Physics
Periodic Wave
A periodic wave is a repetitive disturbance that travels through a medium, characterized by a regular pattern of oscillations. It exhibits a consistent cycle of motion, with the wave shape repeating at regular intervals. Periodic waves are fundamental to understanding the behavior of various phenomena, such as sound, light, and water waves.
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10 Key excerpts on "Periodic Wave"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
If the spatial period of this wave is referred to as its wavelength, then during every period, one wavelength of the wave passes the observer. If the wave propagates with unchanging shape and the velocity in the medium is uniform, this period implies the wavelength is: ________________________ WORLD TECHNOLOGIES ________________________ Near-Periodic Waves over shallow water have sharper crests and flatter troughs than those of a sinusoid. This duality of space and time is expressed mathematically by the fact that the wave's behavior does not depend independently on position x and time t , but rather on the combination of position and time x − vt . A wave's amplitude u is then expressed as u ( x − vt ) and in the case of a periodic function u with period λ , that is, u ( x + λ − vt ) = u ( x − vt ), the periodicity of u in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ . In a similar fashion, this periodicity of u implies a periodicity in time as well: u ( x − v(t + T) ) = u ( x − vt ) using the relation vT = λ described above, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ / v . Traveling waves with non-sinusoidal wave shapes can occur in linear dispersionless media such as free space, but also may arise in nonlinear media under certain circum-stances. For example, large-amplitude ocean waves with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium. An example is the cnoidal wave, a periodic traveling wave named because it is described by the Jacobian elliptic function of m -th order, usually denoted as cn (x; m) .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
A wave's amplitude u is then expressed as u ( x − vt ) and in the case of a periodic function u with period λ , that is, u ( x + λ − vt ) = u ( x − vt ), the periodicity of u in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ . In a similar fashion, this periodicity of u implies a periodicity in time as well: u ( x − v(t + T) ) = u ( x − vt ) using the relation vT = λ described above, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ / v . Traveling waves with non-sinusoidal wave shapes can occur in linear dispersionless media such as free space, but also may arise in nonlinear media under certain circum-stances. For example, large-amplitude ocean waves with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium. An example is the cnoidal wave, a periodic traveling wave named because it is described by the Jacobian elliptic function of m -th order, usually denoted as cn (x; m) . Envelope waves The term wavelength is also sometimes applied to the envelopes of waves, such as the traveling sinusoidal envelope patterns that result from the interference of two sinusoidal waves close in frequency; such envelope characterizations are used in illustrating the derivation of group velocity, the speed at which slow envelope variations propagate. ________________________ WORLD TECHNOLOGIES ________________________ Wave packets A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves. Localized wave packets, bursts of wave action where each wave packet travels as a unit, find application in many fields of physics; the notion of a wavelength also may be applied to these wave packets.
eBook - PDFElectronic Properties of Crystalline Solids
An Introduction to Fundamentals
- Richard Bube(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Chapter i Classical Waves: A Review O n e of the unifying themes that runs t h r o u g h o u t physical p h e n o m e n a of m a n y different kinds is that of wave motion. S o u n d waves a n d light waves are part of the subject matter of classical physics, a n d a classical type of wave mechanics is sufficient for at least a partial treatment of the interaction of these kinds of wave with crystalline solids. M a n y of the properties of these classical wave systems are found also in a variety of forms in q u a n t u m wave mechanics. It is primarily for this reason that we begin with a brief review of the classical treatment of waves suitable for a partial description of sound a n d light. 1.1 General Properties of Waves A wave is any periodic disturbance in time a n d position, characterized by a velocity, a wavelength, a n d a frequency. Such a disturbance m a y have quite a general distribution in space as long as these characteristics are definable. T h e relationship between velocity, wavelength, a n d frequency is ν = λν (1.1) Values of these parameters vary widely for different types of wave. Representative values are indicated in Table 1.1. O n e of the simplest a n d most analytically useful wave forms is that of a h a r m o n i c wave which can be represented in terms of sine a n d cosine functions. A n y general wave form can be expressed in terms of a Fourier 1 2 / Classical Waves: A Review T A B L E 1.1 TYPICAL VALUES OF W A V E PARAMETERS Ψ Velocity, ν (cm/sec) Wavelength, λ (cm) Frequency, ν (sec -1 ) Sound (in air) 3 χ 10 4 1 to 3 χ 10 3 10 to 2 χ 10 4 Visible light (in vacuum) 3 χ 10 10 4 x 10 5 to 7 x 10 5 10 15 Free electron (300°K) 5 χ 10 6 8 χ 10-7 6 x 10 12 expansion of ha rmo n i c waves. A h a r m o n i c wave can be expressed mathe-matically in the following kinds of relationship between the displacement ξ, the angular frequency ω = 2πν, and the wavenumber k = 2π/λ.
eBook - PDFFrom Atoms to Galaxies
A Conceptual Physics Approach to Scientific Awareness
- Sadri Hassani(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
Frequency The number of cycles a simple harmonic oscillator undergoes in one second. Frequency is the inverse of the period. Hertz The unit of frequency. Huygens’ Principle states that the motion of a wave can be determined by assuming that each wave front is composed of infinitely many point sources each producing spherical waves. Interference A property of waves in which two specially prepared sources (coherent sources) construct a pattern at some points of which the waves oscillate with double ampli-tude (constructive interference) and at other points the wave disappears (destructive interference). Longitudinal Wave A wave for which the medium oscillates along the direction of wave motion. Oscillation A motion that repeats itself. Period The time required for a simple harmonic oscillator to return to its “original” posi-tion, which could be any position during the course of its motion. Polarization A property of transverse waves whereby certain materials block the wave when held in a certain orientation in front of the wave, and allow the wave to pass when rotated 90 degrees from the blocking orientation. Pulse A single disturbance that travels in a medium. Simple Harmonic Motion (SHM) An oscillatory motion described mathematically in terms of trigonometric functions. A mass attached to one end of a spring while the other end is held fixed, describes a simple harmonic motion when the mass is displaced slightly and then released. Simple Harmonic Oscillator (SHO) An object undergoing simple harmonic motion. Simple Harmonic Wave (SHW) A wave produced in a medium whose source undergoes a simple harmonic motion. Superposition The property of waves whereby two waves reaching a single point add to give the oscillation of the medium at that point. Transverse Wave A wave for which the medium oscillates perpendicular to the direction of wave motion. Wave A continuous succession of pulses traveling in a medium.
eBook - PDF- Md Nazoor Khan, Simanchala Panigrahi(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
2 The waves move with a velocity depending upon the properties of the medium. The waves remain stationary and do not move. 3 Each particle of the medium executes periodic motion about their mean position with the same amplitude. Except the node, all the particles of the medium execute SHO with varying amplitude. 4 There is a continuous change of phase from particle to particle. All the particles between two consecutive nodes are at the same phase, but differ in phase by p from those in the preceding as well as succeeding similar segments. 5 At any instant all the particles do not come together in the mean position, they pass their mean position in succession but with the same velocity. All the particles pass their mean position at a time, but with different velocities. Oscillations and Waves 57 6 Each particle of the medium undergoes similar change of pressure and density There is no change of pressure and densities at the antinodes while there is maximum change of pressure and densities at the nodes. 7 There is transmission of energy across every plane in the direction of propagation of waves. There is no flow of energy across any plane. 8 A complete wavelength contains a compression and rarefaction in the case of longitudinal waves and crest and trough in the case of transverse waves. The wavelength is the distance between two alternate nodes and anti nodes. 9 Compression and rarefaction move from point to point throughout the medium. The compression and rarefaction do not move from point to point; they simply appear at and disappear at certain equidistance fixed points. 10 No particle of the medium is permanently at rest.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
In Figures 16.2 and 16.3 the repetitive patterns occur as a result of the simple harmonic motion of the left end of the Slinky, so that every segment of the Slinky vibrates in simple harmonic motion. Sections 10.1 and 10.2 discuss the simple harmonic motion of an object on a spring and introduce the concepts of cycle, ampli- tude, period, and frequency. This same terminology is used to describe Periodic Waves, such as the sound waves we hear (discussed later in this chapter) and the light waves we see (discussed in Chapter 24). Figure 16.5 uses a graphical representation of a transverse wave on a Slinky to review the terminology. One cycle of a wave is shaded in color in both parts of the drawing. A wave is a series of many cycles. In part a the vertical position of the Slinky is plotted on the vertical axis, and the corresponding distance along the length of the Slinky is plotted on the horizontal axis. Such a graph is equivalent to a photograph of the wave taken at one instant in time and shows the disturbance that exists at each point along the Slinky’s length. As marked on this graph, the amplitude A is the maximum excursion of a particle of the medium (i.e., the Slinky) in which the wave exists from the particle’s undisturbed position. The amplitude is the distance between a crest, or highest point on the wave pattern, and the undisturbed position; it is also the distance between a trough, or lowest point on the wave pattern, and the undisturbed position. The wavelength l is the horizontal length of one cycle of the wave, as shown in Figure 16.5a. The wavelength is also the horizontal distance between two successive crests, two successive troughs, or any two successive equivalent points on the wave. Part b of Figure 16.5 shows a graph in which time, rather than distance, is plot- ted on the horizontal axis. This graph is obtained by observing a single point on the Slinky.
eBook - PDF- Julian L. Davis(Author)
- 2021(Publication Date)
- Princeton University Press(Publisher)
C H A P T E R O N E Physics of Propagating Waves INTRODUCTION In this chapter we shall discuss the physics of propagating waves, starting with simple physical models and then giving an elementary combined physical and mathematical treatment of waves traveling in continuous media. A mathematical treatment is reserved for subsequent chapters. For our purposes a continuous medium is one in which there is a continuous distribution of matter in the sense that a differential volume of material (in the mathematical sense) has the same properties as the material in the large. This means that molecular and crystalline struc-tures are neglected. It is known that electromagnetic (EM) waves travel in a vacuum with the speed of light. (A vacuum is a continuous medium with zero density of matter.) A propagating medium involves oscillations of the material through which the wave travels, with a wave velocity characteristic of the material and the temperature. For example, sound waves travel with a wave velocity that depends on the temperature and the density of the medium (air or fluid). For EM waves traveling in a vacuum, we have an oscillating electric intensity vector and an oscillating magnetic intensity vector normal to the electric vector. DISCRETE WAVE-PROPAGATING SYSTEMS Although the main thrust of this book is a treatment of waves traveling in continuous media, it is useful to construct a physical model composed of a discrete set of oscillating masses coupled by springs. We shall neglect friction. The limit as the number of masses and springs becomes infinite in a finite region yields a continuous medium. The simplest oscillating system consists of a spring fixed at one end and coupled to a mass. The small-amplitude oscillations of the mass
eBook - PDFLet There Be Light: The Story Of Light From Atoms To Galaxies (2nd Edition)
The Story of Light from Atoms to Galaxies
- Alex Montwill, Ann Breslin(Authors)
- 2013(Publication Date)
- ICP(Publisher)
Chapter 6 Introducing Waves What is a wave? Waves are so diverse that we might tend to regard different types of waves as separate entities, unrelated to one another. Do events such as the devastation of a region by an earthquake and the melodic tones of a musical instrument have anything in common? In this chapter we explore the com-mon ground between these and many other phenomena. Waves carry energy from one place to another. They also act as messengers, transmitting information. We look at different types of waves, how they are created, what they are ‘made of ’ and how they carry out their functions of carrier and messenger. To deal with some aspects of wave behaviour, we need to express the properties of the waves mathematically. We can then make quantitative predictions regarding many wave-related phe-nomena seen in nature. In a mathematical analysis of waves, it is most convenient to use the simplest type of wave form — a continuous sine wave. It is gratifying, if rather startling, to find that even the most complicated Periodic Waves can be constructed simply by adding a number of these sine waves. 6.1 Waves — the basic means of communication When talking about waves , we most likely picture a seaside scene with ocean waves approaching the shore or perhaps a ship at sea tossed in a storm. A survey asking the question ‘What serves as the most common means of communication?’ would be unlikely to favour the answer ‘ Waves ’. Yet, heat and light from 149 150 Let There Be Light 2nd Edition the sun are carried by waves. We could neither see nor speak to one another were it not for light waves and sound waves. Even the sensation of touch relies on the transmission of nerve impulses, which are composed of wave packets. Electromagnetic waves, which pervade all space, and with-out which the universe could not exist, are the basic theme of this book. Light is just one member of that family. These waves propagate in a mysterious way, as we shall see in later chapters.
eBook - PDFOcean Waves and Oscillating Systems: Volume 8
Linear Interactions Including Wave-Energy Extraction
- Johannes Falnes, Adi Kurniawan(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
In the present chapter, we shall consider waves which vary sinusoidally with time. Such waves are called ‘harmonic’ or ‘monochromatic’. When daling with sea waves, they are also characterised as ‘regular’ if their time variation is sinusoidal. Let p = p(x, y, z, t) = Re ˆ p(x, y, z) e iωt (3.1) represent a general harmonic wave, and let p denote the dynamic pressure in a fluid. (The total pressure is p tot = p stat + p, where the static pressure p stat is independent of time.) The complex pressure amplitude ˆ p is a function of the spatial coordinates x, y and z. For a plane acoustic wave propagating in a direction x, we have p = p(x, t) = Re Ae i (ωt−kx) + Be i (ωt+kx) , (3.2) where k = 2π/λ is the angular repetency (wave number) and λ is the wave- length. The first and second terms represent waves propagating in the positive and negative x direction, respectively. Assume that an observer moves with a velocity v p = ω/k in the positive x direction. Then he or she will experience a constant phase (ωt − kx) of the first right-hand term in Eq. (3.2). If the observer moves with same speed in the opposite direction, he or she will experience a constant phase (ωt + kx) of the last term in Eq. (3.2). For this reason, v p = ω/k is called the phase velocity. At a certain instant, the phase is constant on all planes perpendicular to the direction of wave propagation. For this reason, the wave is called plane. In contrast, an acoustic wave p(r, t) = Re (C/r) e i (ωt−kr) (3.3) radiated from a spherical loudspeaker in open air may be called a spherical wave, because the phase (ωt − kr) is the same everywhere on an envisaged sphere with a radius r from the centre of the loudspeaker. Note that in this geometrical case, the pressure amplitude | C| /r decreases with the distance from the loudspeaker.
eBook - PDFWaves and Oscillations in Nature
An Introduction
- A Satya Narayanan, Swapan K Saha(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
Differences in frequency imply differences in wavelength and velocity. The superposition of these two waves, with wave crests moving at different speeds, exhibits period-ically large and small amplitudes. The resultant wave is expressed as U ( z, t ) = Ae i ( κ 1 z − ω 1 t ) + Ae i ( κ 2 z − ω 2 t ) = A e i { [( κ 1 − κ 2 ) / 2] z − [( ω 1 − ω 2 ) / 2] t } + e i { [( κ 2 − κ 1 ) / 2] z − [( ω 2 − ω 1 ) / 2] t } × e i [( κ 2 − κ 1 ) / 2] z − i [( ω 2 − ω 1 ) / 2] t = 2 A cos κ 1 − κ 2 2 z − ω 1 − ω 2 2 t e i { [( κ 1 + κ 2 ) / 2] z − [( ω 1 + ω 2 ) / 2] t } = 2 A cos( κ g z − ω g t ) e i (¯ κz − ¯ ωt ) (1.45) with ¯ κ = κ 1 + κ 2 2 , ¯ ω = ω 1 + ω 2 2 κ g = κ 1 − κ 2 2 , ω g = ω 1 − ω 2 2 (1.46) 22 Waves and Oscillations in Nature — An Introduction as the mean frequency and the mean wave number, respectively. FIGURE 1.9 : One-dimensional plot (in time domain) of the cosine factor of V ( z, t ) = 2 A cos(¯ κz − ¯ ωt ) cos( κ g z − ω g t ); the low frequency wave serves as an envelope modulating the high frequency wave. The dashed lines of this curve display the envelope of the resulting wave disturbance. By this transformation the wave is split into an amplitude factor slowly oscillating at ω 1 − ω 2 and a phase factor rapidly oscillating at ω 1 + ω 2 . Equa-tion (1.45) is interpreted as representing a plane wave of frequency, ¯ ω , and wavelength, 2 π/ ¯ κ , propagating in the z -direction. The amplitude of this wave varies with time and position, between 2 A and 0 (see Figure 1.9), exhibiting the phenomenon of beats. The beat is produced by the superposition of two equal amplitude harmonic waves of different frequency, which can be perceived as periodic variations in volume whose rate is the difference between the two frequencies.
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