String Waves Physics 202 Professor Lee Carkner Lecture 8

Exam #1 Friday, Dec 12  10 multiple choice  4 problems/questions  You get to bring a 3”X5” card of equations and/or notes  Start making it now  I get my inspiration from your assignments  Make sure you know how to do homework, PAL’s/Quizdom, discussion questions  Bring calculator – be sure it works for you

Velocity and the Medium  The speed at which a wave travels depends on the medium  If you send a pulse down a string what properties of the string will affect the wave motion?  Tension (  )  The string tension provides restoring force  If you force the string up, tension brings it back down & vice versa  Linear density (  = m/l =mass/length)  The inertia of the string  Makes it hard to start moving, makes it keep moving through equilibrium

Wave Tension in a String

Force Balance on a String Element  Consider a small piece of string  l of linear density  with a tension  pulling on each end moving in a very small arc a distance R from rest  There is a force balance between tension force:  F = (   l)/R  and centripetal force:  F = (   l) (v 2 /R)  Solving for v,  v = (  ) ½  This is also equal to our previous expression for v v = f

String Properties  How do we affect wave speed?  v = (  ) ½ = f  A string of a certain linear density and fixed tension has a fixed wave speed  Wave speed is solely a property of the medium  We set the frequency by how fast we shake the string up and down  The wavelength then comes from the equation above  The wavelength of a wave on a string depends on how fast you move it and the string properties

Tension and Frequency

Energy  A wave on a string has both kinetic and elastic potential energy  We input this energy when we start the wave by stretching the string  Every time we shake the string up and down we add a little more energy  This energy is transmitted down the string  This energy can be removed at the other end  The energy of a given piece of string changes with time as the string stretches and relaxes  The rate of energy transfer is this change of energy with time  Assuming no energy dissipation

Power Dependency  The average power (energy per unit time) is thus:  P=½  v  2 y m 2  If we want to move a lot of energy fast, we want to add a lot of energy to the string and then have it move on a high velocity wave  v and  depend on the string  y m and  depend on the wave generation process

Equation of a Standing Wave  Equation of standing wave:  y r = [2y m sin kx] cos  t  The amplitude varies with position  e.g. at places where sin kx = 0 the amplitude is always 0 (a node)

Nodes and Antinodes  Consider different values of x (where n is an integer)  For kx = n , sin kx = 0 and y = 0  Node:  x=n ( /2)  Nodes occur every 1/2 wavelength  For kx=(n+½) , sin kx = 1 and y=2y m  Antinode:  x=(n+½) ( /2)  Antinodes also occur every 1/2 wavelength, but at a spot 1/4 wavelength before and after the nodes

Resonance?  Under what conditions will you have resonance?  Must satisfy = 2L/n  n is the number of loops on a string  fractions of n don’t work  v = (  ) ½ = f  Changing, , , or f will change  Can find new in terms of old and see if it is an integer fraction or multiple