Physics 2111 Unit 23 - College of DuPageTypes of Waves Longitudinal: The medium oscillates in the same direction as the wave is moving. Mechanics Lecture 23, Slide 3 Transverse: The

Physics 2111

Unit 23

Today’s Concept:

Waves

Wave Equation

Resonance

Energy

Mechanics Lecture 23, Slide 1

What is a Wave?

A wave is a traveling disturbance that transports energy but not matter.

Examples: Sound waves (air moves back & forth) Stadium waves (people move up & down) Water waves (water moves up & down) Light waves (what moves?)


Types of Waves

Longitudinal: The medium oscillates in the same direction as the wave is moving.


Transverse: The medium oscillates perpendicular to the direction the wave is moving.

• Waves on a string

• Water waves

• Light Waves

• Sound Waves

How to make a Function Move

Suppose we have some function y = f (x):

x

y

0


f (x - a) is just the same shape moveda distance a to the right:

x = ax

y

0

Let a = vt Then

f (x - vt) will describe the same shape moving to the right with speed v.

x = vt

v

x

y

0

If a function moving to the right with speed v is described by f (x - vt) then what describes the same function moving to the left with speed v?

v

x

y

0

y = f (x - vt)

v A) y = - f (x - vt)

B) y = f (x + vt)

C) y = f (-x + vt)

Question


x

y

0x

y

0x

y

Harmonic Wave

Consider a wave that is harmonic in xand has a wavelength of .


Has the functional form:

xAxy

2cos)(

Harmonic Wave


2

( ) cosy x A x vt

-

cos( )A kx t -

Give it speed v

=Acos(2𝜋

𝜆𝑥 −

2𝜋

𝜆𝑣𝑡)

v=/P

Amplitude:

The maximum displacement A of a point on the wave.

A

Wavelength: The distance between identical points on the wave.

Period: The time P it takes for an element of the medium to make one complete oscillation.

cos( )y A kx t -

Pv

k

2k

f2

P

2

Wave Properties


Wavelength

Not the spring constant!

position(x)

NOTE:

Wave Properties


PeriodP

Same plot but vs time

time(t)

Question


A boat is moored in a fixed location and waves make it move up

and down. If the spacing between wave crests is L and the

speed of the waves is v, how much time Δt does it take the boat

to go from the top of one wave to the top of the next?

A. Δt = L/v

B. Δt = Lv

C. Δt = v/L

You shake a rope up and down a distance of 40cm with a frequency of 10Hz. The wave formed has a velocity of 5m/sec. What is the equation for this wave?

Example 23.1 (Traveling Wave)


Example 23.2: Wave Graph


The figure to the left shows

height vs. displacement plot for a

string which has a wave traveling

in the positive x direction at time

t=5.0 sec with a velocity of 12.0

m/sec.

a) What is the amplitude of this wave?

b) What is the wavelength of this wave?

c) What is the frequency of this wave?

d) What is the equation of motion of this string? (i.e. y(x,t)=?)

e) What is the vertical (y) velocity of a piece of string at the

point labeled 1?

f) What is the vertical (y) acceleration of a piece of string at

the point labeled 1?

You have a 2m long piece of string that has a mass of 0.6 grams. If you loop it over a pulley and hang a 200gram mass from one end, what is the velocity of wave in the string?

Example 23.2 (Wave Speed)


Linear Superposition

What happens when two waves “collide”?

x


x

Constructive Interference

Or……

Destructive Interference

x


x

Demo

http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html

Y(x,t) = 2*cos(kx)*cos(t)

In the equation we just derived, when is the value of Y always equal to zero?

A. When kx = 0

B. When kx = /2

C. When kx =

D. (A) and (C)

E. Y varies with time and isn’t always zero anywhere

Question


Resonance


Fundamental

1st harmonic

1st Overtone

2nd harmonic

2nd Overtone

3rd harmonic

L

L = n* /2Resonant frequencies for strings always node at both ends

CheckPoint


L

The wave length of the above wave is:

A. 2 L

B. 1.5 L

C.1 L

D.2/3 L

E. 0.5 L

L = n* /2

Resonant frequencies


L = n* (v/f)/2

f= v*n/(L 2)

Velocity -

determined

by tension

and density

of string

(v2 = T/m

Length of

string

But what is n?

Resonance


Fundamental

1st harmonic

1st Overtone

2nd harmonic

2nd Overtone

3rd harmonic

L

f= v*n/(L 2)

n = 1

n = 2

n = 3

When you tune a guitar, you are adjusting what resonant frequency at which each string will vibrate. You do this by:

A. Adjusting the length of the string

B. Adjusting the density of the string

C. Adjusting the tension in the string

D. Adjusting the speed of sound near the string

Question


When you play a guitar, you are also adjusting what resonant frequency at which each string will vibrate. You do this by:

A. Adjusting the length of the string

B. Adjusting the density of the string

C. Adjusting the tension in the string

D. Adjusting the speed of sound near the string

Question


In our previous example problem, we had a 2m long piece of string that has a mass of 0.6 grams. We looped it over a pulley and hung a 200gram mass from one end. What is fundamental frequency of the string?

Example 23.3 (fundamental frequency)


5m

The velocity we find using the formula

is the velocity of the wave.

Velocity


2 Tvm

The velocity of any little piece of the string varies with time. We get it by taking the derivative of y(x,t) = A*cos(kx-t)

Energy


Documents

Physics 2111 Unit 23 - College of DuPageTypes of Waves Longitudinal: The medium oscillates in the same direction as the wave is moving. Mechanics Lecture 23, Slide 3 Transverse: The