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Waves Tanya Liu

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What is a wave? A wave is a disturbance or oscillation that is a function of space and time, involves a transfer of energy Surface wave Sound wave

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Types of Waves MechanicalElectromagnetic Caused by oscillations/ deformations in a physical medium Sound waves Water waves Seismic waves Consists of periodic oscillations of electric/ magnetic fields Microwaves X-rays Visible light vs

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Mechanical Waves

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Properties of Waves We can visualize waves as a function of space and time by considering a wave pulse A single up and down motion generates a pulse that travels with velocity v

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Properties of Waves Now, with repeated up and down motions, we get a traveling sinusoidal wave

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Properties of Waves: Period Waves x t Variance with space Variance with time y y wavelength λ period T

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Properties of Waves wave at time t wave at time t+Δt ΔxΔx y x

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We can characterize a sine wave traveling in the +x direction with an equation of the following form: y m = amplitude = maximum displacement k = angular wave number = 2π/λ ω = angular frequency = 2π/T=kv λ= 2π/k = vT Properties of Waves: Periodic Waves amplitude wave number angular frequency position time

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Concept Question How many seconds does this wave take per cycle, and what is its amplitude? a.3.14 sec, 5/3 b.3.14 sec, 5 c.2 sec, 5 d.2 sec, 2/3 e.None of the above

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Concept Question What is the wavelength of a 1 MHz wave with a speed of 344 m/s? (1 MHz = 10 6 Hz) a.3.44 x 10 -4 b.2.16 x 10 -3 c.1 x 10 -6 d.Not enough information given

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Concept Question What determines the speed of a wave, v in a medium? a.Angular frequency b.Amplitude c.a and b d.The speed is a function of the properties of the medium

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Some Expectations Participate! Please don’t be scared, I would like to hear everyone’s voice Once again, if you don’t understand me, just raise your hand There may be some short assignments given All my slides will be uploaded to Professor Dodero’s site: http://my.liceti.it/prof/dodero/

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Some Pronounciations C 1 = “C sub one” or “C one”

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Some Pronounciations a + b = “a plus b” a – b = “a minus b” a x b = “a times b” a/b = “a over b” a b = “a to the b” a 2 = “a to the second/ a squared” a 3, a 4 = “a to the third, a to the fourth, etc” 0.05 = “zero point zero five” 2x10 -5 = “two times ten to the negative fifth”

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Review from last time The dotted line represents the equilibrium position when there is no disturbance in the medium peaks or crests troughs

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Review from last time y m = amplitude / maximum displacement from the equilibrium position λ= wavelength, distance between 2 consecutive similar points on a wave (peak to peak, trough to trough) λ =2π/k = vT

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Review from last time ω = angular frequency, number of times a wave passes a fixed point over a certain amount of time – Note: this is for a periodic wave only, meaning the wave repeats itself ω = 2πf = 2π/T = kv v= speed of the wave v = λ/T = λ f → distance/time k = angular wave number k =2π/λ

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Concept Review Which letter represents the amplitude of the wave? a.A b.B c.C d.B and D B A C D y x

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Concept Review Which letter represents the wavelength of the wave? a.A b.B c.C d.B and D B A C D y x

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Wave Pulse Applet

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What determines v ? Velocity of a mechanical wave is typically a function of the medium’s physical properties What physical properties of the string could the speed of a transverse wave on the string depend on?

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Group Problem: Speed of Wave on a String Given the following free body diagram for a portion of a wave on a string, write an expression for the net force in the radial direction on the Δl portion of the string using τ, R, and Δl. = tension in the string = the string’s linear mass density, mass per unit length

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Group Problem Solution Since the wave is transverse, there is acceleration only in the radial direction

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Speed of Wave on a String Now we have Combining,

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What happens when two waves meet? Consider two wave pulses traveling towards each other, both with equal positive displacements =?

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What happens when two waves meet? The result is known as constructive interference Two waves with displacement in the same direction add to produce a resulting wave of greater amplitude. This is known as the superposition of the two waves.

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What happens when two waves meet? Now consider two wave pulses traveling towards each other with equal displacements in opposite directions =?

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What happens when two waves meet? This is deconstructive interference Two waves with displacement in opposite directions add to produce a resulting wave of smaller amplitude. In this case, since the magnitude of each amplitude is the same, they cancel each other out completely

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Interference: A more general case If the interfering wave pulses do not have amplitudes with equal magnitudes, the result is something like this: constructive interferencedestructive interference

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Concept Question: Interference for a traveling wave Two waves are traveling towards each other with equal frequency, wavelength, and amplitude A If the waves arrive at point P at the same time, what is the resulting motion of point P? a.P doesn’t move b.P oscillates with amplitude A c.P oscillates with amplitude 2A

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Hooke’s Law F s = - k x F s is the spring force k is the spring constant It is a measure of the stiffness of the spring A large k indicates a stiff spring.

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