Vibrations & Waves

Chapter 25 - This will be phun!

2 Types of WavesMechanical Wave:

Requires a mechanical mediumSound, water, air, springs, or ropes are examples.

Electromagnetic Waves (EM):Does not require a medium for motion to occurLight, Radio, and X-rays are examples.

“Making Waves”

Transverse WavesCauses the particles of the medium to vibrate perpendicularly to the direction of motion of the wave.Piano and guitar strings are examples

Longitudinal WavesWhen particles of the medium move parallel to the direction of the waves.Fluids, liquids, gases, or plasma usually transmit only longitudinal waves.

Longitudinal and Transverse

Longitudinal vs Transverse Waves

Compression = CrestRarefaction = TroughEnergy Movement:parallel vs perpendicularWavelength: compression + rarefaction

crest + trough

Surface Waves

They are a mixture of transverse and longitudinal waves. (water & Rayleigh)The particles move both parallel and perpendicular to the direction of the wave.

Wave Pulse and Traveling WaveWave Pulse:

A single disturbance that travels through a medium.

Traveling Wave:Moving, periodic disturbances in a medium or field.

PeriodThe shortest time interval during which the motion repeats itself.Abbreviated with the capital letter,TSI Unit: seconds (s)

FrequencyThe number of complete revolutions per second.Frequency is abbreviated with a fancy ƒ.Frequency is measured in Hertz, Hz.A Hertz is one vibration per second (1/s).

EquationFrequency and the period of a wave are

related by the following equation.

Frequency and Period are reciprocals of each other.

1T

Wavelength

The shortest distance between points where the wave pattern repeats itself.The wavelength is abbreviated with the Greek letter, lambda,

A: ?B: ?C: ?D: ?E: ?

Wavelength

The shortest distance between points where the wave pattern repeats itself.The wavelength is abbreviated with the Greek letter, lambda,

A: 1 WavelengthB: 2X AmplitudeC: NodesD: AmplitudeE: ½ Wavelength

VocabularyCrests:

The high points of each wave motion.

Troughs:The low points of each wave motion

Amplitude:The maximum displacement from the rest or equilibrium position.

Nodes:Where the wave crosses the equilibrium line.

Antinodes:The bottom of the trough and the top of the crest

VocabularyCrests:

The high points of each wave motion.

Troughs:The low points of each wave motion

Amplitude:The maximum displacement from the rest or equilibrium position.

Nodes:Where the wave crosses the equilibrium line.

Antinodes:The bottom of the trough and the top of the crest

A&F: Crests (Antinodes)D&I: Troughs (Antinodes)B,E,G,J: Nodes

To find the velocity of a waveWave velocity (v) is the product of the frequency (f) and wavelength ().To find out how fast a wave moves, you would use this equation…

=or v T

=v

Amplitude and EnergyIn order to produce a wave with a larger amplitude, more energy is needed.Waves with larger amplitudes transfer more energy.Amplitude does not affect frequency nor velocity.

Waves Changing MediumsWaves passing from one medium to another have the same frequency.The wavelength change depends on the velocity change so that f is constant.

If the velocity increases, the wavelength increases (direct relationship).

v/

Superposition and InterferencePrinciple of Superposition:

Two or more waves occupying the same space.

Interference:The result from two or more waves occupying the same space.

Constructive InterferenceOccurs when the wave displacements are in phase (crest meets crest or trough meets trough).The result is a wave with a larger amplitude than the individual waves.

Destructive InterferenceOccurs when the wave displacements are out of phase (crest meets trough).The result is a wave with a smaller amplitude than the individual waves. Red: wave moving right

Blue: wave moving leftGreen: superposition

(Red + Blue wave)

Destructive InterferenceIf the pulses have unequal amplitudes, destructive interference is not complete. The pulse of the overlap is the algebraic sum of the two pulses.

Red: wave moving rightBlue: wave moving leftGreen: superposition

(Red + Blue wave)

Standing Wave

When the nodes and antinodes are stationary, the wave appears to be standing still.If you increase the frequency of a standing wave, you will see more nodes.

Superposition of Waves

A. Two pulses traveling in opposite directionsB. Two sine waves traveling in the same direction, but at different speedsC. Two sine waves traveling in opposite directions.

http://paws.kettering.edu/~drussell/Demos/superposition/superposition.html

Nodes and AntinodesNode:

The point in the medium that is completely undisturbed at all times. A node is produced by destructive interference of waves

Antinode:The point of maximum displacement. An antinode is formed from constructive interference.

Harmonics

Let’s check for understanding…The number of nodes in the standing wave shown in the diagram

at the right isa. 6b. 7c. 8d. 14

Let’s check for understanding…The number of nodes in the standing wave shown in the diagram

at the right is

c. 8

Let’s check for understanding…The number of

antinodes in the standing wave shown in the diagram at the

right isa. 6b. 7c. 8d. 14

Let’s check for understanding…The number of

antinodes in the standing wave shown in the diagram at the

right is

b. 7

Let’s check for understanding…In the standing wave shown,

a. What is the amplitude?b. What is its wavelength?

c. How many nodes are there?d. How many antinodes are there?

Let’s check for understanding…In the standing wave shown,

a. What is the amplitude? 10 cmb. What is its wavelength?

c. How many nodes are there?d. How many antinodes are there?

Let’s check for understanding…In the standing wave shown,

a. What is the amplitude? 10 cmb. What is its wavelength? 1 mc. How many nodes are there?

d. How many antinodes are there?

Let’s check for understanding…In the standing wave shown,

a. What is the amplitude? 10 cmb. What is its wavelength? 1 m

c. How many nodes are there? 6d. How many antinodes are there?

Let’s check for understanding…In the standing wave shown,

a. What is the amplitude? 10 cmb. What is its wavelength? 1 m

c. How many nodes are there? 6d. How many antinodes are there? 5

Reflection of WavesNormal:

A line that is drawn perpendicular to the barrier (green).

Angle of Incidence:The angle between the incidence ray and the normal.

Angle of Reflection:The angle between the normal and the reflected ray.

>I = >R

Refraction of WavesRefraction:

The change in the direction of waves at the boundary between two different media.

Diffraction of WavesDiffraction:

The spreading of waves around the edge of a barrier. Diffraction occurs when waves meet a small obstacle.They can bend around the obstacle, producing waves behind it.

Problem-Solving

Springs

Spring Constant

Spring Constant (stiffness)A spring stretches 18 centimeters when a 56 Newton weight is suspended from it. What is the spring constant?Find: kGivens: d (x) = 18 cm = 0.18 mF = 56 NFormula: k = F dSolution: 310 N/m

Springs

Potential Energy in a Spring

Period of a Pendulum

Pendulum

Using a PendulumA pendulum with a length of 36.9 centimeters has a period of 1.22 seconds. What is the acceleration due to gravity at the pendulum’s location?Find: a (g)Givens: d = 36.9 cm = 0.369 m

T = 1.22 sFormula: g = 42L

T2

Solution: 9.78 m/s2

Velocity, Wavelength, Frequency and Period Relationships

Wavelength

WavelengthAn 855 Hertz disturbance moves through an iron rail at a speed of 5130 meters per second. What is the wavelength of the disturbance?Find: Givens: f = 855 Hz

v = 5130 m/sFormula: = v

fSolution: 6.00 m

Velocity, Wavelength, Frequency and Period Relationships

Period

PeriodAn 855 Hertz disturbance moves through an iron rail at a speed of 5130 meters per second. What is the period of the disturbance?Find: TGivens: f = 855 HzFormula: T = 1

fSolution: 0.00117 s

Velocity, Wavelength, Frequency and Period Relationships

Velocity

VelocityA sound wave has a frequency of 192 Hertz and travels the length of a football field, 91.4 meters, in 0.271 seconds. What is the speed of the wave?Find: vGivens: f = 192 Hz

d = 91.4 mt = 0.271 s

Formula: v = d t

Solution: 337 m/s

VelocityA sonar signal of frequency 1.00 X 106 Hertz has a wavelength of 1.50 millimeters in water. What is the speed of the signal?Find: vGivens: f = 1.00 X 106 Hz

= 1.50 mm = 0.00150 mFormula: v = fSolution: 1.50 X 103 m/s