Physics
Superposition of Two Waves
The superposition of two waves refers to the phenomenon where two waves overlap and combine to form a new wave. This new wave is the sum of the individual displacements of the original waves at each point in space and time. Superposition is a fundamental principle in wave mechanics and is used to explain various wave phenomena, such as interference and diffraction.
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10 Key excerpts on "Superposition of Two Waves"
No longer available |Learn more- Steve Adams(Author)
- 2023(Publication Date)
- Mercury Learning and Information(Publisher)
15SUPERPOSITION EFFECTS15.0 SUPERPOSITION EFFECTSSuperposition 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 INTERFERENCEIf 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.
eBook - PDF- Charles A. Bennett(Author)
- 2022(Publication Date)
- Wiley(Publisher)
213 7 Superposition and Interference Twinkle, twinkle, little star. How I wonder what you are… Jane Taylor 7.1 Introduction In superposition, waves are combined, and when waves are combined they interfere. We begin by considering the superposition of harmonic monochromatic waves. The mathematical description of superposition is facilitated by the ideas of Fourier analysis, and we shall take time to develop these ideas fully. In doing so, we will discover that waves must be modeled as packets with a finite spread in frequency and wavelength and an associated group and phase velocity. There are many applications of interferometry that not only are technically useful but also reveal some of the most subtle aspects of electromagnetic waves and photons. Specific applications of thin film technology are discussed for both single and multiple layers. The effects of interference provide the foundation for nearly all aspects of physical optics, including effects of diffraction and image formation to be discussed subsequently. 7.2 Superposition of Harmonic Waves A wave is any function that satisfies the differential wave equation. As discussed in Section 1.5, the superposition principle guarantees that we may combine waves algebraically to give other valid wave solutions. Figure 7.1 illustrates the algebraic combination of two harmonic waves with equal amplitude, frequency, and wavelength; in other words, the two wave being combined differ only in phase. As discussed in Section 1.4, we may plot a harmonic traveling wave as a function of either position or time, so the horizontal axes of the plots of Figure 7.1 are not labeled to allow for either possibility. When waves are combined in phase, they combine to give a larger amplitude wave, and when they are 𝜋 rad out of phase, they tend to cancel.
eBook - ePub- 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
eBook - PDF- 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.
eBook - PDF- 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 .
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
In situations where waves with the same frequency overlap, we have seen how the principle of linear superposition leads to constructive and destructive interference and how it explains diffraction. We will see in this section that two overlapping waves with slightly different frequencies give rise to the phenomenon of beats. However, the principle of linear superposition again provides an explanation of what happens when the waves overlap. FIGURE 17.13 Two tuning forks have slightly different frequencies of 440 and 438 Hz. The phenomenon of beats occurs when the forks are sounded simultaneously. The sound waves are not drawn to scale. Destructive Small piece of putty 440 Hz 438 Hz Destructive Constructive Constructive A tuning fork has the property of producing a single‐frequency sound wave when struck with a sharp blow. Figure 17.13 shows sound waves com- ing from two tuning forks placed side by side. The tuning forks in the draw- ing are identical, and each is designed to produce a 440‐Hz tone. However, a small piece of putty has been attached to one fork, whose frequency is lowered to 438 Hz because of the added mass. When the forks are sounded simulta- neously, the loudness of the resulting sound rises and falls periodically — faint, then loud, then faint, then loud, and so on. The periodic variations in loudness are called beats and result from the interference between two sound waves with slightly different frequencies. CHAPTER 17 The principle of linear superposition and interference phenomena 461 For clarity, figure 17.13 shows the condensations and rarefactions of the sound waves separately. In reality, however, the waves spread out and overlap. In accord with the principle of linear superposition, the ear detects the combined total of the two. Notice that there are places where the waves interfere constructively and places where they interfere destructively. When a region of constructive interference reaches the ear, a loud sound is heard.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
17.1 | The Nature of Waves 405 17 | The 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. Chapter | 17 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. (a) Overlap begins (b) Total overlap; the Slinky has twice the height of either pulse (c) The receding pulses Figure 17.1 Two transverse “up” pulses passing through each other. A 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 Figures 17.1 and 17.2, which show two transverse pulses of equal heights moving toward each other along a Slinky. In Figure 17.1 both pulses are “up,” whereas in 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.
eBook - ePub- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
CHAPTER 17The Principle of Linear Superposition and Interference PhenomenaThis 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
- Tamer Bécherrawy(Author)
- 2013(Publication Date)
- Wiley-ISTE(Publisher)
Chapter 2Superposition of Harmonic Oscillations, Fourier AnalysisThe 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
eBook - PDF- 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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