Chapter 11 Waves. MFMcGrawCh-11b-Waves - Revised 4-3-102 Chapter 11 Topics Energy Transport by Waves Longitudinal and Transverse Waves Transverse Waves

Chapter 11

Waves

MFMcGraw Ch-11b-Waves - Revised 4-3-10 2

Chapter 11 Topics • Energy Transport by Waves

• Longitudinal and Transverse Waves

• Transverse Waves on Strings

• Periodic Waves

• Superposition

• Reflection and Refraction of Waves

• Interference and Diffraction

• Standing Waves on a String


Waves and Energy Transport

• A wave is a disturbance that travels, in a medium, outward from its source.

• Waves carry energy.

• The energy is transported outward from the source to the destination.

• The matter in the medium expieriences no net transport.


When a stone is dropped into a pond, the water is disturbed from its equilibrium positions as the wave passes; it returns to its equilibrium position after the wave has passed.

The water moves up and down as the disturbance moves outward but is not transported by the wave.



A Simple Water Wave

The wave travels to the right. The individual water molecules move up and down locally but do not travel with the wave. A surfer CAN travel with the wave, at least for a while.

Trough Peak


3-D Spherical Wave

Energy spreads out, from a point source, uniformly over a 3 dimensional area

Intensity = Power/Area

Units: Watts/m2


Intensity is a measure of the amount of energy/sec that passes through a square meter of area perpendicular to the wave’s direction of travel.

22 r4r4

Power

P

I Intensity has units of watts/m2 .

This is an inverse square law. The intensity drops as the inverse square of the distance from the source.

(Light sources appear dimmer the farther away from them you are.)



Example: At the location of the Earth’s upper atmosphere, the intensity of the Sun’s light is 1400 W/m2. What is the

intensity of the Sun’s light at the orbit of the planet Mercury?

2es

sune 4 r

PI

2ms

sunm 4 r

PI

Divide one equation by the other (take a ratio):

2em

2

10

112

ms

es

2es

sun

2ms

sun

e

m

W/m920057.6

57.6m 1085.5

m 1050.1

r 4

r 4

II

r

rP

P

I

I



Longitudinal WaveAmplitude change parallel to propagation direction

Transverse WaveAmplitude change perpendicular to propagation direction


The Excitation of a Transverse Wave

The speed with which the string is moved vertically is independent of the speed with the wave travels horizontally down the string.

Boundary Condition:The end of the string is stationary


Transverse Waves on a String

M

Attach a wave driver here

L

Attach a mass to a string to provide tension. The string is then shaken at one end with a frequency f.


A wave traveling on this string will have a speed of

F

v

where F is the force applied to the string (tension) and is the mass/unit length of the string (linear mass density).



Example (text problem 11.8): When the tension in a cord is 75.0 N, the wave speed is 140 m/s. What is the linear mass density of the cord?

F

v The speed of a wave on a string is

kg/m 108.3

m/s 140

N 0.75 322

v

F

Solving for the linear mass density:



Speed of Sound in Various Materials

F Forcev

Inertia

Bv

Yv

Incr

easi

ng a

ttra

cti v

e fo

rce

String Liquid Solid

High mass Low Mass


Periodic Waves

A periodic wave repeats the same pattern over and over.

• For periodic waves: v = f

• v is the wave’s speed

• f is the wave’s frequency

is the wave’s wavelength


A Simple Harmonic Oscillator

Its motion, depicted as a function of time, is a wave.

Apeak-peak = Ap-p = 2Apeak = 2Ap; Ap = Ap-p /2


The period T is measured by the amount of time it takes for a point on the wave to go through one complete cycle of oscillations. The frequency is then f = 1/T.

Periodic Waves


The maximum displacement from equilibrium is amplitude (A) of a wave.

One way to determine the wavelength is by measuring the distance between two consecutive crests.

Periodic Waves


Example (text problem 11.13): What is the wavelength of a wave whose speed and period are 75.0 m/s and 5.00 ms, respectively?

m 3750s 10005m/s 075 3 ...vT

Solving for the wavelength:

Tfv

Periodic Waves


Wave Properties


The Principle of Superposition

For small amplitudes, waves will pass through each other and emerge unchanged.

Superposition Principle: When two or more waves overlap, the net disturbance at any point is the sum of the individual disturbances due to each wave.


Two traveling wave pulses: left pulse travels right; right pulse travels left.


Reflection and Refraction

At an abrupt boundary between two media, a reflection will occur. A portion of the incident wave will be reflected backward from the boundary.


When you have a wave that travels from a “low density” medium to a “high density” medium, the reflected wave pulse will be inverted. (180o phase shift.)

The frequency of the reflected wave remains the same.

The Reflected Wave & Phase Change


When a wave is incident on the boundary between two different media, a portion of the wave is reflected, and a portion will be transmitted into the second medium. Reflected ray is 180o out of phase.

Light Ray Example


The frequency of the transmitted wave remains the same. However, both the wave’s speed and wavelength are changed such that:

2

2

1

1

vv

f

The transmitted wave will also suffer a change in propagation direction (refraction).

The Frequency is Constant


Example (text problem 11.36)

Light of wavelength 0.500 m in air enters the water in a swimming pool. The speed of light in water is 0.750 times the speed in air. What is the wavelength of the light in water?

m3750m 50007500

air

air

airair

waterwater

water

water

air

air

..v

v.

v

v

vvf

Since the frequency is unchanged in both media:


Interference

Two waves are considered coherent if they have the same frequency and maintain a fixed phase relationship.

Two waves are considered incoherent if the phase relationship between them varies randomly.


When waves are in phase, their superposition gives constructive interference.

When waves are one-half a cycle out of phase, their superposition gives destructive interference.

This is referred to as:

“exactly out of phase” or “180o out of phase.”


Constructive Interference


Destructive Interference


Interference


Constructive Interference. Means that the waves ADD together and their amplitudes are in the same direction

Destructive Interference. Means that the waves ADD together and their amplitudes are in the opposite directions.

Interference = Bad choice of words

• The two waves do not interfere with each other.

• They do not interact with each other.

• No energy or momentum is exchanged.


When two waves travel different distances to reach the same point, the phase difference is determined by:

2

difference phase21 dd

Note: This is a ratio comparison. λ is not equal to 2π

Phase Difference


Diffraction is the spreading of a wave around an obstacle in its path and it is common to all types of waves.

The size of the obstacle must be similar to the wavelength of the wave for the diffraction to be observed.

Larger by 10x is too big and

smaller by (1/10)x is too small.

Diffraction


Standing Waves

Pluck a stretched string such that y(x,t) = A sin(t + kx)(This is a wave that moves to the left.)

When the wave strikes the wall, there will be a reflected wave that travels back along the string.


The reflected wave will be 180° out of phase with the wave incident on the wall.

Its form is y(x,t) = A sin (t kx). (This is a wave that moves to the right.)

Apply the superposition principle to the two waves on the string:

kxtA

kxtkxtA

txytxytxy

sincos2

sinsin

),(),(),( 21

The time variation and the spatial variation have been separated.


The previous expression is the mathematical form of a standing wave.

N

NN

N

AAA

A node (N) is a point of zero oscillation. An antinode (A) is a point of maximum displacement. All points between nodes oscillate up and down.


The nodes occur where y(x,t) = 0.

0sincos2, kxtAtxy

The nodes are found from the locations where sin kx = 0, which happens when kx = 0, , 2,…. That is when kx = n where n = 0,1,2,…

The antinodes occur when sin kx = 1; that is where

,,,nn

kx

kx

2 1 0 and 2

12

,2

3,

2


If the string has a length L, and both ends are fixed, then y(x = 0, t) = 0 and y(x = L, t) = 0. (Boundary conditions)

n

L

nL

nkL

kLtLxy

ktxy

2

2

0sin,

00sin,0

The wavelength of a standing wave:

where n = 1, 2, 3,…


n

Ln

2 These are the permitted wavelengths of

standing waves on a string; no others are allowed.

The speed of the wave is: nn fv

The allowed frequencies are then:

L

nvvf

nn 2

n =1, 2, 3,…


The n = 1 frequency is called the fundamental frequency.

122nf

L

vn

L

nvvf

nn

All allowed frequencies (called harmonics) are integer multiples of f1.


Example (text problem 11.51): A Guitar’s E-string has a length 65 cm and is stretched to a tension of 82 N. It vibrates with a fundamental frequency of 329.63 Hz. Determine the mass per unit length of the string.

kg/m 1054

m 650*2Hz63329

N 82

2

422

221

211

2

.. .

Lf

F

f

F

v

F

F

v For a wave on a string:

Solving for the linear mass density:



Two Methods of Classification

Just to make life interesting

1st Harmonic

2nd Harmonic

3rd Harmonic

Fundamental

1st Overtone

2nd Overtone

The labels on the right are more general. If the frequencies are integer multiples of one another then they are referred to as harmonics


Summary

• Energy Transport by Waves

• Longitudinal and Transverse Waves

• Transverse Waves on Strings

• Periodic Waves

• Superposition

• Reflection and Refraction of Waves

• Interference and Diffraction

• Standing Waves on a String


Extra


Wave Properties

The sound in an acoustic instrument comes from the vibrating strings moving the air and coupling into the resonant cavity of the instrument whose walls vibrate and in turn cause vibrations in the surrounding air pressure that we interpret as sound. It acts as a sound amplifier

Documents

Chapter 11 Waves. MFMcGrawCh-11b-Waves - Revised 4-3-102 Chapter 11 Topics Energy Transport by Waves Longitudinal and Transverse Waves Transverse Waves