# Traveling Waves

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02-Jan-2016Category

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Traveling Waves

Wave motion.

Periodic waves: on a string, Sound and electromagnetic waves

Waves in Three Dimensions. Intensity

Waves encountering barriers: Reflection, Refraction and Difraction,

The Doppler Effect

Superposition, Interference Standing waves

INTRODUCTI0N. TRAVELING WAVES

a wave is a disturbance that travels through space and time, usually accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the mediumthat is, with little or no associated mass transport. They consist instead of oscillations or vibrations around almost fixed locations. For example, a cork on rippling water will bob up and down, staying in about the same place while the wave itself moves onwards

INTRODUCTION.TYPE OF WAVE

One type of wave is a mechanical wave, which propagates through a medium in which the substance of this medium is deformed. The deformation reverses itself owing to restoring forces resulting from its deformation. For example, sound waves propagate via air molecules bumping into their neighbors. This transfers some energy to these neighbors, which will cause a cascade of collisions between neighbouring molecules. When air molecules collide with their neighbors, they also bounce away from them (restoring force). This keeps the molecules from continuing to travel in the direction of the wave.

Another type of wave can travel through a vacuum, e.g. electromagnetic radiation (including visible light, ultraviolet radiation, infrared radiation, gamma rays, X-rays, microwaves and radio waves). This type of wave consists of periodic oscillations in electrical and magnetic fields.

Transverse waves: The oscillations occur perpendicularly to the direction of energy transfer. Exemple: a wave in a tense string. Here the varying magnitude is the distance from the equilibrium horizontal positionLongitudinal waves: Those in which the direction of vibration is the same as their direction of propagation. So the movement of the particles of the medium is either in the same or in the opposite direction to the motion of the wave. Exemple: sound waves, what changes in this case is the pressure of the medium (air, water or whatever it be).

Pulses Speed of waveThe shape of pulse is described by the function f(x)The wave function provides the mathematical description of the traveling pulseThe wave function are particular solutions of the differential equation called wave equation, which can derived from Newtons Lawy: disturbance of medium from the equilibrium positionv: speed of propagation of wave

Wave functionPlotting for differents values of time Traveling pulses. An examplet = 0t = 2t = 4This pulse moves to the right (positive direction of X axis) with a velocity of 0.50 m/swhere x, y are in meter, t in seconds, v = 0.50 m/sLet us to write the wave equation in such a way that the group x+vt appears explicitly.

Speed of wavesA general property of waves is that their speed relative to medium depends on the properties of medium but is independent of the motion of the source of waves. If the observer is in motion with respect to the medium, the velocity of wave propagation relative to the observer wil be different. A remarkable exception is encountered in the case of light Speed of a wave on a StringThe 25-m-long string has a mass of 0.25 kg and is kept taut by a hanging object of mass 10 kg. What is the speed of the pulse?. If the 10-kg mass is replaced with 20-kg mass, what is the speed on the string?Transverse waves travel at 150 m/s on a wire of length that is under a tension of 550 N. What is the mass of the wire?A steel piano wire is 0,7 m long and has a mass of 5 g. It is stretched with a tension of 500N. What is the speed of transverse waves on the wire?

Speed of waves (2)Sound (in a elastic material) bulk modulus densitySound (in air) adiabatic coefficient, for air 1,4 R universal gas constant 8.314 J/(mol.K) M: molar mass of gas, for air 28.96x10-3 Kg/mol T: absolute temperature For sound waves in a gas such air, the pressure changes occur too rapidly for appreciable heat transfer, and so the process is adiabatic.Calculate the speed of sound in air at (a) 0C and (b) 20CThe bulk modulus for water is 2.0x109 N/m2. Use it to find the speed of sound in water (b) The speed of sound in mercury is 1410 m/s What is the bulk modulus for mercury ( = 13.6 x 103 Kg/m3 )Solids density of the solid (kg/m3)Young modulus

PERIODIC WAVES

Harmonic waves Harmonic waves are the most basic type of periodic waves. All waves, wether they are periodic or not, can be modeled as a superposition of harmonic waves.If one end of a string is attached to a vibrating point that is moving up and down with simple harmonic motion, a sinusoidal wave train propagates along the string. If a harmonic wave is traveling through a medium , each point of the medium oscillates in simple harmonic motion.

Harmonic waves: The harmonic function Harmonic waves are the most basic type of periodic waves. All waves, wether they are periodic or not, can be modeled as a superposition of harmonic waves., wavelength: the minimun distance after which the wave repeat (distance between crests, per example) crestBasic relationship between wavelength, , speed,v, period, T, and frequency, fThe sinusoidal shape is described by the sine functionFor a wave traveling in the direction of increasing x, with a speed v, replace x by x vt, with = 0k: wave number

Harmonic waves: Energy transfer on a stringThe energy on one vibrating point, considering that describes a harmonic motion, isFor the string where a harmonic wave has been generated, the energy of a particle of mass dm will beEnergy is being transferred from the initial vibrating point to the whole string, because when the wave reaches new portions of the string, they begin to oscillate gaining energy. The energy transferred by the unit time, that is, the power, will be Energy transfer

Harmonic Waves: Energy on Sound WavesThe wave function of harmonic sound waves can be writen considering longitudinal displacements of air mollecules around the equilibrium position s(x,t), The average energy of a harmonic sound wave in a volume element dV, will be that corresponding a vibrating particle with a mas dm = dV, that is

Energy transferThe vibration of air mollecules lead to variation of pressureEnergy per unit of volume

Waves in Three Dimensions. Intensity

Wave Intensity. Case study: Sound WaveThe Wave Intensity, I, is the average power per unit area that is incident perpendicular to the direction of the propagationWave intensity for a sound waveA loudspeaker diafragm 30 cm in diameter is vibrating at 1 kHz, with an amplitude of 0.020 mm. Assuming that the close air mollecules vibrates with the same amplitude, find (a) the pressure amplitude (b) the sound intensity in front of diaphragm (c) the acoustic power being radiated (d) if the sound is radiated uniformly in the hemisphere, find the intensity at 5 m from the loudspeakerFor the case of point source that emits waves uniformly in all directionsThe rate of transfer of energy is the passing into the shell

Range of human ear response to sound wave intensity: Threshold of hearing 10-12 W/m2 Pain 1 W/m2Intensity level and loudness. The human earThe perception of loudness is not proportional to the intensity but varies logaritmically. We use a logaritmic scale to describe the intensity level for the human ear, which is measured in decibels, (dB)Estimate the sound pressure variations for the range of sound intensity in the case of human ear

Waves encountering barriers: Reflection, refraction and Difraction Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism

Waves encountering barriers: refractionSinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results.Refraction is the phenomenon of a wave changing its speed. Typically, refraction occurs when a wave passes from one medium into another. The amount by which a wave is refracted by a material is given by the refractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by Snell's law.

Waves encountering barriers: DifractionA wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.

The Doppler effect (a)The Doppler effect (or Doppler shift), is the change in frequency of a wave for an observer moving relative to the source of the wave The received frequency is higher (compared to the emitted frequency) during the approach, it is identical at the instant of passing by, and it is lower during the recession. Stationary receiverThe number of wave crests passing the receiver per unit timeIn front of the source the minus sign applies. Behind the source the plus sign applies.During time Ts, -period of the source- the source moves a distance usTs and the 5th wavefront travels a distance vTs . The wavelength in front of so

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