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Waves. By Sandrine Colson-Inam, Ph.D. References: Conceptual Physics, Paul G. Hewitt, 10 th edition, Addison Wesley publisher http://www.physicsclassroom.com/Class/waves/wavestoc.html http://www.physicsclassroom.com/Class/sound/soundtoc.html. Outline. The Nature of a Wave

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Waves l.jpg

Waves

By Sandrine Colson-Inam, Ph.D

References:

  • Conceptual Physics, Paul G. Hewitt, 10th edition, Addison Wesley publisher

  • http://www.physicsclassroom.com/Class/waves/wavestoc.html

  • http://www.physicsclassroom.com/Class/sound/soundtoc.html


Outline l.jpg

Outline

  • The Nature of a Wave

  • Properties of a Wave

  • Behavior of Waves

  • Standing Waves


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The Nature of a Wave

  • Waves are everywhere: sound waves, light waves, radio waves, microwaves, water waves, sine waves, cosine waves, telephone chord waves, stadium waves, earthquake waves, waves on a string, and slinky waves.

  • Water ripples form waves. The water wave

    has a crest and a through and travels from

    one location to another

  • Slinky waves.


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What is a Wave?

  • A wave can be described as a disturbance that travels through a medium from one location to another location.

    • When the slinky is stretched from end to end and is held at rest, it assumes a natural position known as the equilibrium or rest position.

    • The act of moving the first coil of the slinky in a given direction and then returning it to its equilibrium position creates a disturbance in the slinky.

    • A pulse is a single disturbance moving through a medium from one location to another location.

    • The repeating and periodic disturbance which moves through a medium from one location to another is referred to as a wave.

    • A medium is a substance or material which carries the wave.

    • Waves are said to be an energy transport phenomenon. As a disturbance moves through a medium from one particle to its adjacent particle, energy is being transported from one end of the medium to the other.

  • In conclusion, a wave can be described as a disturbance which travels through a medium, transporting energy from one location (its source) to another location without transporting matter. Each individual particle of the medium is temporarily displaced and then returns to its original equilibrium positioned.


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Categories of Waves

  • A transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction which the wave moves. If a slinky is stretched out in a horizontal direction across the classroom, and a pulse is introduced into the slinky on the left end by vibrating the first coil up and down, then energy will begin to be transported through the slinky from left to right. As the energy is transported from left to right, the individual coils of the medium will be displaced upwards and downwards. In this case, the particles of the medium move perpendicular to the direction which the pulse moves. This type of wave is a transverse wave. Transverse waves are always characterized by particle motion being perpendicular to wave motion. EX: ROPE

  • A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction which the wave moves. If a slinky is stretched out in a horizontal direction across the classroom, and a pulse is introduced into the slinky on the left end by vibrating the first coil left and right, then energy will begin to be transported through the slinky from left to right. As the energy is transported from left to right, the individual coils of the medium will be displaced leftwards and rightwards. In this case, the particles of the medium move parallel to the direction which the pulse moves. This type of wave is a longitudinal wave. Longitudinal waves are always characterized by particle motion being parallel to wave motion. FOR EXAMPLE : SOUND WAVE


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Categories of Waves continues

  • A surface wave is a wave in which particles of the medium undergo a circular motion. Surface waves are neither longitudinal nor transverse. In longitudinal and transverse waves, all the particles in the entire bulk of the medium move in a parallel and a perpendicular direction (respectively) relative to the direction of energy transport. In a surface wave, it is only the particles at the surface of the medium which undergo the circular motion. The motion of particles tend to decrease as one proceeds further from the surface.

  • An electromagnetic wave is a wave which is capable of transmitting its energy through a vacuum (i.e., empty space). Electromagnetic waves are produced by the vibration of electrons within atoms on the Sun's surface. These waves subsequently travel through the vacuum of outer space, subsequently reaching Earth. Were it not for the ability of electromagnetic waves to travel to Earth, there would undoubtedly be no life on Earth. All light waves are examples of electromagnetic waves. Light waves are the topic of another unit at The Physics Classroom. While the basic properties and behaviors of light will be discussed, the detailed nature of an electromagnetic wave is quite complicated and beyond the scope of The Physics Classroom

  • A mechanical wave is a wave which is not capable of transmitting its energy through a vacuum. Mechanical waves require a medium in order to transport their energy from one location to another. A sound wave is an example of a mechanical wave. Sound waves are incapable of traveling through a vacuum. Slinky waves, water waves, stadium waves, and telephone chord waves are other examples of mechanical waves; each requires some medium in order to exist. A slinky wave requires the coils of the slinky; a water wave requires water; a stadium wave requires fans in a stadium; and a telephone chord wave requires a telephone chord.


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Properties of Waves The Anatomy of a Transverse Wave

  • Dashed line = equilibrium or rest position. The position of the rope if there was no disturbance.

  • Crest = the point on the medium which exhibits the maximum amount of positive or upwards displacement from the rest position

  • Trough = the point on the medium which exhibits the maximum amount of negative or downwards displacement from the rest position

  • Amplitude = refers to the maximum amount of displacement of a a particle on the medium from its rest position. In a sense, the amplitude is the distance from rest to crest.

  • Wavelength of a wave is simply the length of one complete wave cycle. A wave has a repeating pattern (wave cycle). the diagram above, the wavelength is the distance from A to E, or the distance from B to G, or the distance from E to J, or the distance from D to I, or the distance from C to H. Any one of these distance measurements would suffice in determining the wavelength of this wave.


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Properties of Waves The Anatomy of a Longitudinal Wave

  • A compression is a point on a medium through which a longitudinal wave is traveling which has the maximum density. A region where the coils are spread apart, thus maximizing the distance between coils, is known as a rarefaction.

  • A rarefaction is a point on a medium through which a longitudinal wave is traveling which has the minimum density. Points A, C and E on the diagram above represent compressions and points B, D, and F represent rarefactions.

  • In the case of a longitudinal wave, a wavelength measurement is made by measuring the distance from a compression to the next compression or from a rarefaction to the next rarefaction. On the diagram above, the distance from point A to point C or from point B to point D would be representative of the wavelength.

                                       \


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Frequency and Period of a Wave

  • The frequency of a wave refers to how often the particles of the medium vibrate when a wave passes through the medium. Frequency refers to how often something happens; period refers to the time it takes something to happen. Frequency is a rate quantity; period is a time quantity.

  • The quantity frequency is often confused with the quantity period. Period refers to the time which it takes to do something. When an event occurs repeatedly, then we say that the event is periodic and refer to the time for the event to repeat itself as the period. The period of a wave is the time for a particle on a medium to make one complete vibrational cycle. Period, being a time, is measured in units of time such as seconds, hours, days or years.


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Energy Transport and the Amplitude of a Wave

  • A wave is an energy transport phenomenon which transports energy along a medium without transporting matter.

  • The energy is imparted to the medium by the person as he/she does work upon the first coil to give it kinetic energy.

  • In fact, a high energy pulse would likely do some rather noticeable work upon your hand upon reaching the end of the medium; the last coil of the medium would displace you hand in the same direction of motion of the coil. For the same reasons, a high energy ocean wave does considerable damage to the piers along the shoreline when it crashes upon it.

  • The amount of energy carried by a wave is related to the amplitude of the wave. A high energy wave is characterized by a high amplitude; a low energy wave is characterized by a low amplitude.


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The Speed of a Wave

  • The speed of an object refers to how fast an object is moving and is usually expressed as the distance traveled per time of travel. In the case of a wave, the speed is the distance traveled by a given point on the wave (such as a crest) in a given interval of time. In equation form,

    EXAMPLE:

    • Reflection phenomenon are commonly observed with sound waves. When you let out a holler within a canyon, you often hear the echo of the holler. The sound wave travels through the medium (air in this case), reflects off the canyon wall and returns to its origin (you); the result is that you hear the echo (the reflected sound wave) of your holler. A classic physics problem goes like this:

    • If an echo is heard one second after the holler and reflects off canyon walls which are a distance of 170 meters away, then what is the speed of the wave?

      • In this instance, the sound wave travels 340 meters in 1 second, so the speed of the wave is 340 m/s. Remember, when there is a reflection, the wave doubles its distance. In other words, the distance traveled by the sound wave in 1 second is equivalent to the 170 meters down to the canyon wall plus the 170 meters back from the canyon wall.


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The Wave Equation

  • The diagrams at the right show several "snapshots" of the production of a wave within a rope. The motion of the disturbance along the medium after every one-fourth of a period is depicted. Observe that it takes that from the first to the last snapshot, the hand has made one complete back-and-forth motion. A period has elapsed. Observe that during this same amount of time, the disturbance has moved a distance equal to one complete wavelength. So in a time of one period, the wave has moved a distance of one wavelength. Combining this information with the equation for speed (speed=distance/time), it can be said that the speed of a wave is also the wavelength/period.

  • Wave speed is dependent upon medium properties and independent of wave properties.

Speed = Wavelength * Frequency

v = f *


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Behavior of Waves – Boundary Behavior

  • As a wave travels through a medium, it will often reach the end of the medium and encounter an obstacle or perhaps another medium through which it could travel.

  • The behavior of a wave (or pulse) upon reaching the end of a medium is referred to as boundary behavior. When one medium ends, another medium begins; the interface of the two media is referred to as the boundary and the behavior of a wave at that boundary is described as its boundary behavior.


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Fixed Rope

  • If a pulse is introduced at the left end of the rope, it will travel through the rope towards the right end of the medium. This pulse is called the incident pulse since it is incident towards (i.e., approaching) the boundary with the pole. When the incident pulse reaches the boundary, two things occur:

    • A portion of the energy carried by the pulse is reflected and returns towards the left end of the rope. The disturbance which returns to the left after bouncing off the pole is known as the reflected pulse.

    • A portion of the energy carried by the pulse is transmitted to the pole, causing the pole to vibrate.

  • One observes the reflected pulse off the fixed end, there are several notable observations. First the reflected pulse is inverted. Other notable characteristics of the reflected pulse include:

    • the speed of the reflected pulse is the same as the speed of the incident pulse

    • the wavelength of the reflected pulse is the same as the wavelength of the incident pulse

    • the amplitude of the reflected pulse is less than the amplitude of the incident pulse

  • Since the speed of a wave (or pulse) is dependent upon the medium through which it travels, two pulses in the same medium will have the same speed.


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    Free-end Rope


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    Less to More Dense Medium

    • Upon reaching the boundary, the usual two behaviors will occur.

      • A portion of the energy carried by the incident pulse is reflected and returns towards the left end of the thin rope. The disturbance which returns to the left after bouncing off the boundary is known as the reflected pulse.

      • A portion of the energy carried by the incident pulse is transmitted into the thick rope. The disturbance which continues moving to the right is known as the transmitted pulse.

  • Comparisons can also be made between the characteristics of the transmitted pulse and those of the reflected pulse. Once more there are several noteworthy characteristics.

    • the transmitted pulse (in the more dense medium) is traveling slower than the reflected pulse (in the less dense medium)

    • the transmitted pulse (in the more dense medium) has a smaller wavelength than the reflected pulse (in the less dense medium)

    • the speed and the wavelength of the reflected pulse are the same as the speed and the wavelength of the incident pulse


  • More to less dense medium l.jpg

    More to Less Dense Medium

    • Comparisons between the characteristics of the transmitted pulse and the reflected pulse lead to the following observations.

      • the transmitted pulse (in the less dense medium) is traveling faster than the reflected pulse (in the more dense medium)

      • the transmitted pulse (in the less dense medium) has a larger wavelength than the reflected pulse (in the more dense medium)

      • the speed and the wavelength of the reflected pulse are the same as the speed and the wavelength of the incident pulse


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    Summary of Boundary Wave Behavior

    • The boundary behavior of waves can be summarized by the following principles:

      • the wave speed is always greatest in the least dense medium,

      • the wavelength is always greatest in the least dense medium,

      • the frequency of a wave is not altered by crossing a boundary,

      • the reflected pulse becomes inverted when a wave in a less dense medium is heading towards a boundary with a more dense medium,

      • the amplitude of the incident pulse is always greater than the amplitude of the reflected pulse.

  • All the observations discussed can be explained by the simple application of these principles.


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    Reflection, Refraction, and Diffraction

    • Reflection:

    • If a linear object attached to an oscillator bobs up and down within the water, it becomes a source of straightwaves. These straight waves have alternating crests and troughs. As viewed on the sheet of paper below the tank, the crests are the bright lines stretching across the paper and the troughs are the dark lines. These waves will travel through the water until they encounter an obstacle - such as the wall of the tank or an object placed within the water. The diagram at the right depicts a series of straight waves approaching a long barrier extending at an angle across the tank of water. The direction which these wavefronts (straight-line crests) are traveling through the water is represented by the blue arrow. The blue arrow is called a ray and is drawn perpendicular to the wavefronts. Upon reaching the barrier placed within the water, these waves bounce off the water and head in a different direction. The diagram below shows the reflected wavefronts and the reflected ray. Regardless of the angle at which the wavefronts approach the barrier, one general law of reflection holds true: the waves will always reflect in such a way that the angle at which they approach the barrier equals the angle at which they reflect off the barrier. This is known as the law of reflection. i = r


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    Reflection on Curved Surfaces

                 The discussion above pertains to the reflection of


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    Refraction

    • Refraction of waves involves a change in the direction of waves as they pass from one medium to another.

    • As water waves are transmitted from deep water into shallow water, the speed decreases, the wavelength decreases, and the direction changes.

    • Law of refraction


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    Diffraction

    • Diffraction involves a change in direction of waves as they pass through an opening or around a barrier in their path.

    • The amount of diffraction (the sharpness of the bending) increases with increasing (longer) wavelength and decreases with decreasing wavelength. In fact, when the wavelength of the waves are smaller than the obstacle, no noticeable diffraction occurs.

    • Diffraction is observed of light waves but only when the waves encounter obstacles with extremely small wavelengths (such as particles suspended in our atmosphere).


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    Interference of Waves - When two waves meet

    • Wave interference is the phenomenon which occurs when two waves meet while traveling along the same medium.

    • There are two types of wave interference:

      • Constructive

      • Destructive


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    Constructive Interference

    • Constructive interference is a type of interference which occurs at any location along the medium where the two interfering waves have a displacement in the same direction. In this case, both waves have an upward displacement; consequently, the medium has an upward displacement which is greater than the displacement of the two interfering pulses. Constructive interference is observed when a crest meets a crest; but it is also observed when a trough meets a trough as shown in the diagram below.


    Destructive interference l.jpg

    Destructive Interference

    • Destructive interference is a type of interference which occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction.

    • In the situation in the diagram above, the interfering pulses have the same maximum displacement but in opposite directions. The result is that the two pulses completely destroy each other when they are completely overlapped. At the instant of complete overlap, there is no resulting disturbance in the medium.

    • If two interfering waves do not need to have equal amplitudes in opposite directions then destructive interference does not occur.


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    After Interference

    • Yet waves meet, produce a net resulting shape of the medium, and then continue on doing what they were doing before the interference.


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    Principle of Superposition (Interference)

    • The task of determining the shape of the resultant demands that the principle of superposition is applied. The principle of superposition is sometimes stated as follows:


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    The Doppler Effect

    • The Doppler effect can be described as the effect produced by a moving source of waves in which there is an apparent upward shift in frequency for observers towards whom the source is approaching and an apparent downward shift in frequency for observers from whom the source is receding. It is important to note that the effect does not result because of an actual change in the frequency of the source.

    The Doppler effect is of intense interest to astronomers who use the information about the shift in frequency of electromagnetic waves produced by moving stars in our galaxy and beyond in order to derive information about those stars and galaxies.


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    Traveling Waves vs. Standing Waves

    • A mechanical wave is a disturbance which is created by a vibrating object and subsequently travels through a medium from one location to another, transporting energy as it moves. The mechanism by which a mechanical wave propagates itself through a medium involves particle interaction; one particle applies a push or pull on its adjacent neighbor, causing a displacement of that neighbor from the equilibrium or rest position. As a wave is observed traveling through a medium, a crest is seen moving along from particle to particle. This crest is followed by a trough which is in turn followed by the next crest. In fact, one would observe a distinct wave pattern (in the form of a sine wave) traveling through the medium. This sine wave pattern continues to move in uninterrupted fashion until it encounters another wave along the medium or until it encounters a boundary with another medium. This type of wave pattern which is seen traveling through a medium is sometimes referred to as a traveling wave.

    • Traveling waves are observed when a wave is not confined to a given space along the medium. The most commonly observed traveling wave is an ocean wave.


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    Standing Wave

    • It is possible however to have a wave confined to a given space in a medium and still produce a regular wave pattern which is readily discernible amidst the motion of the medium. For instance, if an elastic rope is held end to end and vibrated at just the right frequency, a wave pattern would be produced which assumes the shape of a sine wave and is seen to change over time. The wave pattern is only produced when one end of the rope is vibrated at just the right frequency. When the proper frequency is used, the interference of the incident wave and the reflected wave occur in such a manner that there are specific points along the medium which appear to be standing still. Because the observed wave pattern is characterized by points which appear to be standing still, the pattern is often called a standing wave pattern.


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    Formation of Standing Waves

    • A standing wave pattern is a vibrational pattern created within a medium when the vibrational frequency of the source causes reflected waves from one end of the medium to interfere with incident waves from the source in such a manner that specific points along the medium appear to be standing still. Because the observed wave pattern is characterized by points which appear to be standing still, the pattern is often called a "standing wave pattern." Such patterns are only created within the medium at specific frequencies of vibration; these frequencies are known as harmonic frequencies, or merely harmonics. At any frequency other than a harmonic frequency, the interference of reflected and incident waves results in a resulting disturbance of the medium which is irregular and non-repeating.

    • A standing wave pattern is an interference phenomenon. It is formed as the result of the perfectly time interference of two waves passing through the same medium. A standing wave pattern is not actually a wave; rather it is the pattern resulting from the presence of two waves (sometimes more) of the same frequency with different directions of travel within the same medium.


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    Standing Wave – Nodes and Anti-Nodes

    • One characteristic of every standing wave pattern is that there are points along the medium which appear to be standing still. These points, sometimes described as points of no displacement, are referred to as nodes. There are other points along the medium which undergo vibrations between a large positive and and large negative displacement. These are the points which undergo the maximum displacement during each vibrational cycle of the standing wave. In a sense, these points are the opposite of nodes, and so they are called antinodes. A standing wave pattern always consist of an alternating pattern of nodes and antinodes.


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    First Harmonic Standing Wave Pattern

    Second Harmonic Standing Wave Pattern

    Harmonics and Patterns

    • A variety of actual wave patterns could be produced, with each pattern characterized by a distinctly different number of nodes. Such standing wave patterns can only be produced within the medium when it is vibrated at certain frequencies. There are several frequencies with which the snakey can be vibrated to produce the patterns. Each frequency is associated with a different standing wave pattern. These frequencies and their associated wave patterns are referred to as harmonics.

    A pattern with three nodes and two antinodes is referred to as the second harmonic

    A pattern with two nodes and one antinode is referred to as the first harmonic


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    Third Harmonic Standing Wave Pattern

    QUESTION!!

    • How many nodes and antinodes in the third harmonic?

    Third Harmonic Standing Wave Pattern


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    Mathematics of Standing Waves

    • Consider the first harmonic standing wave pattern for a vibrating rope as shown below.

    • The pattern for the first harmonic reveals a single antinode in the middle of the rope. This antinode position along the rope vibrates up and down from a maximum upward displacement from rest to a maximum downward displacement as shown. The vibration of the rope in this manner creates the appearance of a loop within the string.

    • In comparing the standing wave pattern for the first harmonic with its single loop to the diagram of a complete wave, it is evident that there is only one-half of a wave stretching across the length of the string. That is, the length of the string is equal to one-half the length of a wave. Put in the form of the equation above.


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    More Harmonics


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    Sinusoidal Nature of Waves

    Physical waves, or mechanical waves, form through the vibration of a medium, be it a string, the Earth's crust, or particles of gases and fluids. Waves have mathematical properties that can be analyzed to understand the motion of the wave.

    A wave having a form which, if plotted, would be the same as that of a trigonometric sine or cosine function. The sine wave may be thought of as the projection on a plane of the path of a point moving around a circle at uniform speed. It is characteristic of one-dimensional vibrations and one-dimensional waves having no dissipation.

    The sine wave is the basic function employed in harmonic analysis. It can be shown that any complex motion in a one-dimensional system can be described as the superposition of sine waves having certain amplitude and phase relationships. The technique for determining these relationships is known as Fourier analysis.


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    Sinusoidal Nature of Waves

    • This wave pattern occurs often in nature, including ocean waves, sound waves, and light waves.

    • A cosine wave is said to be "sinusoidal", because cos(x) = sin(x + π / 2), which is also a sine wave with a phase-shift of π/2. Because of this "head start", it is often said that the cosine function leads the sine function or the sine lags the cosine.

    • The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics; some sounds that approximate a pure sine wave are whistling, a crystal glass set to vibrate by running a wet finger around its rim, and the sound made by a tuning fork.

    • To the human ear, a sound that is made up of more than one sine wave will either sound "noisy" or will have detectable harmonics; this may be described as a different timbre.


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    Sinusoidal Nature of Waves

    • A simple travelling wave with a single frequency is sinusoidal.

    • At t = 0, y = A sin (2p/l x) where y is the displacement of the wave (longitudinal or transverse) at position x, A is the amplitude of the wave, and l is the wavelength.

    • If the wave is moving to the right with velocity v. At time t, each part of the wave has moved to the right at distance vt.

      y = A sin (2p/l (x – vt))

    • If the wave is moving left

      y = A sin (2p/l (x + vt))


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