Ch11 Waves

• Period (T): The shortest time interval during which motion repeats.

Measures of a Wave

Time (s)

1 2 3 4 5 6 7 8

• Frequency (f): The number of complete vibrations per second.

Measures of a Wave

0.2 0.4 0.6 0.8 1.0 1.2 1.6 1.8

Time (s)

Number of vibration (cycles)

1 2 3 4 5

Waves• When these oscillations between two extremes are graphed wrt time, we

see the following profile emerge.

• The Wavelength () is the distance from the “same” point on two consecutive oscillations.

• The Amplitude (A) is the maximum displacement from zero.

• The Period (T) is the time between the same position on consecutive “humps.”

• The Frequency (f) describes how often an oscillation occurs.

• The high points on the wave are known as “crests.”

• The low points on the wave are known as “troughs.”

0

+A

-A

Wave Examples• Can two waves can have the same

wavelength and frequency, but different amplitudes?

.01 .02 .03 .04 .05 .06 .07 .08

The greater the amplitude the greater the energy.

Wave Boundaries

• What happens when a wave hits a boundary between two mediums?– Part of the wave is transmitted– Part is reflected

• The amount that gets transmitted versus reflected depends on the difference between two mediums.

Waves at BoundariesWaves at Boundaries

Reflected pulse

Transmitted pulse

Low Density Medium High Density Medium

Note: Both amplitudes get smaller

Waves at BoundariesWaves at Boundaries

High Density Medium Low Density Medium

Reflected pulse

Transmitted pulse

Wave Boundaries

• The frequency of a wave being transmitted from one medium to another does NOT change.

• e.g. If I’m moving a string up and down, I don’t change the velocity that I vibrate it.

Wave Types

Mechanical Waves: require a medium (material) to propagate.

Water Rope

Springs Sound

3 types of Mechanical Waves– Transverse– Longitudinal

Longitudinal Waves• A Longitudinal: A wave in which the vibration is in the same

direction that the wave is traveling.

• Notice how the atom in the box below never leaves the box even though the wave is obviously traveling to the right.

Animation courtesy of Dr. Dan Russell, Kettering University

Transverse Mechanical Waves• A transverse wave is one in which the individual atoms or

particles vibrate in a direction perpendicular to the direction of motion of the wave.

• Notice how the atoms in the box below never leave the box even though the wave is obviously traveling to the right.

Animation courtesy of Dr. Dan Russell, Kettering University

Rotations

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2

t

As the Wheel Turns• Watch how the sine function traces out as a wheel turns.

• The vertical axis represents horizontal position and the horizontal axis represents time.

• Note that one revolution (2π radians) is one sine wave cycle

Rotations

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2

t

Simple Harmonic Motion

2 ms kT

Period of a swing pendulum

2l

Tg

Simple Harmonic MotionSimple Harmonic Motion: Motion caused by a linear restoring force that has a period independent of amplitude.

Period: The time required to repeat

one complete cycle

Amplitude: Maximum displacement

from equilibrium.

Position versus time

Derive v, in terms of x and A

sin(2 )x A ft

x AKE EPE EPE

Simple Harmonic Motion

Derive v in terms of x and A

sin(2 )x A ft

maxKmv A

2

max 21x

v vA

v f

Doppler Effect• Sound waves propagate out from the source in

all direction. • If the source isn’t moving, the wavelengths are

constant

Police

Sonic boom from real player library

Doppler Effect

• Source moving towards you:1

( )1

o ss

f fvv

Police

• Source moving away from you:1

( )1

o ss

f fvv

Doppler Effect Example

A train is approaching you at 31m/s and blows its whistle of 305hz.

a)What frequency do you hear?

1( )1 so s v

s

f f

31 /343 /

1305 ( )

1o m sm s

f Hz

335of Hz

Doppler Effect ExampleA train is approaching you at 31m/s and blows its whistle of 305hz.

b) What frequency does your friend hear if the train has past him and continues moving away?

1( )1

o ss

f fvv

1305 ( )

31 /1 343 /of Hz

m sm s

279.7of Hz

Doppler Effect Example

1( )1 so s v

v

f f

1( )1 so s v

v

f f

(1 )ovo s vf f (1 )ovo s vf f

Towards Away

( )

( )s o

os

f v vf

v v

Superposition of Waves

• Principle of superposition: The displacement of a medium caused by two or more waves is the algebraic sum of each wave.

• Waves pass each other so the original wave continues unaltered.

• Interference is the result of the superposition of two or more waves.

Superposition of Waves

• Constructive Interference: Occurs when the displacements are in the same direction

• Destructive Interference: Occurs when the displacements are on opposite sides of equilibrium.

• Show excel demo

Wave Superposition

Graph A

Wave Sum

Constructive InterferenceConstructive InterferenceDestructive InterferenceDestructive Interference

Antinodes

AntinodesAntinodes

NodesNodes

Think First then ACT!

B

A

C

• Lets think about what is happening in terms of position, speed, and acceleration as the particle moves in simple harmonic motion.

• Which way does acceleration act at all times during the motion of an object moving in either circular or simple harmonic motion?

• Center seeking acceleration.• At what point(s) on the graph below is the

blue dot moving at the fastest speed?

• Slowest speed?

• What is the point physically doing as it approaches these two points?

Simple Harmonic and Circular Motion

• Simple harmonic motion may be thought of as the projection of circular motion into a linear scale.

• Notice how the laser strikes the mass at all times while the mass oscillates and while the disc rotates

• This demonstrates that both the rotating disk and the oscillating spring have the same period and angular frequency.

Phasors – the Displacement Vector • Any vector that rotates is known as a

Phasor.• The displacement vector “A” that

always points to the yellow point in the animation below is an example of a phasor.

• The horizontal and vertical components of the phasor A may be determined using the following equation. A

cosx A

siny A

Mathematical Description of a Wave

t

siny A

siny A t

1t 2t

t

2 f

sin 2y A ft

• The equation for a sinusoidal wave is as follows.

• Recall that the angular distance is and can be expressed in terms of frequency

• After substitution we get an expression for the vertical position of a particle y in terms of the horizontal position x and time t.

Simple Harmonic Motion and Phase Shifts

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0 0.2 0.4 0.6 0.8

Time

Am

plitu

de

f 0 f /2

The Phase Angle• In order to better understand the physical meaning of the phase angle, we will look at the graphs generated by

the black and blue points on the edge of the circle below.• Recall that the phase angle of the black point was f = 0 radians and the phase angle of the blue point was f =

/2.• This phase angle can be thought of as the lag or lead angular distance of one point when compared to another.• The blue graph leads the black graph by /2 radians.• Conversely, the black graph lags the blue graph by /2 radians.• This fact is true at every point on the graph.

cosx A t f

2

f

cosx t

cos sin2

x t t

Amplitude Variations

• Variations in amplitude only change the height of the waveform.• In the graph below, the three waves are in phase with each other (f = 0),

and they have the same angular frequency.• The amplitude of the black line is 1.• What is the amplitude of the green line?• What is the amplitude of the red line?• The equations of the three lines are as follows.

cosx t

2cosx t

1cos

2x t

Variations in Amplitude

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

0 0.2 0.4 0.6 0.8

Time

Am

pli

tud

e

Standing Waves

• Standing Wave: has stationary nodes and antinodes. It is the results of identical waves traveling in opposite direction.

• Node: The medium is not displaced as the waves pass through

• Antinode: The displacement caused by interfering waves is largest.