Superposition of Waves

12 Key excerpts on "Superposition of Waves"

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  Foundations of Physics

  • Steve Adams(Author)
  • 2019(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  15

  Superposition Effects

  15.0   Superposition Effects

   

  Superposition is what happens when two or more waves are present at the same point. This might occur because they originate from different sources or because of reflection. If the waves are of the same type, the resultant disturbance at that point is the vector sum of the disturbances from each individual wave. This is called the principle of superposition. We can determine the effects of superposition graphically, by adding phasors or by calculation. Many important phenomena are linked to superposition, including interference and diffraction and the formation of standing (stationary) waves.

  15.1   Two-Source Interference

   

  If waves of the same type with equal wavelength, frequency, and amplitude are emitted from two sources placed a short distance apart (comparable to a few wavelengths), then a regular interference pattern is formed. The pattern consists of regions where the waves reinforce to produce maximum intensity (constructive interference) and regions where they cancel to produce minimum intensity (destructive interference).

  The famous double slit experiment carried out by Thomas Young in 1801 provided strong evidence for the wavelength of light and enabled Young to calculate its wavelength. A similar setup with sound can be used to demonstrate superposition patterns and, in a modified form, to create noise-cancelling headphones.

  In order for stable clear interference effects to be created, the two sources must be coherent.

  Coherent sources maintain a constant phase relationship.

  This means that they must be the same type of wave and have the same wavelength and frequency. The sources do not have to be in phase, but the phase difference between them must be constant. They must also have comparable amplitudes; if one wave has a much greater amplitude than the other, then variations in intensity will be hard to detect.

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  Optics for Materials Scientists

  • Myeongkyu Lee(Author)
  • 2019(Publication Date)
  • Apple Academic Press

    (Publisher)

  CHAPTER 3 Superposition of Waves In the previous chapter, we described various phenomena occurring when a light wave of a given amplitude, wavelength, and frequency passes from one medium to another. We are here interested in what happens when two or more waves are superposed at some point in space. Interference and diffraction, which will be treated in subsequent chapters, also result from the Superposition of Waves. Of course, the combined effects of two or more waves are influenced by the specifications (frequency, amplitude, phase, etc.) of each constituent wave. When harmonic waves of different amplitudes and phases but with the same frequency are combined, the composite wave is another harmonic wave having the same frequency. While the principle of superposition is equally applicable to waves differing in frequency, the resultant may not be expressed by a single harmonic wave. The Superposition of Waves with some range of frequencies leads to the concepts of coherence and bandwidth. The term coherence is used to describe the phase correlation of monochromatic waves. If the phase of any portion of a wave is predictable from any other portion of the wave, it is said to be perfectly coherent. Since there are no perfectly monochromatic sources, no light waves are perfectly coherent. A perfectly coherent wave expressible by a single harmonic wave (sine or cosine wave) of infinite extent is just an ideal case far from reality. A real wave exists only over a finite duration of time and can be represented as a sequence of harmonic wave trains of finite length. The average time duration and length of the wave trains are referred to as coherent time and length, respectively. For an electromagnetic wave, the coherent length is the distance over which a propagating wave may be considered coherent. A more monochromatic wave has a longer coherent length

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  Foundations of Physics

  • Steve Adams(Author)
  • 2023(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  15

  SUPERPOSITION EFFECTS

  15.0 SUPERPOSITION EFFECTS

  Superposition is what happens when two or more waves are present at the same point. This might occur because they originate from different sources or because of reflection. If the waves are of the same type the resultant disturbance at that point is the vector sum of the disturbances from each individual wave. This is called the “principle of superposition.” We can determine the effects of superposition graphically, by adding phasors or by calculation. Many important phenomena are linked to superposition including interference and diffraction and the formation of standing (stationary) waves.

  15.1 TWO-SOURCE INTERFERENCE

  If waves of the same type with equal wavelength, frequency, and amplitude are emitted from two sources placed a short distance apart (comparable to a few wavelengths) then a regular interference pattern is formed. The pattern consists of regions where the waves reinforce to produce maximum intensity (constructive interference) and regions where they cancel to produce minimum intensity (destructive interference).

  The famous double slit experiment carried out by Thomas Young in 1801 provided strong evidence for the wavelength of light and enabled Young to calculate its wavelength. A similar set up with sound can be used to demonstrate superposition patterns and, in a modified form to create noise-canceling headphones.

  In order for stable clear interference effects to be created the two sources must be coherent. Coherent sources maintain a constant phase relationship.

  This means that they must be the same type of wave, and have the same wavelength and frequency. The sources do not have to be in phase but the phase difference between them must be constant. They must also have comparable amplitudes; if one wave has a much greater amplitude than the other then variations in intensity will be hard to detect.

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  Let There Be Light

  The Story of Light from Atoms to Galaxies

  • Alex Montwill, Ann Breslin(Authors)
  • 2008(Publication Date)
  • ICP

    (Publisher)

  6.3 The Superposition of Waves

  6.3.1 The superposition principle

  The superposition principle states that the total displacement of any particle, simultaneously disturbed by more than one wave, is simply the linear sum of the displacements due to the individual waves.

  Raindrops. Courtesy of Piotr Pieranski.

  When droplets of rain fall on the surface of a pool, they create circular surface waves which expand and overlap one another. Each wave is unaffected by the presence of the others, and each independently displaces particles of water. To get the total displacement of any particle, we simply add the displacements due to individual waves.

  6.4 Applying the superposition principle

  6.4.1 The superposition of two waves travelling in the same direction

  Two identical sine waves travel in the same direction: Waves in phase: the individual waves combine to give a wave with a total amplitude twice that of either wave. Waves out of phase: the individual waves combine to give a total amplitude of zero.

  6.4.2 Path difference and phase difference

  If two sources emit periodic waves in phase, the total amplitude of the disturbance at any point where the waves overlap depends on the phase difference between them.

  This phase difference depends on how far each wave has travelled from its source. Waves from the two sources will be in phase provided that the difference in the length of the path is zero or some whole number of wavelengths (λ, 2λ, 3λ, etc.). The waves will be completely out of phase if the path difference is one half wavelength or any odd number of half wavelengths , as illustrated in Figure 6.5 .

  Figure 6.5 Path difference and phase difference.

  6.4.3 When two waves travelling in opposite directions meet

  If a transverse pulse is sent down a string tied at one end, it will be reflected and come back upside down. This is according to Newton's third law of motion, which states that action and reaction at the point of reflection are opposite. (The fact that it is reflected upside down is not particularly important as far as the argument that follows is concerned; what is more relevant is that the pulse comes back with the same speed with which it was sent.) If the support is rigid, very little energy is absorbed and the amplitude of the pulse will not be significantly diminished.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  513 CHAPTER 17 Each raindrop that strikes the water’s surface creates waves that propagate outward in a circular pattern. When two or more of these sets of waves meet on the surface, they overlap, or interfere with each other, in a process called linear superposition. Where the waves meet crest-to-crest or trough- to-trough, the amplitude of the resultant wave increases, and where the waves meet trough-to-crest, the amplitude of the resultant wave decreases. These phenomena are known as constructive and destructive interference, respectively, and they are two important topics in this chapter. The Principle of Linear Superposition and Interference Phenomena LEARNING OBJECTIVES After reading this module, you should be able to... 17.1 Express the principle of linear superposition. 17.2 Solve spatial interference problems for sound waves. 17.3 Apply wave interference concepts to the diffraction of sound waves. 17.4 Explain beats as a wave interference phenomenon. 17.5 Analyze transverse standing waves. 17.6 Analyze longitudinal standing waves. 17.7 Define the harmonic content of complex sound waves. Sandid/1642 images/Pixabay 17.1 The Principle of Linear Superposition Often, two or more sound waves are present at the same place at the same time, as is the case with sound waves when everyone is talking at a party or when music plays from the speakers of a stereo system. To illustrate what happens when several waves pass simultaneously through the same region, 514 CHAPTER 17 The Principle of Linear Superposition and Interference Phenomena let’s consider Animated Figures 17.1 and 17.2, which show two transverse pulses of equal heights moving toward each other along a Slinky. In Animated Figure 17.1 both pulses are “up,” whereas in Animated Figure 17.2 one is “up” and the other is “down.” Part a of each figure shows the two pulses beginning to overlap. The pulses merge, and the Slinky assumes a shape that is the sum of the shapes of the individual pulses.

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  Interferometry

  Recent Developments and Contemporary Applications

  • Mithun Bhowmick, Bruno Ullrich, Mithun Bhowmick, Bruno Ullrich(Authors)
  • 2019(Publication Date)
  • IntechOpen

    (Publisher)

  2. Differentiating the superposition principle from the measurable superposition effect 2.1 Formulating the basic superposition equation The most neglected issue in current books and literature is that the co-propagating and cross-propagating wave amplitudes pass through each other completely unperturbed (uninfluenced) by each other ’ s presence in the absence of interacting materials. In other words, the superposed wave fronts by themselves do not generate observable interferometric fringes whenever they are superposed. This is noninteraction of wave (NIW) amplitudes [1]. Alhazen observed this phenome-non almost 1000 years ago using a set of candles and a pinhole camera [2]. Huygens underscored this in his book [3] around 1667. This is why it is important to remem-ber that even quantum electrodynamics acknowledges that photon-photon interac-tion cross section is immeasurably small [4]. In fact, Dirac mathematically found that “ different photons do not interfere with each other ” (NIW?). Unfortunately, he introduced the noncausal notion that “ a photon interferes only with itself ” [5]. This assertion is noncausal because interference fringes always appear as some 32 Interferometry -Recent Developments and Contemporary Applications physical transformation in a detector induced by more than one amplitude signals carrying different phase information. Further, the dark fringes are not due to nonarrival of “ photons. ” It is because the joint stimulations by the out-of-phase E-vectors (only when equal amplitude!) fail to stimulate the detecting dipoles, and hence, the field energy cannot enter into the detecting dipoles ’ quantum cups . We always represent the superposition equation by two separate amplitude terms, each containing its own phase factor representing separate and independent oscillations of the E-vectors. A single stable elementary particle (here, a “ photon ” ) could not be multivalued in its critical dynamic parameters at any single moment.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 17The Principle of Linear Superposition and Interference Phenomena

  This performer is playing a wind instrument known as a didgeridoo, which is thought to have originated in northern Australia at least 1500 years ago and has been likened to a natural wooden trumpet. The didgeridoo and virtually all musical instruments produce sound in a way that involves the principle of linear superposition. All of the topics in this chapter are related to this principle.

  LEARNING OBJECTIVES

  After reading this module, you should be able to…

  • 17.1  Express the principle of linear superposition.
  • 17.2  Solve spatial interference problems for sound waves.
  • 17.3  Apply wave interference concepts to the diffraction of sound waves.
  • 17.4  Explain beats as a wave interference phenomenon.
  • 17.5  Analyze transverse standing waves.
  • 17.6  Analyze longitudinal standing waves.
  • 17.7  Define the harmonic content of complex sound waves.

  17.1 The Principle of Linear Superposition

  Often, two or more sound waves are present at the same place at the same time, as is the case with sound waves when everyone is talking at a party or when music plays from the speakers of a stereo system. To illustrate what happens when several waves pass simultaneously through the same region, let's consider Animated Figures 17.1 and 17.2 , which show two transverse pulses of equal heights moving toward each other along a Slinky. In Animated Figure 17.1 both pulses are “up,” whereas in Animated Figure 17.2 one is “up” and the other is “down.” Part a of each figure shows the two pulses beginning to overlap. The pulses merge, and the Slinky assumes a shape that is the sum of the shapes of the individual pulses. Thus, when the two “up” pulses overlap completely, as in Animated Figure 17.1 b, the Slinky has a pulse height that is twice the height of an individual pulse. Likewise, when the “up” pulse and the “down” pulse overlap exactly, as in Animated Figure 17.2 b, they momentarily cancel, and the Slinky becomes straight. In either case, the two pulses move apart after overlapping, and the Slinky once again conforms to the shapes of the individual pulses, as in part c

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  Sound & Hearing

  A Conceptual Introduction

  • R. Duncan Luce(Author)
  • 2013(Publication Date)
  • Psychology Press

    (Publisher)

  PART V

  Descriptive Physics of Complex Sounds

         

  This part concerns how several pure tone waves combine physically to form more complex sounds, such as notes with overtones and chords in music and, potentially, even speech. Various aspects of their interaction play an important role in our hearing. But, there is much about the hearing of complex sounds that cannot be accounted for by the physics of the situation or by that knowledge coupled with what we know about the peripheral nervous system.

  1. SUPERPOSITION

  1.1 Basic Principle of Superposition

  If we have two or more simultaneous waveforms, then at each point in space and time a very simple superposition principle holds, namely amplitudes add. For sound waves, this means that the pressures add. This result is a consequence of the simple principle that if two forces are applied in the same direction, the resultant force is their sum.

  1.1.1 Pulses . A simple example of such Superposition of Waves can be demonstrated with a rope or wire. Suppose it is held taut at opposite ends by two people and they simultaneously snap it in the same direction, each thereby initiating an identical one-sided pulse. So the two pulses are moving in opposite directions on the wire. Figure V.1 illustrates several “snapshots” of the situation. One sees that as the pulses come together, the one can be thought of as climbing over the other, with the amplitudes adding, and then they come apart and continue on their separate ways.

  FIG. V.1 Two positive pulses of identical size and shape propagate along a wire toward each other at time t 1 reaching each other at time t 2 , then superimposing so that at time t 3 the two amplitudes simply add, and at time t 4 they are moving away from each other.

  Consider what happens if the two people snap the string to the same degree, but in opposite directions. One creates a pulse with a positive amplitude and the other one with an equal, but negative, amplitude, as in Fig. V.2

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  The curves are labeled by their intensity levels at 1000 Hz and are known as Fletcher-Munson curves. Concept Summary 17.1 The Principle of Linear Superposition The principle of linear superposition states that when two or more waves are present simultaneously at the same place, the resultant disturbance is the sum of the disturbances from the individual waves. 17.2 Constructive and Destructive Interference of Sound Waves Constructive interference occurs at a point when two waves meet there crest-to-crest and trough-to-trough, thus reinforcing each other. Destruc- tive interference occurs when the waves meet crest-to-trough and cancel each other. When waves meet crest-to-crest and trough-to-trough, they are exactly in phase. When they meet crest-to-trough, they are exactly out of phase. For two wave sources vibrating in phase, a difference in path lengths that is zero or an integer number (1, 2, 3, . . .) of wavelengths leads to construc- tive interference; a difference in path lengths that is a half-integer number ( 1 2 , 1 1 2 , 2 1 2 , . . .) of wavelengths leads to destructive interference. For two wave sources vibrating out of phase, a difference in path lengths that is a half-integer number ( 1 2 , 1 1 2 , 2 1 2 , . . .) of wavelengths leads to constructive interference; a difference in path lengths that is zero or an integer number (1, 2, 3, . . .) of wavelengths leads to destructive interference. 17.3 Diffraction Diffraction is the bending of a wave around an obstacle or the edges of an opening. The angle through which the wave bends depends on the ratio of the wavelength  of the wave to the width D of the opening; the greater the ratio /D, the greater the angle. When a sound wave of wavelength  passes through an opening, the first place where the intensity of the sound is a minimum relative to that at the center of the opening is specified by the angle .

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  Mechanical and Electromagnetic Vibrations and Waves

  • Tamer Bécherrawy(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  Chapter 2Superposition of Harmonic Oscillations, Fourier Analysis

  The oscillations of a system can never be exactly periodic since they always have a beginning and an end. However, the oscillations may be treated as approximately periodic if they last a very long time, compared to the period of a single oscillation. On the other hand, even if the oscillations of a system are approximately periodic, they are never exactly simple harmonic (i.e. represented by a sinusoidal function with a single frequency). For instance, even if light is a single line of the discrete atomic spectrum or a laser beam, it is always a superposition of monochromatic waves in a more or less wide band. The superposition of oscillations and waves is so real and important that it is often raised to the rank of a principle (called the superposition principle). It plays a very important part in the study of interference, diffraction and quantum mechanics. The validity of this principle relies on the linearity of the mechanics and electromagnetism equations.

  Our purpose in this chapter is to study the superposition of simple harmonic oscillations. The Fourier analysis considers any function as a superposition of simple harmonic functions. We study the case of periodic functions and of non-periodic functions, namely signals of short duration.

  2.1. Superposition of two scalar and isochronous simple harmonic oscillations

  Consider the oscillations

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  Principles of Engineering Physics 1

  • Md Nazoor Khan, Simanchala Panigrahi(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Interference 2.1 Introduction In Section 1.16 of the previous chapter we learned that two beams of light waves can cross each other without either one producing any modification on the other after passing beyond the region of crossing. However, from the concepts explained in Section 1.16.2, we expect some modifications in the amplitudes or intensity (since intensity ∝ amplitude 2 ) of the two waves inside the region of crossing. The intensity of the resultant wave becomes a function of the position of the point. At certain points intensity is maximum and at other points it is minimum. In other words, we say that the two waves interfere with each other inside the region of crossing. This modification of intensity obtained by the superposition of two or more beams of light waves is called interference of light. The phenomenon of interference of light complements the validity of the concept that light is a wave. As a result of the short wavelength and disordered phase relationships of the interfering light waves, the interference pattern is not visible to the naked eye without special arrangements. It was in the year 1801 that Thomas Young for the first time demonstrated the interference of sunlight experimentally. Before discussing the interference phenomenon, let us discuss Huygens’ principle, a helpful tool and an early concept in favour of the wave theory of light when the scientific world was mesmerized by Newton’s corpuscular theory of light. 2.2 Huygens’ Principle Huygens, a Dutch mathematician, in 1678, propounded a theory regarding the propagation of light wave in any medium. According to this theory, light is a sort of disturbance in the medium in which it propagates in all direction from a point source. To explain the propagation of light in vacuum, he postulated an all-pervading medium called ‘ether’ (Later on, in the year 1881, Michelson and Morley, American scientists, performed a

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  Sneaking a Look at God's Cards

  Unraveling the Mysteries of Quantum Mechanics - Revised Edition

  • Giancarlo Ghirardi, Gerald Malsbary(Authors)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  79 C H A P T E R F O U R The Superposition Principle and the Conceptual Structure of the Theory The assumption of superposition relationships between the states leads to a mathematical theory in which the equations that define a state are linear in the unknowns. In consequence of this, people have tried to establish analogies with systems in classical mechanics, such as vibrating strings or membranes, which are governed by linear equations and for which, therefore, a superposition principle holds. Such analogies have led to the name “Wave Mechanics” being sometimes given to quantum mechanics. It is important to remember, however, that the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory, as is shown by the fact that the quantum superposition principle demands indeterminacy in the results of observations in order to be capable of a sensible physical interpretation. The analogies are thus liable to be misleading. —Paul Adrien Maurice Dirac W e can now deepen our analysis of the most innovative point of the new theory: the superposition principle. In the preceding chapters we have shown that the formal structure of the theory is such as to permit the “sum-mation” of quantum states. In particular, our attention has been drawn to the fact that, for instance, the polarization state |45° > of a photon is the “sum” of the states |V > and |H > (for “vertical” and “horizontal,” respec- tively). Analogously, it was asserted that the state of a particle with upward spin along the direction of the x axis is the “sum” of the states correspond-ing to upward and downward spin along the z axis.

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