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Phys 0174 Fall 2008 - Chapter 16-17, 4 slides

# Phys 0174 Fall 2008 - Chapter 16-17, 4 slides - Chapter 16...

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1 Chapter 16 & 17 Waves I & II ¾ Types of waves ¾ Amplitude, phase, frequency, period, propagation speed of a wave ¾ Mechanical waves propagating along a stretched string ¾ Wave equation ¾ Principle of superposition of waves ¾ Standing waves, resonance ¾ Speed of sound waves ¾ Sound intensity and sound level ¾ Beats ¾ The Doppler effect 2 A wave is defined as a disturbance that is self-sustained and propagates in space with a constant speed Waves can be classified in the following three categories: 1. Mechanical waves. 2. Electromagnetic waves. 3. Matter waves. Waves can be divided into the following two categories depending on the orientation of the disturbance with respect to the wave propagation velocity . Transverse and Longitudinal waves v G 3 y x ( ) ( ) , sin m y x t y kx t ω = 2 2 f T π ω π = = 2 k π λ = The equation for a wave is: ω is the angular frequency: k is the wave number: λ is the wavelength λ This gives you the displacement y at a position x and at a time t . The frequency f is the number of crests that pass a given point per second. The units are Hertz (Hz) where 1 Hz = 1/s. The period T is the time for one complete oscillation. ( ) ( ) , sin m y x t y kx t ω = 4 The figure shows two snapshots of a harmonic wave taken at times and . During the time interval the wave has traveled a distance . The wave speed is . O t t x v t t t x + ∆ = The speed of a traveling wave ne method of finding is to imagine that we move with the same speed as the wave along the -axis, so that the wave appears frozen. v x

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5 ( ) If ( , ) sin appears constant, then: constant Take the derivative with respect to . 0 m y x t y kx t kx t t dx k dt dx dx k dt dt k v ω ω ω ω ω = = = = = = v k ω = 6 Assume that a string has a linear mass density of and the tension on the string is . Consider a small section of the string of length . The shape of the element c µ τ Wave speed on a stretched string G A ( ) an be approximated as the arc of a circle of radius R whose center is at O. The net force in the direction of O is 2 sin . Assume that 1 so that si n 2 F R R F τ θ θ θ τ θ = = = (equation 1) ± A A 7 ( ) 2 2 The force is also given by Newton's second law: Set equations 1 and 2 equal and solve for : v v F m R R v µ µ = ∆ = (equation 2) A A ( ) 2 v R τ = A The speed depends on the tension and the mass density but not on the wave frequency . R v f v τ µ τ µ = Note : T v µ = F R τ = (equation 1) A 8
10 ( ) ( ) Consider a transverse wave propagating along a string which is described by the equation: The transverse velocity is is ( , equal to ) sin - . cos - m m y x t y kx t y u y kx u t t ω ω ω = = = − Rate of energy transmission zero at point a. is maximum at point b. u 11 ( ) ( ) ( ) 2 2 2 In general the kinetic energy of an element of mass is given by: 1 1 2 2 1 - cos - 2 The rate at which kinetic energy propagates along the string is equal to 1 2 m dm dK v dm v dx dK y kx t dx dK dx dt µ ω ω µ µ = = = = ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 1 cos - cos - 2 The average rate 1 1 1 1 cos - 2 2 2 4

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Phys 0174 Fall 2008 - Chapter 16-17, 4 slides - Chapter 16...

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