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Chapter 11 Waves

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Waves l A wave is a disturbance/oscillation generated from its source and travels over long distances. l A wave transports energy but not matter. l Examples: è Water waves: formed when you throw a stone in water. Water moves up and down. è Sound waves (air moves back & forth) è Electromagnetic waves: Light waves, Radio waves, TV waves, X-rays etc. (what moves?).

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All waves are produced by vibrating (oscillating) sources. If the source vibrates sinusoidally in SHM, the wave will have sinusoidal shape in space and time. When waves travel through a medium, the particles of the medium vibrate. The manner in which medium particles vibrate defines two types of waves: Transverse and Longitudinal waves.

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Types of Waves l Transverse and Longitudinal 1. Transverse Waves: The medium oscillates perpendicular to the direction the wave is moving. è Water (more or less) è Electromagnetic waves: Light, radio, TV, X- rays …) è Slinky demo.

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Transverse Waves: The medium oscillates perpendicular to the direction the wave is moving. Wave velocity (v) Particles of medium x y

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Transverse Waves… Crest – highest point Trough – Lowest point Wavelength ( ) – distance from crest to next crest. Or from a trough to next trough. x y Crest trough Crest trough

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Oscillation of medium particles about their equilibrium position with time. Period (T) – Time for a medium particle to oscillate through one cycle. Within this time, the wave will have traveled through a distance equal to one wavelength. Amplitude (A) = maximum displacement from equilibrium position. t y T T¾T ¼T ½T A

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Period (T) –Within one period (T), the wave will have traveled through a distance equal to one wavelength ( ) Thus velocity of the wave v = distance/time v = /T = f t y T T¾T¾T ¼T¼T ½T A x y Crest trough Crest trough

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2. Longitudinal Waves: The medium oscillates in the same direction as the wave is moving. Examples: è Sound è Slinky demo Wave velocity (v) Particles of medium

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l Longitudinal: The medium oscillates in the same direction as the wave is moving Wave velocity (v) Particles of medium Series of regions of compressions and expansions (rarefactions) are formed.

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Longitudinal One wavelength = distance from one region of compression to the next region of compression. OR distance from one rarefaction to the next rarefaction. Time taken to make one wavelength is the period T.

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The speed of sound in air is a bit over 300 m/s, and the speed of light in air is about 300,000,000 m/s. Suppose we make a sound wave and a light wave that both have a wavelength of 3 meters. What is the ratio of the frequency of the light wave to that of the sound wave? Example V = f

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Reflection of Waves When a wave meets an obstacle in its path, some of it will be reflected, some will be absorbed (as heat energy) and some will be transmitted through the obstacle. Medium 1 Medium 2 Incident ray Reflected ray Transmitted (refracted) ray v i, i, f i v r, r, f r v t, t, f t

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Reflection of Waves Reflection of waves – When waves bounce off of an obstacle and take a different path of propagation. Eg. Echo – is reflection of sound waves.

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Reflection l When a traveling wave meets a boundary, reflection occurs. l Part of the wave travels backwards from the boundary.

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Wave fronts: Representation of a whole width of a wave crest. Rays: Lines drawn perpendicular to the wave fronts. Arrows point to direction of travel. Rays Wave fronts

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When waves are reflected, they obey the law of reflection: Angle of incidence = angle of reflection. i = r Incident ray Reflected ray ii rr

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When a wave crosses from one medium to another, its speed and direction changes. However, its frequency remains the same. Will the wavelength be the same or different across the media? Refraction

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Recall: Wave velocity: v = /T = f. In a given medium, the velocity (v) of a wave stays constant. So for a wave traveling in the same medium, if its frequency changes, the wavelength will change accordingly to maintain constant velocity. If the wave crosses one medium into a different medium, its velocity (both magnitude and direction) will change.

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Velocity: v i, v t - different Frequency: f i, f t - same Wavelength: i, t - different Refraction Medium 1 Medium 2 Incident ray Reflected ray Transmitted (refracted) ray v i, i, f i v r, r, f r v t, t, f t

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Two waves are in phase if they reach identical positions at the same time. In-phase Wave 1 t y Wave 2

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Two waves are out of phase if they reach identical positions at different times. t y Wave 2 Wave 3 Out of phase Wave 2 and wave 3 are out of phase by 180 o

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Coherent Waves Coherent Waves: two or more waves that have the same frequency and keep the same phase relationship between them. Wave 1 t y Wave 3 Wave 2

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Incoherent Waves Incoherent waves: Waves for which the phase relationship varies randomly. Wave 1 t y Wave 2

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Interference and Superposition Interference: What happens when 2 or more waves pass through a point at the same time. Overall displacement = algebraic sum of the separate displacements of each wave – this is principle of superposition.

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Interference and Superposition When too waves overlap, the amplitudes add. Constructive: increases amplitude Destructive: decreases amplitude

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Standing Waves Interference of a wave and its reflection creates standing wave. Node Antinode Node – displacement is zero. Corresponds to destructive interference. Antinode – where cord moves with maximum amplitude – Constructive interference.

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Standing Waves Interference (superposition) of a wave and its reflection creates standing wave. The two superimposed waves have equal amplitudes, equal frequencies, equal wavelengths, equal speeds but traveling opposite to each other.

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Standing wave patterns can be set up in almost any structure. At its fundamental frequency, all the particles oscillate in phase with the same frequency. A harmonic driving force with the same frequency can very efficiently pump energy into this mode. At resonance, the amplitude of the standing wave increases without limit, until the structure is damaged. A famous example of a structure driven into resonance in a windstorm and collapsing is the Tacoma Narrows bridge failure. Tacoma Narrows bridge failure The 5,939-foot-long Tacoma Narrows Bridge collapsed during a 42-MPH wind storm on November 7,

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Resonant Modes of vibration Second Harmonic n = 2 Third Harmonic n = 3 Fourth Harmonic n = 4 Fundamental Frequency First Harmonic, n = 1

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Resonant Modes of vibration v = f # Loops Wavelength ( Frequency (f) Name 1 L = L/1 f 1 = v/ f 1 = v/ L Fundamental 1 st Harmonic 2 L = 2L/2 f 2 = v/ f 2 = 2v/ L = 2f 1 2 nd Harmonic 1 st Overtone 3 L = 3 L f 3 = v/ f 3 = 3v/ L = 3f 1 3 rd Harmonic 2 nd Overtone 4 L = 2 L/4 f 4 = v/ f 4 = 4v/ L = 4f 1 4 th Harmonic 3 rd Overtone n L = n /2 n L n f n = v/ f n = nv/ L = nf 1 n = 2L/n f n = nv/(2L) f n = nf 1

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Example A string with both ends fixed resonates in five loops at a frequency of 525 Hz. If the velocity of the waves in the string is 84 m/s (a) How far apart are two adjacent nodes? (b) What is its fundamental frequency? (c) What is the length of the string?

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Example A standing wave is produced in a vibrating string as shown below. If the length of the string is 1.5 m and the frequency of the vibrating motor is 60 Hz, the speed of the wave is (A) 15 m/s (B) 20 m/s (C) 40 m/s (D) 60 m/s (E) 90 m/s

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