Physics
Stationary Waves
Stationary waves, also known as standing waves, are formed by the interference of two waves traveling in opposite directions. In a stationary wave, certain points called nodes remain stationary while other points called antinodes oscillate with maximum amplitude. These waves are commonly observed in musical instruments, such as strings and pipes, and have applications in various fields including acoustics and optics.
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9 Key excerpts on "Stationary Waves"
eBook - PDF- Md Nazoor Khan, Simanchala Panigrahi(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
At all the points, P 1 , P 2 , P 3 , P 4 , P 5 , P 6 , P 7 , P 8 , P 9 , the particles of the medium have zero amplitude. See Fig. 1.17(e). 56 Principles of Engineering Physics 1 1.10.2 Characteristics of Stationary Waves The followings are the characteristics of Stationary Waves. i. A stationary wave is produced when two identical progressive waves (i.e., waves having the same wavelength, the same time period, the same frequency, and the same speed) travelling in the same line but in opposite directions are superimposed. ii. The points at which particles are not oscillating are called nodes. iii. The points at which particles are oscillating with maximum displacements are called anti nodes iv. The distance between any two consecutive nodes is half the wavelength. v. The distance between any two consecutive anti nodes is half the wavelength vi. The distance between any two consecutive nodes and anti nodes is one-fourth of the wavelength. vii. Except at nodes, each particle of the medium executes simple harmonic oscillation about its mean position with a time period equal to the time period of the wave motion. viii. Amplitudes of all the particles of the medium are not the same. It is zero at the node and maximum at the anti node. 1.10.3 Differences between progressive and Stationary Waves Progressive waves Stationary Waves 1 A progressive wave is produced due to the oscillation of the particles of the medium. A stationary wave is produced when two identical progressive waves travelling in the same line but opposite directions are superimposed. 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.
eBook - ePub- Robert H. Randall(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
CHAPTER 7Stationary Waves. VIBRATING SOURCES. MUSICAL INSTRUMENTS
7–1 Introduction
. The fundamental relations for plane and for spherical waves developed in Chapters 2 and 3 have assumed that a disturbance, once set up, travels out from the source an indefinite distance. This picture of a medium infinite in extent is useful for any fundamental description of the physics of wave propagation since, at any one point in space, we are concerned with waves traveling in one direction only. For sound sources radiating into the open air, and with few obstacles to reflect or scatter the energy, the medium may be considered virtually infinite in extent. However, when sound waves strike hard, relatively rigid structures, an appreciable fraction of the incident energy may be deflected and perhaps returned in the direction of the source. In the region where this occurs there will be two wave trains moving in opposite directions, each contributing to the deformation of the medium.Under certain conditions this situation may give rise to stationary or standing waves, with a whole new set of features quite foreign to waves of the unidirectional type. In this case the “pattern” of the deformation in the medium remains fixed in space, with no evidence at all of propagating crests or troughs. It is with this general phenomenon that we shall be mainly concerned in this chapter.Stationary Waves may occur in any medium having definite boundaries. In air, such waves may be of primary importance within a room, where the medium is confined by the surrounding walls. Wave reflection, and the consequent production of a standing wave pattern, may sometimes take place without an actual change of medium. This is the case when waves traveling down a cylindrical pipe reach an end open to the surrounding air. In this case the reflection is associated with the change in the acoustic impedance as the wave passes from the region within the pipe to the region of free space beyond it. This phenomenon is very similar to that taking place in an electrical transmission line whenever the line characteristics change abruptly and we shall have more to say about it later.
eBook - PDFWaves and Oscillations in Nature
An Introduction
- A Satya Narayanan, Swapan K Saha(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
If all the amplitudes are equal, we find a 2 0 = N 2 a 2 0 (1.82) The resultant irradiance of N identical coherent sources, radiating in phase with each other, is equal to N 2 times the irradiance of the individual sources. For both coherent and incoherent cases, the total energy does not change, but the distribution of the energy does change. 1.4.3 Standing Wave Standing waves (also called the Stationary Waves) are set up as a con-sequence of superposition of two waves of same amplitude and frequency propagating at the same speed in a unidimensional space, but in opposite directions. Unlike the traveling waves, standing waves transmit no energy. Standing waves are formed in a medium that has boundaries where a wave is reflected back. The waves bounce against these boundaries and partly reflect back into the medium and partly transmit outside the medium. The reflected waves superimpose on incident waves, which may, in turn, cancel each other. Having boundary means that the waves are confined to a specific length of the medium. The waves are moving, but the same places have a very large ampli-tude oscillation while others have zero amplitude and continuous destructive interference. The Stationary Waves may be set up when a wave reflects back from a surface and the reflected wave interferes with the wave still traveling in the original direction. The reflected wave and the incoming wave interfere. At the reflecting surface, the two waves are equal but opposite and get canceled out. Such a place is called a node of the wave. At other points along the waves, these waves the same. Therefore, they add together or interfere constructively. Such points are called the anti-nodes of the wave. If the boundary is closed, there will be a node at the boundary, while if the boundary is open, there will be an anti-node at the boundary.
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
As a result, the waves are in phase and there is a maximum intensity at point P. 17.2 Standing Waves The sound waves from the pair of loudspeakers in Example 17.1 leave the speak- ers in the forward direction, and we considered interference at a point in front of the speakers. Suppose we turn each speaker by 908 so that they face each other as in Figure 17.6, and then have them emit sound of the same frequency and ampli- tude. In this situation, two identical waves travel in opposite directions in the same medium. These waves combine in accordance with the waves in interference model. We can analyze such a situation by considering wave functions for two transverse sinusoidal waves having the same amplitude, frequency, and wavelength but travel- ing in opposite directions in the same medium: y 1 5 A sin ( kx 2 vt) y 2 5 A sin ( kx 1 vt) where y 1 represents a wave traveling in the positive x direction and y 2 represents one traveling in the negative x direction. Adding these two functions according to the superposition principle gives the resultant wave function y: y 5 y 1 1 y 2 5 A sin ( kx 2 vt) 1 A sin ( kx 1 vt) When we use the trigonometric identity sin ( a 6 b) 5 sin a cos b 6 cos a sin b, this expression reduces to y 5 (2A sin kx) cos vt (17.1) Equation 17.1 represents the wave function of a standing wave. A standing wave such as the one on a string shown in Figure 17.7 is an oscillation pattern with a sta- tionary outline that results from the superposition of two identical waves traveling in opposite directions. Notice that Equation 17.1 does not contain a function of kx 2 vt . Therefore, it is not an expression for a traveling wave. When you observe a standing wave, there is no sense of motion in the direction of propagation of either original wave. If you were to observe the motion of the string in Figure 17.7, you would not see any motion to the left or right. You would only see up and down motion of the elements of the string.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
This is constructive interference. If the phase difference is 180°, the waves interfere in destructive interference (part (c)). The resultant wave has an amplitude of zero. Any other phase difference results in a wave with the same wave number and angular frequency as the two incident waves but with a phase shift of ϕ/2 and an amplitude equal to 2A cos(ϕ/2). Examples are shown in parts (b) and (d). Chapter 16 | Waves 827 Figure 16.24 Superposition of two waves with identical amplitudes, wavelengths, and frequency, but that differ in a phase shift. The red wave is defined by the wave function y 1 (x, t) = A sin(kx − ωt) and the blue wave is defined by the wave function y 2 (x, t) = A sin ⎛ ⎝kx − ωt + ϕ ⎞ ⎠ . The black line shows the result of adding the two waves. The phase difference between the two waves are (a) 0.00 rad, (b) π/2 rad, (c) π rad, and (d) 3π/2 rad . 16.6 | Standing Waves and Resonance Learning Objectives By the end of this section, you will be able to: • Describe standing waves and explain how they are produced • Describe the modes of a standing wave on a string • Provide examples of standing waves beyond the waves on a string Throughout this chapter, we have been studying traveling waves, or waves that transport energy from one place to another. Under certain conditions, waves can bounce back and forth through a particular region, effectively becoming stationary. These are called standing waves. 828 Chapter 16 | Waves This OpenStax book is available for free at http://cnx.org/content/col12031/1.5 Another related effect is known as resonance. In Oscillations, we defined resonance as a phenomenon in which a small- amplitude driving force could produce large-amplitude motion. Think of a child on a swing, which can be modeled as a physical pendulum. Relatively small-amplitude pushes by a parent can produce large-amplitude swings. Sometimes this resonance is good—for example, when producing music with a stringed instrument.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Standing waves can arise with transverse waves, such as those on a guitar string, and also with longitudinal sound waves, such as those in a flute. In any case, the principle of linear superposition provides an explanation of the effect, just as it does for diffraction and beats. Interactive Figure 17.15 shows some of the essential features of transverse standing waves. In this figure the left end of each string is vibrated back and forth, while the right end is attached to a wall. Regions of the string move so fast that they appear only as a blur in the photographs. Each of the patterns shown is called a transverse standing wave pattern. Notice that the pat- terns include special places called nodes and antinodes. The nodes are places that do not vibrate at all, and the antinodes are places where maximum vibration occurs. To the right of each pho- tograph is a drawing that helps us to visualize the motion of the string as it vibrates in a standing wave pattern. These drawings freeze the shape of the string at various times and emphasize the maximum vibration that occurs at an antinode with the aid of a red dot attached to the string. Each standing wave pattern is produced at a unique frequency of vibration. These frequen- cies form a series, the smallest frequency f 1 corresponding to the one-loop pattern and the larger frequencies being integer multiples of f 1 , as Interactive Figure 17.15 indicates. Thus, if f 1 is 10 Hz, the frequency needed to establish the 2-loop pattern is 2f 1 or 20 Hz, whereas the frequency needed to create the 3-loop pattern is 3f 1 or 30 Hz, and so on. The frequencies in this series (f 1 , 2 f 1 , 3 f 1 , etc.) are called harmonics. The lowest frequency f 1 is called the first harmonic, and 17.5 Transverse Standing Waves 475 the higher frequencies are designated as the second harmonic (2 f 1 ), the third harmonic (3 f 1 ), and so forth.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
17.6 | Longitudinal Standing Waves Standing wave patterns can also be formed from longitudinal waves. For example, when sound reflects from a wall, the forward- and backward-going waves can produce a standing wave. Figure 17.19 illustrates the vibrational motion in a longitudinal standing wave on a Slinky. As in a transverse standing wave, there are nodes and antinodes. At the nodes the Slinky coils do not vibrate at all; that is, they have no displacement. At the antinodes the coils vibrate with maximum amplitude and, thus, have a maximum displacement. The red dots in Figure 17.19 indicate the lack of vibration at a node and the maximum vibration at an antinode. The vibration occurs along the line of travel of the individual waves, as is to be expected for longitudinal waves. In a standing wave of sound, at the nodes and antinodes, the molecules or atoms of the medium behave as the red dots do. Musical instruments in the wind family depend on longitudinal standing waves in pro- ducing sound. Since wind instruments (trumpet, flute, clarinet, pipe organ, etc.) are modi- fied tubes or columns of air, it is useful to examine the standing waves that can be set up in such tubes. Figure 17.20 shows two cylindrical columns of air that are open at both ends. Sound waves, originating from a tuning fork, travel up and down within each tube, since they reflect from the ends of the tubes, even though the ends are open. If the frequency f of the tuning fork matches one of the natural frequencies of the air column, the downward- and upward-traveling waves combine to form a standing wave, and the sound of the tuning N N N N A A A A Figure 17.19 A longitudinal standing wave on a Slinky showing the displacement nodes (N) and antinodes (A). 470 Chapter 17 | The Principle of Linear Superposition and Interference Phenomena fork becomes markedly louder.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
(16.2.5) Power The average power of, or average rate at which energy is transmitted by, a sinusoidal wave on a stretched string is given by P avg = 1 _ 2 μvω 2 y m 2 . (16.3.7) Superposition of Waves When two or more waves traverse the same medium, the displacement of any particle of the medium REVIEW & SUMMARY is the sum of the displacements that the individual waves would give it. Interference of Waves Two sinusoidal waves on the same string exhibit interference, adding or canceling according to the principle of superposition. If the two are traveling in the same direction and have the same amplitude y m and frequency (hence the same wavelength) but differ in phase by a phase constant ϕ, the result is a single wave with this same frequency: y′(x, t) = [ 2y m cos 1 _ 2 ϕ ] sin ( kx − ωt + 1 _ 2 ϕ ) . (16.5.6) If ϕ = 0, the waves are exactly in phase and their interference is fully constructive; if ϕ = π rad, they are exactly out of phase and their interference is fully destructive. Phasors A wave y(x, t) can be represented with a phasor. This is a vector that has a magnitude equal to the amplitude y m of the wave and that rotates about an origin with an angular speed equal to the angular frequency ω of the wave. The projection of the rotating phasor on a vertical axis gives the displacement y of a point along the wave’s travel. Standing Waves The interference of two identical sinusoidal waves moving in opposite directions produces standing waves. For a string with fixed ends, the standing wave is given by y′(x, t) = [2y m sin kx] cos ωt. (16.7.3) Standing waves are characterized by fixed locations of zero displacement called nodes and fixed locations of maximum dis- placement called antinodes. Resonance Standing waves on a string can be set up by reflection of traveling waves from the ends of the string. If an end is fixed, it must be the position of a node. This limits the frequen- cies at which standing waves will occur on a given string.
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 dots show successive positions of string particles at equal intervals of time, indicated by the different shades. Particles situated at antinodes such as A vibrate with maximal amplitude, and those at nodes such as B do not vibrate at all. The contemporaneous positions of adjacent particles lie on curves of the same shade. The unique feature of standing waves is that energy is not transmitted but stored as vibrational energy of the particles. A standing wave stores energy in the oscillations of the particles disturbed by the two waves. (In practice, there is no such thing as a completely rigid support and a small amount of energy will ‘escape’ from the string at each reflection.) A string of fixed length has a number of normal modes of vibration corresponding to whole numbers of half wavelengths which ‘fit exactly’ into that length of string. The amplitude of vibration must be zero at both ends of the string. The first harmonic (lowest frequency mode) of a string of length L is produced when just one half wavelength ‘fits’ on the string. The second and third harmonics correspond to two and A B Time sequence of the positions of the particles of a vibrating string. 164 Let There Be Light 2nd Edition three half wavelengths respectively, as shown in the figure below. Wavespeed = frequency × wavelength = constant, so if the wavelength gets smaller the frequency increases accordingly. The frequencies of the harmonics (or normal modes) are called the natural frequencies of the string. 1st harmonic 2nd harmonic 3rd harmonic λ 1 = 2 L λ 2 = λ 1 / 2 f 2 = 2 f 1 f 3 = 3 f 1 λ 3 = λ 1 / 3 Natural frequencies of a vibrating string. A fuller mathematical treatment does not give any more physical insight, but it does make a very clear distinction between travelling and standing waves and allows us to calcu-late the position of any particle at any time.
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