Constructive Interference

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1 PHYSICS 0 Results Test 1 & Lecture 0 Waves > Mean +sd = 14.6 very good performance Mean = < mean - sd = not too good Lecture 0 Purdue University, Physics 0 1 Much of lower test average due to one problem 86% missed Constructive Interference Escape velocity from a planet v E means KE E = 1/ mv E is enough to overcome gravitational potential PE G and escape If an object falls to planet it picks up -PE G which turns into!ke which is same as KE E as gravity is a conserved field If object had an initial velocity v i far from planet it has initial KE i = 1/ mv i When it hits it has final KE = KE i +!KE 1/ mv f = 1/ mv i + 1/ mv E v f = sqrt(v i + v E ) The net disturbance is the sum of the individual disturbances. When waves cross in a point and have the same sign amplitude, they add constructively. Since intensity is proportional to amplitude squared, the resulted intensity is more than the sum of the intensities of individual waves. Lecture 0 Purdue University, Physics 0 4

2 Destructive Interference Interference The net disturbance is the sum of the individual disturbances. When waves cross in a point and have the opposite sign amplitude, they add destructively. Since intensity is proportional to amplitude squared, the resulted intensity is less than the sum of the intensities of individual waves. Lecture 0 Purdue University, Physics 0 5 Constructive In phase #=0 Out of phase by #=$ Fully Destructive Destructive Out of phase by #=(/3)$ Destructive Lecture 0 Purdue University, Physics 0 6 Coherence and Interference Two waves are coherent when their frequencies are same and they maintain a constant phase difference. A path difference!l of " corresponds to a phase difference # of $! = "L # $ Lecture 0 Purdue University, Physics 0 7 For constructive interference: Amplitudes add: A=A 1 +A For destructive interference: Interference Coherent waves may interfere constructively or destructively.! = m" or #L = m$ m = 0,1,,! $! = (m + 1)" or #L = m + 1 ' % & ( ) * m = 0,1,,! Amplitudes subtract: A= A 1 -A Lecture 0 Purdue University, Physics 0 8

3 Interference Interference The total amplitude (thus intensity) at each location depends on phase shift between two waves. Constructive: Destructive: y 1 = Acos(!t) y 1 = Acos(!t) y = Acos(!t) y = Acos(!t + " ) y = y 1 + y = Acos(!t) y = y 1 + y = Acos(!t) + Acos(!t + " ) Lecture 0 Purdue University, Physics 0 9 Maxima with twice the amplitude occur when phase shift between two waves is 0,!, 4!, 6! (or path difference is 0, ", " ) Minima with zero amplitude occur when phase shift between two waves is!, 3!, 5! (or path difference is 0, "/, 3"/ ) Lecture 0 Purdue University, Physics 0 10 Intensity Constructive interference: I > I 1 + I I = A Diffraction Diffraction is the spreading of a wave around an obstacle in its path. Destructive interference: I < I 1 + I Incoherent superposition: I = I 1 + I Lecture 0 Purdue University, Physics 0 11 Lecture 0 Purdue University, Physics 0 1

4 Reflection! When a wave strikes an obstacle or comes at the end of the medium it is traveling in, it is reflected (at least in part).! If the end of the rope is fixed the reflected pulse is inverted! If the end of the rope is free the reflected pulse is not inverted Why? Energy is conserved and a new wave has to form leading to a reflected wave The total motion at the end is then the supperposition of incoming and reflected wave At fixed end must have destructive interference to have no motion so phase shift for outgoing wave At free end don t have fixed point and phase is conserved Lecture 0 Purdue University, Physics 0 13 Reflection and Transmission When a wave travels from one boundary to another, reflection occurs. Some of the wave travels backwards from the boundary Traveling from fast to slow inverted Traveling slow to fast upright Reflection When the end of the rope is fixed to a support the pulse reaching the fixed end exerts a force (upward) on the support. The support exerts a force equal and opposite (downward) on the rope. This downward force generates the inversion that can be thought as a change of phase of 180. sin(!) = "sin(! + # ) Incoming outgoing reflected If the end of the rope is free it can overshoot. The overshoot exerts an upward pull on the rope and the reflected pulse is not inverted. Lecture 0 Purdue University, Physics 0 15 Lecture 0 Purdue University, Physics 0 16

5 Speed of Waves Depends on Medium When a wave reaches a boundary between two different media the speed and the wavelength change, but the frequency remains the same. Law of Refraction & 1 f = v 1! 1 = v! & sin! 1 sin! = v 1 v Lecture 0 Purdue University, Physics 0 17 Lecture 0 Purdue University, Physics 0 18! Shake the end of a string! If the other end is fixed, a wave travels down and it is reflected back % interference! The two waves interfere => standing wave! Nodes when sin (kx)=0: Standing Waves y(x,t) = Asin(!t + kx) y(x,t) =! Asin("t! kx) y(x,t) =!" Asin(kx) # $ cos(%t) x = n! k = n", n = 0,1,,!! Anti-nodes sin (kx)=+/- 1, half way between nodes nodes Anti-nodes Lecture 0 Purdue University, Physics 0 19 Standing Waves The natural frequencies are related to the length of the string L. The lowest frequency (first harmonic) has one antinode L =! 1 "! = L 1 The second harmonic has two antinodes! = L The n-th harmonic! n = n L Lecture 0 Purdue University, Physics 0 0

6 Standing Waves Demo 1S - 41 Fixed endpoints f = v/" Fundamental n=1 " n = L/n f n = n v / (L) f n = n f 1 No energy is transmitted by a standing wave Antinode and node at ends Lecture 0 Purdue University, Physics 0 1