Physics 151: Lecture 34, Pg 1 Physics 151: Lecture 34 Today’s Agenda l Topic - Waves (Chapter 16 ) ç1-D traveling waves çWaves on a string çWave speed

Physics 151: Lecture 34, Pg 2 What is a wave ? l A definition of a wave: çA wave is a traveling disturbance that transports energy but not matter. l Examples: çSound waves (air moves back & forth) çStadium waves (people move up & down) çWater waves (water moves up & down) çLight waves (what moves ??) Animation

Physics 151: Lecture 34, Pg 3 Types of Waves l Transverse: The medium oscillates perpendicular to the direction the wave is moving. çWater (more or less) çString waves See text: 16.2 see Figures l Longitudinal: The medium oscillates in the same direction as the wave is moving çSound çSlinky Animation

Physics 151: Lecture 34, Pg 4 Wave Properties Wavelength Wavelength: The distance between identical points on the wave. Amplitude A l Amplitude: The maximum displacement A of a point on the wave. A See text: 16.2 Animation

Physics 151: Lecture 34, Pg 5 Wave Properties... l Period: The time T for a point on the wave to undergo one complete oscillation. Speed: The wave moves one wavelength in one period T so its speed is v =  / T. See text: 16.2 see Figure 16.6

Physics 151: Lecture 34, Pg 6 Wave Properties... We will show that the speed of a wave is a constant that depends only on the medium, not on amplitude, wavelength or period and T are related ! See text: 16.2 v = / T = v T or = 2  v /  (since  T = 2  /   or  v / f (since T = 1/ f ) l Recall f = cycles/sec or revolutions/sec  rad/sec = 2  f

Physics 151: Lecture 34, Pg 7 Which of the graphs on the right shows a wave where the frequency and wave velocity are both doubled ? Example The figure on the right shows a sine wave on a string at one instant of time.

Physics 151: Lecture 34, Pg 8 Lecture 34, Act 1 Wave Motion l 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. l Suppose we make a sound wave and a light wave that both have a wavelength of 3 meters. çWhat is the ratio of the frequency of the light wave to that of the sound wave ? (a) About 1,000,000 (b) About.000,001 (c) About 1000

Physics 151: Lecture 34, Pg 9 Lecture 34, Act 1 Solution l What are these frequencies ??? For sound having = 3m : (low humm) For light having = 3m : (FM radio)

Physics 151: Lecture 34, Pg 10 Wave Forms continuous waves l So far we have examined “continuous waves” that go on forever in each direction ! v v l We can also have “pulses” caused by a brief disturbance of the medium: v l And “pulse trains” which are somewhere in between. See text: 16-1 see Figure 16.3

Physics 151: Lecture 34, Pg 11 Mathematical Description l Suppose we have some function y = f(x): x y l f(x-a) is just the same shape moved a distance a to the right: x y x=a 0 0 l Let a=vt Then f(x-vt) will describe the same shape moving to the right with speed v. x y x=vt 0 v See text: 16.3 see Figure 16.7

Physics 151: Lecture 34, Pg 12 Math... Consider a wave that is harmonic in x and has a wavelength of. If the amplitude is maximum at x=0 this has the functional form: y x A l Now, if this is moving to the right with speed v it will be described by: y x v See text: 16.3

Physics 151: Lecture 34, Pg 13 Math... l By usingfrom before, and by defining l So we see that a simple harmonic wave moving with speed v in the x direction is described by the equation: we can write this as: (what about moving in the -x direction ?) See text: 16-2

Physics 151: Lecture 34, Pg 14 Math Summary l The formula describes a harmonic wave of amplitude A moving in the +x direction. y x A Each point on the wave oscillates in the y direction with simple harmonic motion of angular frequency . l The wavelength of the wave is l The speed of the wave is l The quantity k is often called “wave number”. Movie (twave) See text: 16-2

Physics 151: Lecture 34, Pg 15 Lecture 34, Act 2 Wave Motion A harmonic wave moving in the positive x direction can be described by the equation y(x,t) = A cos ( kx -  t ) l Which of the following equation describes a harmonic wave moving in the negative x direction ? (a) y(x,t) = A sin (  kx   t ) (b) y(x,t) = A cos ( kx   t ) (c) y(x,t) = A cos (  kx   t )

Physics 151: Lecture 34, Pg 16 Lecture 34, Act 3 Wave Motion A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 meters and the speed of the waves is 5 m/s, how long  t does it take the boat to go from the top of a crest to the bottom of a trough ? (a) 2 sec (b) 4 sec (c) 8 sec t t +  t

Physics 151: Lecture 34, Pg 17 Waves on a string l What determines the speed of a wave ? l Consider a pulse propagating along a string: v l “Snap” a rope to see such a pulse l How can you make it go faster ? See text: 16.5 Animation

Physics 151: Lecture 34, Pg 18 Waves on a string... l The tension in the string is F The mass per unit length of the string is  (kg/m) l The shape of the string at the pulse’s maximum is circular and has radius R R  F Suppose: See text: 16.5

Physics 151: Lecture 34, Pg 19 Waves on a string... v x y l Consider moving along with the pulse l Apply F = ma to the small bit of string at the “top” of the pulse which is moving with Uniform Circular Motion. See text: 16.5 see Figure 16-11

Physics 151: Lecture 34, Pg 20 Waves on a string...   F F x y l The total force F TOT is the sum of the tension F at each end of the string segment. l The total force is in the -y direction. F TOT = 2F  (since  is small, sin  ~  ) See text: 16.5

Physics 151: Lecture 34, Pg 21 Waves on a string...  m = R 2  R x y The mass m of the segment is its length (R x 2  ) times its mass density .   See text: 16.5

Physics 151: Lecture 34, Pg 22 Waves on a string... R v x y l The acceleration a of the segment is v 2 / R (centripetal) in the -y direction. a See text: 16.5

Physics 151: Lecture 34, Pg 23 Waves on a string... l So F TOT = ma becomes: F TOT m a See text: 16.5 v tension F mass per unit length 

Physics 151: Lecture 34, Pg 24 Waves on a string... l So we find: l Making the tension bigger increases the speed. l Making the string heavier decreases the speed. l As we asserted earlier, this depends only on the nature of the medium, not on amplitude, frequency etc of the wave. v tension F mass per unit length  See text: 16.5 Animation-1 Animation-2 Animation-3

Physics 151: Lecture 34, Pg 25 Lecture 34, Act 4 Wave Motion l A heavy rope hangs from the ceiling, and a small amplitude transverse wave is started by jiggling the rope at the bottom. l As the wave travels up the rope, its speed will: (a) increase (b) decrease (c) stay the same v l Can you calcuate how long will it take for a pulse travels a rope of length L and mass m ?

Physics 151: Lecture 34, Pg 26 Recap of today’s lecture l Chapter 16 çDefinitions ç1-D traveling waves çWaves on a string çWave speed l Next time: Finish Chapter 16