Name: modeling a wave on a string
Uploaded: 2023-04-30T11:09:41.000Z
Duration: 1 min 2 s
Channel: JoVE

K12 Standards

north-carolina-k-12-science-standard-course-of-study

North Carolina K-12 Science Standard Course of Study

9-12

Physical Science

Waves and Their Applications

PS.PSc.8.1

Equations of Wave Motion

Волны можно описать движением и положением частиц среды, через которую они движутся.
Если мы построим график вертикальной высоты волны до ее положения, то увидим, что каждая частица в среде колеблется вверх и вниз в простом гармоническом движении с одинаковой частотой.
Равновесное положение волны проходит по прямой линии и считается осью x системы координат.
Рассмотрим частицу среды, в которой движется волна. Она смещается из положения равновесия на определенную высоту в положительном и отрицательном направлении оси y в период Т. Волновое движение имеет синусоидальную форму и со временем переходит в положительное направление x.
Величина смещения частиц из положения равновесия является функцией как положения, так и времени и известна как волновая функция.
Знаки плюс и минус указывают на то, движется ли волна в отрицательном или положительном направлении x соответственно.

Уравнения волнового движения

Математически движение волны может быть изучено с помощью волновой функции. Рассмотрим струну, колеблющуюся вверх и вниз с гармоническим движением, имеющим период T. Волна на струне синусоидальна и смещается в положительном направлении оси x по мере продвижения времени. Синус представляет функцию угла θ, колеблющуюся между +A и −A и повторяющуюся каждые 2π радиан. Для построения волновой модели рассматривается отношение угла θ и положения x.

<img alt="Figure1" src="/files/ftp_upload/12775/12775_Equation_1.png" style="text-align: center; height: 30px;" />

Из этого отношения, используя значение угла θ и умножая функцию синуса на амплитуду A, мы можем моделировать положение y-координаты струны как функцию положения x.

<img alt="Figure2" src="/files/ftp_upload/12775/12775_Equation_2.png" style="text-align: center; height: 35px;" />

Волна на струне движется с постоянной скоростью и перемещается на расстояние, равное vt, за время t, поэтому мы можем модифицировать волновую функцию, используя это. Кроме того, умножение на термин 2π/λ упрощает уравнение.

<img alt="Figure3" src="/files/ftp_upload/12775/12775_Equation_3.png" style="text-align: center; height: 35px;" />

Вспомним выражения для волнового числа и угловой частоты, которые модифицируют первый и второй члены вышеприведенного уравнения соответственно. Таким образом, получим упрощенное выражение для волновой функции гармонической волны на струне. p style="text-align: center

<img alt="Figure4" src="/files/ftp_upload/12775/12775_Equation_4.png" style="text-align: center; height: 18px;" />

К волновой функции добавляется угол φ, который соответствует начальной фазе волны. Он указывает на начальное положение частицы непосредственно перед началом волнового движения и используется для определения различных параметров волны.

Этот текст адаптирован из <a href="https://openstax.org/books/university-physics-volume-1/pages/16-2-mathematics-of-waves">Openstax, University Physics Volume 1, Section 16.2: Mathematics of Waves.</a>

Математически движение волны можно изучать с помощью волновой функции. Представьте себе струну, колеблющуюся вверх и вниз в простом гармоническом движении, ...

Education

JoVE Core

Physics

Физика

Unit 1

Раздел 1

Waves

Волны

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

modeling a wave on a string

A wave on a string can be modeled with a wavefunction. The string moves up and down in simple harmonic motion and has a period T. As time passes, the sinusoidal wave shifts in the positive x-direction.

The sine function is used because it repeats every 2π radians and varies between +A and −A, where A is the amplitude. To describe the string’s y-position, the angle θ is related to position x. The sine function is then multiplied by amplitude A to build the wave model.

The model must also account for motion over time. Since the wave travels with constant velocity, it moves a distance equal to vt in time t. The equation is then modified by the factor 2π/λ, where λ is the wavelength. This leads to the wave number and the angular frequency, which adjust the first and second parts of the equation.

The reduced wavefunction describes a simple harmonic wave on a string. An initial phase angle φ is added to show the wave’s starting position just before motion begins. That phase helps determine different wave parameters.

Modeling a Wave on a String

travelling waves

Mechanical Waves in Air, Water, and Earth

Mechanical waves carry a disturbance through a medium. They repeat periodically and are often linked with simple harmonic motion. These waves follow Newton's laws and need a material medium to move through. A medium is the substance that carries the wave and provides an elastic restoring force when it is deformed. Water waves, sound waves, and seismic waves are examples of mechanical waves. In water waves, water is the medium. In sound waves, the medium is usually air, but it can also be water.

wave parameters

Reading Wave Motion by Its Features

Wave motion can be described by a few key parameters. Simple mechanical waves repeat over several cycles and can be modeled with sine and cosine functions. A simplified surface water wave is a good example. The water moves up and down, while the wave disturbance travels horizontally through the medium. A floating seagull would move up and down as the crests and troughs pass below it. Crest means the peak of the wave, and trough means the valley. This motion follows simple harmonic motion,...

equations of wave motion

A wave on a string can be modeled with a wavefunction. The string moves up and down in simple harmonic motion and has a period T. As time passes, the sinusoidal wave shifts in the positive x-direction. The sine function is used because it repeats every 2π radians and varies between +A and −A, where A is the amplitude. To describe the string’s y-position, the angle θ is related to position x. The sine function is then multiplied by amplitude A to build the wave model. The model must also...

graphing the wave function

Wave Snapshots, Period, and Phase

Wave snapshots, period, and phase help describe a sinusoidal wave moving in the positive x-direction. The wave equation depends on both position and time. That is why it can be shown in more than one way. If time is fixed, such as at t = 0, the graph becomes a snapshot of the wave. This is a displacement versus position graph. It shows the displacement of the particle from equilibrium as a function of position. From this graph, the wavelength can be found. For a transverse wave on a string,...

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velocity and acceleration of a wave

Wave Motion: Particle Speed and Curvature

Wave motion in a medium has a constant wave velocity. The particles in the medium do not move at that same constant speed. Their motion is perpendicular to the direction of the wave, which helps separate wave speed from particle speed. Because particle speed changes, the particles must also have acceleration. The particle velocity can be found by taking the partial derivative of the position equation with respect to time. Then the acceleration comes from taking the partial derivative of the...

speed of a transverse wave

How String Tension Changes Wave Speed

The speed of transverse waves on a string depends on the string’s tension and its mass per unit length. A string with more tension has stronger restoring forces. Those forces help pull the string back to straight after it is disturbed, so the wave moves faster. Mass per unit length is the mass of a string divided by its length. It is also called linear density. Strings on a guitar can have different thicknesses even when they are made of similar material, so they can have different linear...

/files/media/science-education/science-education-thumbs/14722.jpg

problem-solving: tuning of a guitar string

problem-solving-tuning-of-a-guitar-string

Guitar String Waves and Musical Notes

Guitar string waves depend on linear mass density, tension, and string length. Linear mass density means the mass per unit length of the string. When the density stays constant along the string, it equals the total mass divided by the total length. Changing this linear density can change the wave speed. Tension also affects the speed of transverse waves along a guitar string. The fundamental frequency of the string depends on its length as well. When a string is held on a fret, its effective...

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kinetic and potential energy of a wave

How Waves Transfer Energy

Waves transfer energy through motion in the air, water, and ground. That energy can have real effects. Earthquake waves can shake large concrete buildings until they fall. Loud sound waves can damage nerve cells in the inner ear and lead to permanent hearing loss. Ocean waves can also wear away beaches. In mechanical waves, energy depends on amplitude and frequency. Amplitude is the height or strength of the wave. Large-amplitude earthquakes cause larger ground movement. Loud sounds come from...

energy and power of a wave

Wave Intensity and Energy Transfer

Wave intensity and energy transfer describe how waves carry energy from one place to another. The total energy in a wavelength is the sum of potential energy and kinetic energy. The rate at which that energy moves is called power, which is the total energy divided by the time needed to transfer it. For a sinusoidal wave, energy and power both depend on the square of the amplitude and the angular frequency. That means a larger amplitude or a higher angular frequency gives a greater energy...

interference and superposition of waves

Wave Interference Patterns in Physics

Wave interference patterns form when two waves of the same nature occupy the same region at the same time. The combined effect is the sum of the individual wave effects. Interference does not mean that one wave changes how the other wave travels. This idea applies to mechanical waves, including sound waves, waves on a string, and surface water waves. Mechanical waves move through a medium, so interference shows up as the algebraic sum of particle displacements in that medium. The same...

reflection of waves

Wave Reflection at Fixed and Free Boundaries

Wave reflection happens when a wave reaches a boundary between two media and bounces back. A familiar example is an echo, where sound waves in air reflect from a cliff. Sound waves are longitudinal waves, while a pulse on a string is a transverse wave. A pulse on a string tied to a wall reflects when it reaches the fixed end. The wall acts as a boundary that cannot move with the oscillation. By Newton’s third law, the string pushes on the wall and the wall pushes back with an equal and...

propagation of waves

Wave Reflection and Transmission

Wave reflection and transmission happen when a wave moves from one medium to another. At the boundary, part of the wave can bounce back into the first medium, and part can pass into the second medium. The interface between the two media then acts like a boundary that is neither fixed nor free. The wave behavior depends on the change in linear mass density of the strings. When a wave travels from a string with low linear mass density to a string with high linear mass density, the reflected wave...

standing waves

Wave Interference in Strings and Air Columns

Wave interference in strings and air columns can create standing waves that seem to vibrate in place. A standing wave may appear on the surface of a glass of milk kept in a refrigerator. The refrigerator motor sends vibrations into the milk, and the wave moves up and down without seeming to travel across the surface. Standing waves form when two or more identical moving waves travel in opposite directions. As the waves pass through each other, their disturbances add together. When the waves...

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modes of standing waves - i

Standing Wave Harmonics in Strings and Buildings

Standing waves and resonance can be seen in both earthquakes and strings. Earthquake waves can make a building vibrate for several seconds when the driving frequency matches the building’s natural frequency of vibration. That resonance can cause one building to collapse while nearby buildings stay standing. Building height also matters because it can match the condition needed to form a standing wave at that height. In many cases, buildings of one height are damaged while taller buildings...

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modes of standing waves: ii

Standing Wave Modes in Sound Tubes

Standing wave modes in sound tubes depend on the boundary conditions at the ends of the tube. These boundary conditions describe how the air particles can move where the tube is open or closed. They set the pattern the standing wave must follow. For a tube that is open at one end and closed at the other, the air has an antinode at the open end and a node at the closed end. An antinode is a point of maximum motion, while a node is a point of no motion. The resonant frequencies of the tube also...

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key terms and concepts

Learn how a sinusoidal wave on a string is modeled with amplitude, wavelength, wave number, angular frequency, and initial phase in simple harmonic motion.

Modeling a Wave on a String | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

PS.PSc.8.2

waves-and-their-applications-in-technologies-for-information-transfer

Waves and Their Applications in Technologies for Information Transfer

PS.Phy.7.1

Pennsylvania State Academic Standards

Waves and Electromagnetic Radiation

3.2.9-12.T

Louisiana Student Standards for Science

High School

Waves And Their Applications

HS-PS4-1

Waves And Their Applications In Technologies For Information Transfer

Tennessee Academic Standards

PHYS.PS4.1

Virginia  Standards of Learning

HIgh School

PS.6

PS.6.a

PH.5

PH.5.a

new-jersey-student-learning-standards-for-science

New Jersey Student Learning Standards for Science

Arizona Science Standards

HS.P4

HS.P4U1.10

Alabama Course of Study Science

9-12.PS.2.2

9-12.P.3.7

9-12.P.3.7.a

Next Generation Science Standards

Earth and Space Science

Earth’s Systems

HS-ESS2-3

Florida Department of Education

SC.912.P.10

SC.912.P.10.20

Texas Essential Knowledge and Skills

PHYS.8

PHYS.8.B

PHYS.8.C

south-carolina-college-and-career-ready-science-standards

South Carolina College and Career Ready Science Standards

P-PS4-1

P-PS4-3

Earth and Space Science (ESS)

E-ESS2-3.

new-york-state-science-learning-standards

New York State Science Learning Standards

HS. Waves and Electromagnetic Radiation

HS-PS4-1.

Georgia Standards of Excellence

SPS9.

electromagnetic-waves-and-mechanical-waves

Electromagnetic Waves and Mechanical Waves

SP4.

mechanical-and-electromagnetic-waves-properties

Mechanical and Electromagnetic Waves Properties

Travelling Waves

A wave is a disturbance that propagates from its source, repeating itself periodically, and is typically associated with simple harmonic motion. ...

Wave Parameters

The simplest mechanical waves are associated with simple harmonic motion and repeat themselves for several cycles. These simple harmonic waves can be ...

Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a ...

Graphing the Wave Function

Consider the wave equation for a sinusoidal wave moving in the positive x-direction. The wave equation is a function of both position and time. From the ...

Velocity and Acceleration of a Wave

A wave propagates through a medium with a constant speed, known as a wave velocity. It is different from the speed of the particles of the medium, which ...

Speed of a Transverse Wave

The speed of a wave depends on the characteristics of the medium. For example, in the case of a guitar, the strings vibrate to produce the sound. The ...

Problem-Solving: Tuning of a Guitar String

In the case of stringed instruments like the guitar, the elastic property that determines the speed of the sound produced is its linear mass density or ...

Kinetic and Potential Energy of a Wave

All forms of waves carry energy; this is directly visualized in nature. For instance, the waves of earthquakes are so intense that they can shake huge ...

Energy and Power of a Wave

The total energy associated with a wavelength is the sum of the potential energy and the kinetic energy. The average rate of energy transfer associated ...

Carry out investigations to explain the quantitative and qualitative relationships among wave frequency, wave velocity, wavelength, and wave energy.

Electromagnetic Waves

James Clerk Maxwell formulated a single theory combining all the electric and magnetic effects scientists knew during that time, calling the phenomena his ...

Electromagnetic Wave Equation

Maxwell&#39;s equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose ...

propagation-speed-of-electromagnetic-waves

Propagation Speed of Electromagnetic Waves

Electromagnetic waves are consistent with Ampere&#39;s law. Assuming there is no conduction current Ampere&#39;s law is given as:


Consider a plane ...

Electromagnetic Waves in Matter

Electromagnetic waves can travel in the vacuum as well as in matter. For example light, which is an electromagnetic wave, can travel through air, water, ...

Energy Carried By Electromagnetic Waves

Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes, this energy is obvious, such as in the summer sun&#39;s ...

Intensity Of Electromagnetic Waves

The energy transport per unit area per unit time, or the Poynting vector, gives the energy flux of an electromagnetic wave at any specific time. For a ...

Use models to compare the characteristics of mechanical and electromagnetic waves.

Interference and Superposition of Waves

When two waves of the same nature occur in the same region simultaneously, they result in interference. Interference of waves implies that the net effect ...

Reflection of Waves

When a wave travels from one medium to another, it gets reflected at the boundary of the second medium. A common example of this is when a person yells at ...

Propagation of Waves

When a wave propagates from one medium to another, part of it may get reflected in the first medium, and part of it may get transmitted to the second ...

Sound Waves: Interference

Sound waves can be modeled either as longitudinal waves, wherein the molecules of the medium oscillate around an equilibrium position, or as pressure ...

Interference: Path Lengths

Consider two sources of sound, that may or may not be in phase, emitting waves at a single frequency, and consider the frequencies to be the same.
Two ...

/files/media/science-education/science-education-thumbs/14727.jpg

Use models to explain the wave interactions of reflection, refraction, diffraction, and interference.

PS.PSc.8.3

Standing Waves

Sometimes waves do not seem to move; rather, they just vibrate in place. Unmoving waves can be seen on the surface of a glass of milk kept in a ...

Modes of Standing Waves - I

A close look at earthquakes provides evidence for the conditions appropriate for resonance, standing waves, and constructive and destructive interference. ...

Modes of Standing Waves: II

The starting point for expressing the modes of standing waves is understanding the boundary conditions that the waves must follow. The boundary conditions ...

Obtain, evaluate, and communicate information to explain how instruments that transmit and detect waves are used in everyday life.

PS.PSc.8.4

Analyze the nature of waves and their applications.

Modeling a Wave on a String

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