Physics
Standing Waves
Standing waves are formed when two waves of the same frequency and amplitude traveling in opposite directions interfere with each other. This results in points along the medium where the wave appears to be standing still, known as nodes, and points of maximum displacement, known as antinodes. Standing waves are commonly observed in musical instruments and are characterized by their fixed patterns of nodes and antinodes.
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10 Key excerpts on "Standing Waves"
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)
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. Each possible frequency is a resonant frequency, and the correspond- ing standing wave pattern is an oscillation mode. For a stretched string of length L with fixed ends, the resonant frequencies are f = v _ λ = n v _ 2L , for n = 1, 2, 3, . . . . (16.7.9) The oscillation mode corresponding to n = 1 is called the funda- mental mode or the first harmonic; the mode corresponding to n = 2 is the second harmonic; and so on. 1 The following four waves are sent along strings with the same linear densities (x is in meters and t is in seconds). Rank the waves according to (a) their wave speed and (b) the tension in the strings along which they travel, greatest first: (1) y 1 = (3 mm) sin(x − 3t), (2) y 2 = (6 mm) sin(2x − t), (3) y 3 = (1 mm) sin(4x − t), (4) y 4 = (2 mm) sin(x − 2t). QUESTIONS 2 In Fig. 16.1, wave 1 consists of a rectangular peak of height 4 units and width d, and a rectangular valley of depth 2 units and width d. The wave travels rightward along an x axis. Choices 2, 3, and 4 are similar waves, with the same heights, depths, and widths, that will travel leftward along that axis and through wave 1. Right-going wave 1 and one of the left-going waves will inter- fere as they pass through each other. With which left-going wave will the interference give, for an instant, (a) the deepest valley, (b) a flat line, and (c) a flat peak 2d wide? Questions 467 (3) (4) (1) (2) FIGURE 16.1 Question 2. 3 Figure 16.2a gives a snapshot of a wave traveling in the direc- tion of positive x along a string under tension. Four string elements are indicated by the lettered points. For each of those elements, determine whether, at the instant of the snapshot, the element is moving upward or downward or is momentarily at rest.
No longer available |Learn morePhysics for Scientists and Engineers
Foundations and Connections, Extended Version with Modern Physics
- Debora Katz(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Standing Waves in Musical Instruments When a musician plays an instrument, a standing wave is established in the instru- ment. How can a solo musician do so when, according to Figure 18.21, we need two traveling waves to make a standing wave? Imagine that you attach one end of a string to a fixed ring as in Figure 18.5, but instead of creating a pulse shaped like a shark’s fin you oscillate your end of the string in simple harmonic motion. A harmonic wave travels from you toward the ring, reflects from the ring, and creates a second wave traveling toward you. The reflected wave has the same amplitude and wavelength as the incident wave you created. The superposition of these two waves creates a standing wave on the rope just as if you had a friend at the other end helping you. The basis of musical instruments is the formation of Standing Waves from the superposition of re- flected waves. In the next three sections, we focus on Standing Waves that form in three instruments in a big-band ensemble: the guitar, the flute, and the clarinet. Making Standing Waves EXAMPLE 18.3 In a large laboratory, two waves with the same amplitude and wavelength travel in opposite di- rections along the same rope. The two waves interfere, and the result is a standing wave on the rope. One traveling wave is described by y 1 1x, t 2 5 0.150 sin 12.50x 2 6.35t 2 (1) where numerical values have the appropriate SI units. A Write an expression for the standing wave and find the amplitude of the antinodes. INTERPRET and ANTICIPATE Equation (1) is the wave function of a wave traveling in the positive x direction. The identical wave traveling in the negative x direction must by given by y 2 1x, t 2 5 0.150 sin 12.50x 1 6.35t 2 . The sum of the two waves y 1 1 y 2 is a standing wave. SOLVE Identify y max , k, and v from Equation (1) and substitute these values into Equation 18.6 to find an expression for the stand- ing wave y(x, t).
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
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). 17.6 | Longitudinal Standing Waves 419 A Frequency = f A N A A A N N Frequency = 2f Figure 17.20 A pictorial representation of longitudinal Standing Waves on a Slinky (left side of each pair) and in a tube of air (right side of each pair) that is open at both ends (A, antinode; N, node). fork becomes markedly louder. To emphasize the longitudinal nature of the standing wave patterns, the left side of each pair of drawings in Figure 17.20 replaces the air in the tubes with Slinkies, on which the nodes and antinodes are indicated with red dots. As an addi- tional aid in visualizing the Standing Waves, the right side of each pair of drawings shows blurred blue patterns within each tube. These patterns symbolize the amplitude of the vibrating air molecules at various locations.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
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. At other times, the effects can be devastating, such as the collapse of a building during an earthquake. In the case of Standing Waves, the relatively large amplitude Standing Waves are produced by the superposition of smaller amplitude component waves. Standing Waves Sometimes waves do not seem to move; rather, they just vibrate in place. You can see unmoving waves on the surface of a glass of milk in a refrigerator, for example. Vibrations from the refrigerator motor create waves on the milk that oscillate up and down but do not seem to move across the surface. Figure 16.25 shows an experiment you can try at home. Take a bowl of milk and place it on a common box fan. Vibrations from the fan will produce circular Standing Waves in the milk. The waves are visible in the photo due to the reflection from a lamp. These waves are formed by the superposition of two or more traveling waves, such as illustrated in Figure 16.26 for two identical waves moving in opposite directions. The waves move through each other with their disturbances adding as they go by. If the two waves have the same amplitude and wavelength, then they alternate between constructive and destructive interference. The resultant looks like a wave standing in place and, thus, is called a standing wave. Figure 16.25 Standing Waves are formed on the surface of a bowl of milk sitting on a box fan. The vibrations from the fan causes the surface of the milk of oscillate. The waves are visible due to the reflection of light from a lamp. Chapter 16 | Waves 829 Figure 16.26 Time snapshots of two sine waves. The red wave is moving in the −x-direction and the blue wave is moving in the +x-direction.
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 - 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.
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)
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 centre of the opening is specified by the angle . If the opening is a rectangular slit of width D, such as a doorway, the angle is given by equation 17.1. If the opening is a circular opening of diameter D, such as that in a loudspeaker, the angle is given by equation 17.2. sin = D (17.1) sin = 1.22 D (17.2) 17.4 Explain beats as a wave interference phenomenon. Beats are the periodic variations in amplitude that arise from the linear superposition of two waves that have slightly different frequencies. When the waves are sound waves, the variations in amplitude cause the loudness to vary at the beat frequency, which is the difference between the frequencies of the waves. 17.5 Analyse transverse Standing Waves. A standing wave is the pattern of disturbance that results when oppositely travelling waves of the same frequency and amplitude pass through each other. A standing wave has places of minimum and maximum vibration called, respectively, nodes and antinodes. Under resonance conditions, Standing Waves can be established only at certain natural frequencies. The frequencies in this series (f 1 , 2f 1 , 3f 1 , etc.) are called harmonics. The lowest frequency f 1 is called the first harmonic, the next frequency 2f 1 is the second harmonic, and so on. For a string that is fixed at both ends and has a length L, the natural frequencies are specified by equation 17.3, where v is the speed of the wave on the string and n is a positive integer. f n = n ( v 2L ) n = 1, 2, 3, 4, . . . (17.3) 17.6 Analyse longitudinal Standing Waves. For a gas in a cylindrical tube open at both ends, the natural fre- quencies of vibration are specified by equation 17.4, where v is the speed of sound in the gas and L is the length of the tube.
eBook - PDFWaves and Oscillations in Nature
An Introduction
- A Satya Narayanan, Swapan K Saha(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
1.2 What Is a Wave? A wave may be described as a periodic disturbance that transports energy from one point to another. Its direction of propagation is the direction in which energy is carried. It is thought to be the result of correlated oscillations occurring at every point along the path of the wave. The direction of these oscillations determines the nature of a particular wave, for example, 1. Transverse wave: The oscillations are known as transverse when the vibration takes place in a direction perpendicular to the direction of propagation. 2. Longitudinal wave: In this case, the vibrations are in the direction of propagation. For a sound wave, in which the pattern of disturbance caused by the movement of energy traveling through a medium as it propagates away from the source, the particles move back and forth parallel to the direction of the propagation of the wave. The wave phenomena and their oscillatory motion possess many degrees of freedom. They are the most important physical feature of a given system. Systems that are much larger than the atomic scales are continuous at scales of small distances. The discrete atomic properties tend to be replaced by their continuous local averages. Their degrees of freedom are denoted by a function of space and time. The infinite continuous set of degrees of freedom is subjected to a set of mutually dependent nonlinear equations of motion. These equations are fused into a partial differential equation, whose temporal dependence and derivatives are inherited from the Newtonian single atom equation of motion or its relativistic generalization. The spatial dependence and derivatives of this equation are byproducts of the discrete atomic indices in the continuum limit. A plane wave may be a simple two-dimensional (2D) or three-dimensional (3D) wave. Its characteristic is that all the points on a plane that is per-pendicular to the direction of propagation have the same phase value.
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