Physics II chapter 15 notes

Physics II chapter 15 notes - Chapters 15: Mechanical Waves...

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v6.01 RS/BQ F’10 117 Chapters 15: Mechanical Waves a mechanical wave is a disturbance of a medium that propagates (travels) through the medium with speed Introduction v medium itself does not propagate individual particles in medium undergo only small displacements overall pattern of the disturbance is what propagates wave transports energy , but not matter , from one region to another wave examples: mechanical wave formation requires a medium having Types of Mechanical Waves (15-1) an equilibrium state straight taut string, level surface of water, air of uniform pressure forces that tend to restore medium to its equilibrium state and through which elements influence each other string tension, surface tension and gravity Disturbance type: Medium: Wake of a boat Water Guitar string vibration Wire or fibers Thunderclap Air Earthquake vibration Earth Radio, visible light, microwaves… “Nothing” (no Aether) ! Steelers fans at a game Fans
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v6.01 RS/BQ F’10 118 there are two common simple types of waves, but we’ll also talk about a third transverse waves disturbance is perpendicular to motion of wave example: wave on rope longitudinal wave disturbance is in same direction of motion of wave example: sound, slinky torsion waves disturbance twists some medium axially to motion of wave waves having combinations of transverse and longitudinal disturbances possible deep water waves: circular motion (up-down, forward-backward) of surface particles A bobbing cork doesn’t move forward a wave is quantitatively described using a wave function Mathematical Representation of Waves (15-3) a wave function for a transverse wave is a function (,) yxt of both position x and time t which describes the displacement from equilibrium y at time t of the element whose equilibrium location is x example: waves on a taut string (along x axis) v v 1 t 2 t v v 1 t 2 t v
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v6.01 RS/BQ F’10 119 equilibrium is straight line (,) 0 yxt = let (,) be the transverse location at time t of the string segment at x textbook discusses sinusoidal waves first, but for contrast we’ll discuss a “Gaussian” wave packet first to learn the same physics as the book presents consider the example of a string with a small Gaussian (“Bell curve”) disturbance centered at 0 x = at time 0 t = with width σ and peak height A if the disturbance moves with speed v in the +x direction, then the wave function remains a Gaussian curve of height A and width , but now the location of the peak is at x vt = at time t if wave moves in the x direction with speed v , then at time t , the peak is located at x vt = − x y(x,t) String displacements at one instant in time v x=vt t=0 later time t x=0 2 2 2 ( , 0) x A e    = = ( ) 2 2 2 x vt A e = ( ) 2 2 2 x vt A e + = y x=0 A x σ y
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v6.01 RS/BQ F’10 120 mathematical interlude if x and t are independent variables, suppose then we can compute partial
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Physics II chapter 15 notes - Chapters 15: Mechanical Waves...

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