,3... Destructive Interference: r (2 n 1) , f (2 n 1) , n 0,1,2,3... 2 In Phase or Out of Phase? A B Constructive or Destructive? A B The interference at point C in the figure at the right is A. maximum constructive. B. destructive, but not perfect. C. constructive, but less than maximum. D. perfect destructive. E. there is no interference at point C. The interference at point C in the figure at the right is A. maximum constructive. B. destructive, but not perfect. C. constructive, but less than maximum. D. perfect destructive. E. there is no interference at point C. Ch 18 HO Problem #2 2 Phase Difference at P: f r Path Difference at P: r f 2 Ch 18 HO Problem #3 You Try Reflected PULSE: Free End Bound End Reflected PULSE: Standing Waves Created by Boundary Conditions Standing Waves on Strings Standing Wave Standing Wave: Transverse Standing Wave Produced by the superposition of two identical waves moving in opposite directions. Standing Waves on a String Harmonics Standing Waves Superposition of two identical waves moving in opposite directions. y12 A sin ( kx - ww t) y A sin ( kx t) y (2 A sin kx )cosw t •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 There are three types of amplitudes used in describing waves – The amplitude of ythe individual(2 A sin waves, kx )cos A w t – 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 n xn0,1, 2 • An antinode occurs at a point of maximum displacement, 2A Antinode #5 Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 2 sin (x) cos (3t) where x is in m and t is in s. What is the distance (in m) between the first two antinodes? a. 8 b. 2 c. 4 y (2 A sin kx )cosw t d. 1 e. 0.5 Standing Waves . Intensity of a wave is proportional to the square of the amplitude: I A2. Intensity is maximum at points of constructive interference and zero at points of destructive interference. Slide 21-30 Standing WavesMode Numberon a String of Standing Waves . m is the number of antinodes on the standing wave. The fundamental mode, with m = 1, has 1 = 2L. The frequencies of the normal modes form a series: f1, 2f1, 3f1, … . The fundamental frequency f1 can be found as the difference between the frequencies of any two adjacent modes: f1 = f = fm+1 – fm. Below is a time-exposure photograph of the m = 3 standing wave on a string. Slide 21-46 QuickCheck 21.4 What is the mode number of this standing wave? A. 4. B. 5. C. 6. D. Can’t say without knowing what kind of wave it is. Slide 21-47 QuickCheck 21.4 What is the mode number of this standing wave? A. 4. B. 5. C. 6. D. Can’t say without knowing what kind of wave it is. Slide 21-48 Standing Waves on a String Harmonics Which harmonics (modes) are present on the string? The Fundamental and third harmonic. Standing Waves on a String Harmonics Standing Waves on a String 1 2L 2 L 2L 3 3 Standing Waves on a String 2L n n fvnn / v fn n 2L Standing Wave on a String T v vf v fn n 2L Longitudinal Standing Wave http://www.kettering.edu/~drussell/Demos.html Standing Sound Waves . Shown are the displacement x and pressure graphs for the m = 2 mode of standing sound waves in a closed-closed tube. The nodes and antinodes of the pressure wave are interchanged with those of the displacement wave. Slide 21-58 Standing Sound Waves Shown are displacement and pressure graphs for the first three standing-wave modes of a tube closed at both ends: Slide 21-60 Standing Sound Waves Shown are displacement and pressure graphs for the first three standing-wave modes of a tube open at both ends: Slide 21-61 Standing Waves in an Open Tube • Both ends are displacement antinodes • The fundamental frequency is v/2L – This corresponds to the first diagram • The higher harmonics are ƒn = nƒ1 = n (v/2L) where n = 1, 2, 3, … Standing Waves in a Tube Closed at One End • The closed end is a displacement node • The open end is a displacement antinode • The fundamental corresponds to ¼ • The frequencies are ƒn = nƒ = n (v/4L) where n = 1, 3, 5, … QuickCheck 21.6 An open-open tube of air has length L. Which is the displacement graph of the m = 3 standing wave in this tube? Slide 21-63 QuickCheck 21.6 An open-open tube of air has length L.