Wave Energy and Superposition

Physics 202Professor Lee

CarknerLecture 7

Consider mark made on a piece of string with a wave traveling down it. At what point does the mark have the largest velocity? : At what point does the mark have the largest acceleration?

a) In the middle : At the topb) At the top : In the middlec) In the middle : In the middled) At the top: At the tope) Velocity and acceleration are

constant

Suppose you are producing a wave on a string by shaking. What properties of the wave do you directly control?

a) Amplitudeb) Wavelengthc) Frequencyd) Propagation velocitye) a and c only

PAL #6 Waves

=2/k so A = , B = /2, C = /3 T = 2/ so TA = , TB = 1/3, TC = 1/4 Which wave has largest transverse velocity?

Wave C: largest amplitude, shortest period Largest wave speed?

v =f = /T, vA = 1, vB = 1.5, vC = 1.3

A: y=2sin(2x-2t), B: y=4sin(4x-6t), C: y=6sin(6x-8t)

PAL #6 Waves (cont.) Wave with y = 2 sin (2x-2t), find time when x=

5.2 cm has max a Happens when y = ym = 2

2 = 2 sin (2x-2t) 1 = sin (2x-2t) arcsin 1 = 2x-2t /2 = (2x - 2t) t = [2x-(/2)]/2 t = 4.4 seconds

Maximum velocity when y = 0 0 = sin (2x-2t) 2x -2t = arcsin 0 = 0 t = x t = 5.2 seconds

Velocity and the Medium

If you send a pulse down a string what properties of the string will affect the wave motion?

Tension ()

If you force the string up, tension brings it back down Linear density ( = m/l =mass/length)

You have to convert the PE to KE to have the string

move

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:

and centripetal force:

Solving for v,

This is also equal to our previous expression for v

v = f

String Properties

How do we affect wave speed?v = ()½ = f

Wave speed is solely a property of the medium

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

Every time we shake the string up and down we add a little more energy

This energy is transmitted down the string

The energy of a given piece of string changes with time as the string stretches and relaxes

Assuming no energy dissipation

Power Dependency

P=½v2ym2

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 ym and depend on the wave generation process

Superposition

When 2 waves overlap each other they add algebraically

Traveling waves only add up as they

overlap and then continue on

Waves can pass right through each other with no lasting effect

Pulse Collision

Interference

The waves may be offset by a phase constant y1 = ym sin (kx - t)

y2 = ym sin (kx - t +)

yr = ymr sin (kx - t +½) What is ymr (the resulting amplitude)?

Is it greater or less than ym?

Interference and Phase

ymr = 2 ym cos (½) The phase constant can be

expressed in degrees, radians or wavelengths Example: 180 degrees = radians = 0.5

wavelengths

Resultant Equation

Combining Waves

Types of Interference Constructive Interference -- when the

resultant has a larger amplitude than the originals No offset or offset by a full wavelength

Destructive Interference -- when the resultant has a smaller amplitude than the originals Offset by 1/2 wavelength

Next Time

Read: 16.11-16.13 Homework: Ch 16, P: 20, 30, 40,

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