Traveling Waves

1. 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 Traveling Waves

2. 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 medium—that 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

3. 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.

4. Transversewaves:Theoscillationsoccurperpendicularlytothedirection of energy transfer. Exemple: a wave in a tense string. Herethevaryingmagnitudeisthedistancefromtheequilibrium horizontal position Longitudinal waves: Those in whichthedirection of vibrationisthesame as theirdirection of propagation. So themovement of theparticles of themediumiseither in thesameor in theoppositedirectiontothemotion of the wave. Exemple: soundwaves, whatchanges in this case isthepressure of themedium (air, waterorwhateveritbe).

5. Pulses Speed of wave The shape of pulse is described by the function f(x) The wave function provides the mathematical description of the traveling pulse y: disturbance of medium from the equilibrium position v: speed of propagation of wave The wave function are particular solutions of the differential equation called wave equation, which can derived from Newton´s Law

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

7. Speed of waves A 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 String The 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?

8. Speed of waves (2) Sound (in a elastic material) β bulk modulus ρ density For sound waves in a gas such air, the pressure changes occur too rapidly for appreciable heat transfer, and so the process is adiabatic. γ 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 Sound (in air)   density of the solid (kg/m3) Solids Young modulus Calculate the speed of sound in air at (a) 0ºC and (b) 20ºC The 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 (ρ = 13.6 x 103 Kg/m3 )

9. 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.

10. 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. The sinusoidal shape is described by the sine function crest For a wave traveling in the direction of increasing x, with a speed v, replace x by x –vt, with δ = 0 λ, wavelength: the minimun distance after which the wave repeat (distance between crests, per example) Basic relationship between wavelength,λ , speed,v, period, T, and frequency, f k: wave number

11. Harmonic waves: Energy transfer on a string The energy on one vibrating point, considering that describes a harmonic motion, is For the string where a harmonic wave has been generated, the energy of a particle of mass dm will be Energy transfer Energy 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

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

13. Waves in Three Dimensions. Intensity

14. Wave Intensity. Case study: Sound Wave The Wave Intensity, I, is the average power per unit area that is incident perpendicular to the direction of the propagation For the case of point source that emits waves uniformly in all directions The rate of transfer of energy is the passing into the shell Wave intensity for a sound wave A 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 loudspeaker

15. Intensity level and loudness. The human ear Range of human ear response to sound wave intensity: Threshold of hearing 10-12 W/m2 Pain 1 W/m2 The 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

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

18. Waves encountering barriers: refraction 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. Sinusoidal 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.

19. Waves encountering barriers: Difraction A 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.

20. 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. All motions are relative to medium During time Ts, -period of the source- the source moves a distance usTs and the 5th wavefront travels a distancevTs . The wavelength in front of source is (v-us)Ts Stationary receiver The number of wave crests passing the receiver per unit time In front of the source the minus sign applies. Behind the source the plus sign applies.

21. The Doppler effect (b) 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. Sign plus is used in the case of receiver moving in the direction opposite to that of the wave All motions are relative to medium Moving receiver The number of wave crests passing the receiver per unit time Source and receiver are moving relative to medium If the receiver is moving toward the source the plus sign is selected in the numerator. If the source is moving to the receiver the minus sign is selected in the denominator. The general rule is that the frequency tends to increase when the source moves toward the receiver and when the receiver moves toward the source

22. The Doppler effect. Summary and exercises 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. All motions are relative to medium The frequency of a train horn is 400 Hz. If the train speed train is 122 km/h, (a) the wavelength and the frequency of the sound passing a stationary receiver placed in front of train; (b) the same if the stationary receiever were placed behind of train; (c). If the receiver is approaching to the train with a speed 120 km/h respect to ground, in the same way but in oppsite direction, what the received frequency will be? ** The trafic stationary radar unit emits waves with a frequency of 1.5x109 Hz. The receiver unit measures the reflected waves from the car moving away. The frequency of this reflected wave differs from the emiting by 500 Hz . What is the car speed?. ** fr receiver frequency; fs source frequency v wave propagation speed ur receiver speed; us source speed; Choose the signs that give an up-shift in frequency for an approaching source or receiver, and vice-versa. If the receiver is moving toward the source the plus sign is selected in the numerator. If the source is moving to the receiver the minus sign is selected in the denominator A ship at rest is equipped with sonar that sends out pulses of sound at 40 MHz. Reflected pulses are received from a submarine directly below with a time delay of 80 ms, at a frequency of 39.958 MHz. Find (a) the depth of the submarine (b) ts vertical speed. Speed of sound in seawater is 1.54 KHz

23. Shock waves (Shock front) If a source moves with speed greater than the wave speed , then there will be no waves in front of the source. Instead, the waves pile up behind the source to form a shock wave. U.S. Navy F/A-18 breaking the sound barrier. The white halo is formed by condensed water droplets thought to result from a drop in air pressure around the aircraft

24. SUPERPOSITION OF WAVES INTERFERENCE STANDING WAVES When two or more waves ovelap in space,their individual disturbances superimpose and add algebraically, creating a resultant wave. This property of waves is called the principle of superposition Under certain circumstances the superposition of harmonic waves of the same frequency produce s sustained wave patterns in the space. This phenomenon is called interference. Interference and difraction are what distinguish between wave motion and particle motion

25. SUPERPOSITION OF WAVES Constant phase 0 Interference Constant phase π/2

26. Interference: Constructive and destructive Phase difference due to path difference

27. STANDING WAVES A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions. The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave. Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut, where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing a node. Halfway between two nodes there is an antinode, where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time

28. STANDING WAVES When waves are confined in spaces, multiple reflections cause superposing waves that interfer according the superposition principle. For a given string or pipe, there are certain frequencies for which superpposition results in a stationary vibration pattern: standing wave