Transcript waves in a string

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 (t)
The string tension provides restoring force
If you force the string up, tension brings it back
down & vice versa
Linear density (m = 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 Dl of linear density m
with a tension t pulling on each end moving in a
very small arc a distance R from rest
 There is a force balance between tension force:
 F = (t Dl)/R
 and centripetal force:
F = (m Dl) (v2/R)
 Solving for v,
 v = (t/m)½
 This is also equal to our previous expression for v
v = lf
String Properties
How do we affect wave speed?
v = (t/m)½ = lf
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=½mvw2ym2
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 m depend on the string
ym and w depend on the wave generation process
Equation of a Standing Wave
Equation of standing wave:
yr = [2ym sin kx] cos wt
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 = np, sin kx = 0 and y = 0
Node:
x=n (l/2)
Nodes occur every 1/2 wavelength
For kx=(n+½)p, sin kx = 1 and y=2ym
Antinode:
x=(n+½) (l/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 l = 2L/n
n is the number of loops on a string
fractions of n don’t work
v = (t/m)½ = lf
Changing, m, t, or f will change l
Can find new l in terms of old l and see if it is
an integer fraction or multiple