Lecture Power Point Physics for Scientists and Engineers

Lecture Power. Point Physics for Scientists and Engineers, 3 rd edition Fishbane Gasiorowicz Thornton © 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials.

Chapter 15 Superposition and Interference of Waves

Main Points of Chapter 15 • The superposition principle • Interference • Coherence • Standing Waves • Beats • Spatial interference • Pulses • Fourier decomposition • Pulses and the uncertainty principle

15 -1 The Superposition Principle The sum of two solutions of the wave equation for the system is also a solution Interference: if more than one wave passing, displacement is arithmetic sum of individual displacements

15 -1 The Superposition Principle Some examples: red wave is the sum of blue and green in each

15 -2 Standing Waves through Interference • Assume wavelength, frequency, phase, and amplitude to be the same for two waves traveling in the same space – only difference is direction of wave motion • Write down each wave; add; find: (15 -6) This is the equation for a standing wave.

15 -2 Standing Waves through Interference Look at incoming wave on string that is tied to a wall: at the wall, the string cannot move. The wave reflects at this point and begins to travel in the opposite direction. This reflected wave is inverted, and interferes with the incoming wave to create a standing wave.

15 -2 Standing Waves through Interference Different stages in the cycle of a standing wave:

15 -3 Beats occur when two waves of almost the same frequency are superposed – the sum has an envelope with a much longer wavelength:

15 -3 Beats The beat frequency can be calculated, assuming that the differences in frequency and wavelength are small. Start with two waves: (15 -7) After calculation and approximation, we find: (15 -15) (15 -16)

15 -4 Spatial Interference Phenomena Any two- or three-dimensional wave can show spatial interference if the waves are coherent. Interference arises because of different path lengths from source(s).

15 -4 Spatial Interference Phenomena If path length difference is an integral number of wavelengths, interference is constructive. If path length difference is an integral number of wavelengths plus a half wavelength, interference is destructive.

15 -4 Spatial Interference Phenomena Constructive interference: (15 -17) Destructive interference: (15 -18)

15 -4 Spatial Interference Phenomena Geometrically, how can the minima and maxima be located? Here, S 1 and S 2 are the sources, and P is either a maximum or a minimum. The difference in path lengths can be written in terms of the angle θ, assuming that R>>d.

15 -4 Spatial Interference Phenomena Geometrically, how can the minima and maxima be located? For maxima: For minima: (15 -20) (15 -21)

15 -5 Pulses A pulse is a single waveform – rather than a continuing wave – traveling through a medium. Pulses obey the usual wave equation, which only requires that they be of the form Pulses traveling in opposite directions may collide; interference occurs as with wave trains.

15 -5 Pulses Collisions between Pulses

15 -5 Pulses Reflection: An incident pulse reflecting from a fixed point will be inverted.

15 -5 Pulses Reflection: An incident pulse reflecting from a movable point will be upright.

15 -5 Pulses Transmission: An incident pulse incident on a denser medium will have a reflection that is inverted.

15 -5 Pulses Transmission: An incident pulse incident on a lighter medium will have a reflection that is upright.

15 -6 Fourier Decomposition of Waves Fourier’s theorem: 1. any periodic wave can be approximated by a sum of sinusoidal waves with different frequencies 2. any pulse can be approximated by an integral of sinusoidal waves 3. General form of Fourier expansion: (15 -22)

15 -6 Fourier Decomposition of Waves Example: Triangular wave. Left: The first two terms in the Fourier expansion Right: The first term and the sum of the first two terms compared to the original waveform

15 -7 Pulses and the Uncertainty Principle The Fourier expansion of a wave pulse can be described in terms either of frequency or of wave number. If a pulse is limited in time or space, it is also limited in the range of angular frequencies or wave numbers that enter into it.

15 -7 Pulses and the Uncertainty Principle We can describe the wave as lasting a time Δt and having a range of frequencies Δf. There is a limit on their product: (15 -26)

15 -7 Pulses and the Uncertainty Principle Alternatively, we can describe the wave as having a width Δt and a range of wavelengths Δλ. A similar limit applies: (15 -27)

15 -7 Pulses and the Uncertainty Principle What these relationships mean is that the shorter in time a pulse is, the more frequencies needed to represent it; and the narrower in space it is, the more wave numbers it takes to represent it.

Summary of Chapter 15 • Superposition of two solutions of wave equation is also a solution • Waves having opposite algebraic signs at a particular point interfere destructively; waves having the same algebraic sign interfere constructively. • Standing waves can be considered to be the superposition of identical waves traveling in opposite directions.

Summary of Chapter 15, cont. • The superposition of two waves very close in frequency results in beats with frequency: (15 -16) • Waves of the same wavelength are coherent if there is a constant phase difference between them. They will interfere constructively or destructively depending on the path length difference from the sources.

Summary of Chapter 15, cont. • Waves reflect from and may be transmitted past boundaries • Sign of reflected wave depends on relative properties of media • Fourier analysis allows both periodic and nonperiodic waves to be described as superpositions of sinusoidal waves