Physics
Sinusoidal Wave
A sinusoidal wave is a type of wave that has a smooth, repetitive oscillation resembling the graph of a sine function. It is characterized by its periodic and uniform oscillation, with the amplitude, frequency, and phase determining its specific properties. Sinusoidal waves are fundamental in describing various natural phenomena and are widely used in physics to model wave behavior.
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eBook - PDFPrinciples of Biological Regulation
An Introduction to Feedback Systems
- Richard Jones(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
The sinusoidal method of analysis is not directly applicable to all systems, and thus is not a panacea. However, it is valid for many physical processes, and for those systems that are not completely amenable to sinusoidal analysis, it still provides an insight not obtainable in other ways (Stoll, 1969). Although many biological processes do in fact oscillate continually, this dis-cussion is not directed solely to that class of systems. The immediate application of sinusoidal methods will be to systems which either show no inherent oscillatory behavior, or if they do exhibit transient oscillations, the transient is rapidly damped out (Stark and Sherman, 1957). The sinusoid is the simplest form of a periodic function, which is a quantity that undergoes a cyclic change with time, and repeats that cyclic variation in-definitely. The sinusoid occupies a distinctive place in the study of periodic functions, that is, any periodic function can be described as the sum of a number of sinusoids. Thus the sinusoid is a fundamental building block in the description of periodic functions of arbitrary shape. 21300°= 2[-βΟ' (α) (b) Fig. 6-1. Directed line segment with its angular measure in (a) degrees, and (b) radians. amplitude, A angular velocity, ώ 2-rr cut X = Asincot = A [ J C J ] Ix'max = lAsincotl = A Fig. 6-2. A sinusoid, X = A sin ωί, and its phasor representation. The fact that JC is a sinusoidal quantity is shown formally by the functional notation, Α[ωί]. The period P is the time required for one cycle, and the frequency in hertz can be expressed by either of the following expressions, / = /P = ω/2π. 166 6 SINUSOIDAL SIGNALS Properties of a Sinusoid: 6.2 Figs. 6-1 and 6-2 a. A geometric interpretation of a sinusoid may be obtained with the aid of a directed line segment emanating from the origin of a coordinate system. Each line segment is defined by its length and the angle it makes with the horizontal axis.
eBook - PDF- Stanley A. Gelfand(Author)
- 2016(Publication Date)
- Thieme(Publisher)
The wave will move out horizontally, but the floating cork bobs up and down (vertically) at right angles to the wave. In contrast, sound waves are longitudinal waves because each air particle oscillates in the same direction in which the wave is propagating (Fig. 1.6) . Although sound waves are longitudi-nal, it is more convenient to draw them with a transverse representation, as in Fig. 1.6 . In such a diagram the vertical dimension represents some measure of the size of the signal (e.g., displace-ment, pressure, etc.), and left to right distance represents time (or distance). For example, the waveform in Fig. 1.6 shows the amount of positive pressure (compression) above the baseline, nega-tive pressure (rarefaction) below the baseline, and distance horizontally going from left to right. The Sinusoidal Function Simple harmonic motion is also known as sinusoidal motion , and has a waveform that is called a sinu-soidal wave or a sine wave . Let us see why. Fig. 1.7 shows one cycle of a sine wave in the center, sur-rounded by circles labeled to correspond to points on the wave. Each circle shows a horizontal line corre-sponding to the horizontal baseline on the sine wave, as well as a radius line (r) that will move around the circle at a fixed speed, much like a clock hand but in a counterclockwise direction. Point a on the waveform in the center of the figure can be viewed as the “starting point” of the cycle. The displacement here is zero because this point is on the horizontal line. The radius appears as shown in circle b when it reaches 45° of rotation, which corresponds to point b on the sine wave. The angle between the radius and the horizontal is called the phase angle ( θ ) and is a handy way to tell loca-tion going around the circle and on the sine wave. In other words, we consider one cycle (one “round trip” of oscillation) to be the same as going around a circle one time.
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)
The function which expresses the value of sin( θ ) in terms of θ is called the sine function and it is this basic function which repre-sents the physical properties of periodic waves. Generating the sine function The simplest sort of wave is described by a single sine function. If such a wave enters a medium which can transmit transverse waves, it displaces particles and sets them in motion. This The waves appear to be coming in but they do not bring in the sea! s l op i ng beac h waves w a t e r p a r t i c l e s A wave propagates particle oscillates x Introducing Waves 157 motion is transmitted to neighbouring particles resulting in the creation of a simple harmonic wave . If, at some instant, we were able to take a snapshot of the vibrating particles, it would show how the transverse displace-ment from equilibrium of successive particles along the string varies. We could then display this in the form of a graph, as illustrated on the blackboard, where y represents the transverse displacement. The result would have the profile of a sine wave. A second snapshot, taken a short time later, when the wave has advanced along the string, shows the profile displaced to the right. The displacement (a) is the phase difference between the two waves. An expression for a sine wave in motion So far our mathematical expression is limited; it represents a wave ‘frozen’ in time — a ‘snapshot’ of the wave at a given instant. 158 Let There Be Light 2nd Edition 6.3 The superposition of waves 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. 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 dis-places particles of water.
eBook - PDF- Stanley A. Gelfand, Lauren Calandruccio(Authors)
- 2022(Publication Date)
- Thieme(Publisher)
The wave will move out horizontally, but the floating cork bobs up and down (vertically) at right angles to the wave. In contrast, sound waves are longitudinal waves because each air particle oscillates in the same direction in which the wave is propagating (Fig. 1.6). Although sound waves are longitudinal, it is more convenient to draw them with a transverse represen- tation, as in Fig. 1.6. In such a diagram, the vertical dimension represents some measure of the size of the signal (e.g., displacement, pressure, etc.), and left to right distance represents time (or distance). For ex- ample, the waveform in Fig. 1.6 shows the amount of positive pressure (compression) above the baseline, negative pressure (rarefaction) below the baseline, and distance horizontally going from left to right. Fig. 1.5 Sound is initiated by transmitting the vibratory pattern of the sound source to nearby air particles, and then the vibratory pattern is passed from particle to particle as a wave. Notice how it is the pattern of vibration that is being transmitted, whereas each particle oscillates around its own average location. Fig. 1.6 Longitudinal and transverse representations of a sound wave. Wave- length (λ) is the distance covered by one replication (cycle) of a wave, and is most easily visualized as the distance from one peak to the next. 1 Acoustics and Sound Measurement 10 The Sinusoidal Function Simple harmonic motion is also known as sinusoi- dal motion, and has a waveform that is called a Sinusoidal Wave or a sine wave. Let us see why. Fig. 1.7 shows one cycle of a sine wave in the center, surrounded by circles labeled to correspond to points on the wave. Each circle shows a horizontal line corresponding to the horizontal baseline on the sine wave, as well as a radius line (r) that will move around the circle at a fixed speed, much like a clock hand but in a counterclockwise direction.
eBook - PDFIntroduction To Digital Signal Processing: Computer Musically Speaking
Computer Musically Speaking
- Tae Hong Park(Author)
- 2009(Publication Date)
- World Scientific(Publisher)
Chapter 4 SINE WAVES 1 Introduction We began the book with a brief introduction of the sine wave as a way to start getting our hands dirty and begin with something that a lot of us (hopefully) are probably familiar with in one way or another. In this chapter, we will revisit some of the concepts related to the sine wave along with the complex number system, which will lead us to the powerful Euler formula . We will later see the importance of this formula as a tool for dealing with trigonometric functions and harmonic oscillation, especially in Chap. 8 when we get to know Mr. Fourier. Towards the end of the chapter, we will learn one of the most simple, yet effective and economically successful sound synthesis algorithms, frequency modulation synthesis. We again conclude Chap. 4 with the introduction of some musical examples and composers who have used some of the concepts we cover here. 97 98 Introduction to Digital Signal Processing 2 Sinusoids Revisited At the beginning of Chap. 1, we started with the sinusoid and briefly discussed the three components that characterize it — amplitude, frequency, and initial phase as shown in Eq. (2.1). y ( t ) = A · sin(2 · π · f · t + φ ) (2.1) It is customary to denote the time varying part of Eq. (2.1) as ω — the radian frequency (rad/sec) where f is the frequency in Hertz. ω = 2 · π · f (2.2) The phase φ in the Eq. (2.1) describes the phase offset with respect to a reference point, and is more precisely referred to as the initial phase at reference point t = 0. In the above equation φ is a constant. To distinguish the initial phase from a time-variant version of the phase, the term instantaneous phase is used and can be expressed as shown in Eq. (2.3) [note that φ itself can be a function of time: φ ( t )].
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 3 Fundamental Physics Concepts 1. Wavelength Wavelength of a sine wave , λ, can be measured between any two points wi th the same phase, such as between crests, or troughs, or corresponding zero crossings as shown. In physics, the wavelength of a Sinusoidal Wave is the spatial period of the wave – the distance over which the wave's shape repeats. It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The concept can also be applied to periodic waves of non -sinusoidal shape. The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. ________________________ WORLD TECHNOLOGIES ________________________ Assuming a Sinusoidal Wave moving at a fixed wave speed, wavelength is inversely proportional to frequency: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. Examples of wave-like phenomena are sound waves, light, and water waves. A sound wave is a periodic variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are periodic variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary periodically in both lattice position and time. Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
A standing wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called nodes, and the wavelength is twice the distance between nodes. The wavelength, period, and wave velocity are related as before, if the stationary wave is viewed as the sum of two traveling Sinusoidal Waves of oppositely directed velocities. Mathematical representation Traveling Sinusoidal Waves are often represented mathematically in terms of their velocity v (in the x direction), frequency f and wavelength λ as: ________________________ WORLD TECHNOLOGIES ________________________ where y is the value of the wave at any position x and time t , and A is the amplitude of the wave. They are also commonly expressed in terms of (radian) wavenumber k ( 2π times the reciprocal of wavelength) and angular frequency ω ( 2π times the frequency) as: in which wavelength and wavenumber are related to velocity and frequency as: or Dispersion causes separation of colors when light is refracted by a prism The relationship between ω and λ (or k ) is called a dispersion relation. This is not generally a simple inverse relation because the wave velocity itself typically varies with frequency. ________________________ WORLD TECHNOLOGIES ________________________ Wavelength is decreased in a medium with higher refractive index In the second form given above, the phase ( kx − ωt ) is often generalized to ( k • r − ωt ), by replacing the wavenumber k with a wave vector that specifies the direction and wavenumber of a plane wave in 3-space, parameterized by position vector r . In that case, the wavenumber k , the magnitude of k , is still in the same relationship with wavelength as shown above, with v being interpreted as scalar speed in the direction of the wave vector. The first form, using reciprocal wavelength in the phase, does not generalize as easily to a wave in an arbitrary direction.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
A standing wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called nodes, and the wavelength is twice the distance between nodes. The wavelength, period, and wave velocity are related as before, if the stationary wave is viewed as the sum of two traveling Sinusoidal Waves of oppositely directed velocities. Mathematical representation Traveling Sinusoidal Waves are often represented mathematically in terms of their velocity v (in the x direction), frequency f and wavelength λ as: ________________________ WORLD TECHNOLOGIES ________________________ where y is the value of the wave at any position x and time t , and A is the amplitude of the wave. They are also commonly expressed in terms of (radian) wavenumber k ( 2π times the reciprocal of wavelength) and angular frequency ω ( 2π times the frequency) as: in which wavelength and wavenumber are related to velocity and frequency as: or Dispersion causes separation of colors when light is refracted by a prism The relationship between ω and λ (or k ) is called a dispersion relation. This is not generally a simple inverse relation because the wave velocity itself typically varies with frequency. ________________________ WORLD TECHNOLOGIES ________________________ Wavelength is decreased in a medium with higher refractive index In the second form given above, the phase ( kx − ωt ) is often generalized to ( k • r − ωt ), by replacing the wavenumber k with a wave vector that specifies the direction and wave-number of a plane wave in 3-space, parameterized by position vector r . In that case, the wavenumber k , the magnitude of k , is still in the same relationship with wavelength as shown above, with v being interpreted as scalar speed in the direction of the wave vector. The first form, using reciprocal wavelength in the phase, does not generalize as easily to a wave in an arbitrary direction.
eBook - PDF- Stanley A. Gelfand(Author)
- 2011(Publication Date)
- Thieme(Publisher)
Parameters of Sound Waves We already know that a cycle is one complete replication of a vibratory pattern. For example, two cycles are shown for each sine wave in the upper frame of Fig. 1.8 , and four cycles are shown for each sine wave in the lower frame. Each of the sine waves in this figure is said to be periodic because it repeats itself exactly over time. Sine waves are the simplest kind of peri-odic wave because simple harmonic motion is the simplest form of vibration. Later we will ad-dress complex periodic waves. The duration of one cycle is called its period . The period is expressed in time (t) because it re-fers to the amount of time that it takes to com-plete one cycle (i.e., how long it takes for one round trip). For example, a periodic wave that repeats itself every one hundredth of a second has a period of 1/100 second, or t = 0.01 second. One hundredth of a second is also ten thou-sandths of a second (milliseconds), so we could also say that the period of this wave is 10 mil-liseconds. Similarly, a wave that repeats itself every one thousandth of a second has a period of 1 millisecond or 0.001 second; and the pe-riod would be 2 milliseconds or 0.002 second if the duration of one cycle is two thousandths of a second. The number of times a waveform repeats itself in one second is its frequency (f ), or the number of cycles per second (cps) . We could say that frequency is the number of cycles that can fit Fig. .8 Each frame shows two sine waves that have the same frequency but different amplitudes. Compared with the upper frame, twice as many cycles occur in the same amount of time in the lower frame, thus the period is half as long and the frequency is twice as high. The Nature of Sound 1 into one second. Frequency is expressed in units called hertz (Hz) , which means the same thing as cycles per second.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
• A wave has a wavelength λ , which is the distance between adjacent identical parts of the wave. Wave velocity and wavelength are related to the wave’s frequency and period by v = λ T = λf . • Mechanical waves are disturbances that move through a medium and are governed by Newton’s laws. • Electromagnetic waves are disturbances in the electric and magnetic fields, and do not require a medium. • Matter waves are a central part of quantum mechanics and are associated with protons, electrons, neutrons, and other fundamental particles found in nature. • A transverse wave has a disturbance perpendicular to the wave’s direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation. 16.2 Mathematics of Waves • A wave is an oscillation (of a physical quantity) that travels through a medium, accompanied by a transfer of energy. Energy transfers from one point to another in the direction of the wave motion. The particles of the medium oscillate up and down, back and forth, or both up and down and back and forth, around an equilibrium position. • A snapshot of a Sinusoidal Wave at time t = 0.00 s can be modeled as a function of position. Two examples of such Chapter 16 | Waves 839 functions are y(x) = A sin ⎛ ⎝kx + ϕ ⎞ ⎠ and y(x) = A cos ⎛ ⎝kx + ϕ ⎞ ⎠. • Given a function of a wave that is a snapshot of the wave, and is only a function of the position x, the motion of the pulse or wave moving at a constant velocity can be modeled with the function, replacing x with x ∓ vt . The minus sign is for motion in the positive direction and the plus sign for the negative direction. • The wave function is given by y(x, t) = A sin ⎛ ⎝kx − ωt + ϕ ⎞ ⎠ where k = 2π/λ is defined as the wave number, ω = 2π/T is the angular frequency, and ϕ is the phase shift. • The wave moves with a constant velocity v w , where the particles of the medium oscillate about an equilibrium position. The constant velocity of a wave can be found by v = λ T = ω k .
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