Department of Physics and Applied Physics95.141, F2010, Lecture 24

Physics I95.141

LECTURE 2412/8/10

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Administrative Notes

• Physics I Final:– TUESDAY 12/14/10– Olney 150 (HERE)– 8:00 A.M.

• Review Session Sunday (5/13), 6:30 pm, OH218.• Practice Exams Posted

– Will try to post solutions in near future

• Practice problems posted– 3+ Will Be on Exam– Everything from the semester is fair game

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Review of Lecture 23• Introduced concept of the pendulum, and determined restoring

Force acting on pendulum for small displacements.

• Natural frequency, Period of pendulum given by

• Introduced damping force to harmonic motion

• Discussed Forced Oscillations

• Used Ao to discuss concept of resonance

gT

g

2,

2

2

4,2,)cos()(m

bmk

mbt tAetx

)sin( oo tAx 2

22222 )(m

bo

oom

FA

)(tan

221

mb

oo

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Chapter 15: Waves

• A wave is a displacement that travels (almost always through a medium) with a velocity and carries energy.– It is the displacement that travels, not the medium!!

– The wave travels over large distances, the displacement is small compared to these distances.

– All forms of waves transport energy

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Characteristics of Waves• A continuous or periodic wave has a source which is

continuous and oscillating– Think of a hand oscillating a piece of rope up and down– Or a speaker playing a note

• This vibration is the source of the wave, and it is the displacement that propagates.

• If we freeze that wave in time (take a picture)

x

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Characteristics of Waves

• We can freeze the wave in time, and plot displacement vs. position:

• Or we can look at a single point in space, and plot displacement vs. time.

x

t

T

T

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Characteristics of Waves

• Amplitude (A): Maximum displacement of medium from equilibrium.

• Wavelength(λ): Distance, in space, between successive crests (or troughs) of wave.

• Period (T): Time between successive crests or troughs at a fixed position

• Frequency (f=1/T): Number of crests passing through a fixed point in a second.

• Velocity (v= λf): Speed at which displacement or disturbance propagates in space.

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Types of waves

• Transverse Wave: A transverse wave is a wave where the direction of the displacement or disturbance is perpendicular to the direction of wave propagation.

• Examples:– Wave on a string, water waves (on ocean, or ripples in pond), light

waves, the wave in a stadium

• Longitudinal Wave: A longitudinal wave is a wave where the direction of displacement is parallel to the direction of wave propagation

• Examples– Sound waves, shock waves

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Properties of Waves

• The speed of a wave depends on the properties of the medium the wave is travelling through– A vibration on a string travels with a velocity given by

– Sound waves will travel with different speeds in water or in air

– Light travels with different speeds through different materials, depending of the material’s index of refraction (n)

m

TensionFF

v TT

string

,,

V

mModulusBulkB

Bvsound

,,

refractionofindexnvacuuminlightspeedcn

cvlight )(,,

)(

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Mathematical Representation of Waves

• Say we freeze a wave in time.

0 1 2 3 4 5 6

-2

-1

0

1

2

D(x)

x (m)

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Mathematical Representation of Waves

• Now we look at the wave 1s later.

• Wave has moved to the right by 0.5m!

0 1 2 3 4 5 6

-2

-1

0

1

2

D(x)

x (m)

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Mathematical Representation of Waves

• Now we look at the wave 1s later (t=2s).

• Wave has moved to the right by 1m!

0 1 2 3 4 5 6

-2

-1

0

1

2

D(x)

x (m)

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Mathematical Representation of Waves

• Now we look at the wave after 6s (t=6s).

• Wave has moved to the right by one wavelength!

0 1 2 3 4 5 6

-2

-1

0

1

2

D(x)

x (m)

vT

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Mathematical Description of a Wave

)(2

sin),( vtxAtxD

v2

2

v

T

2

k

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Mathematical Description of a Wave

• Equation for a forward travelling wave:

• Equation for a backward travelling wave:

tkxAtxD sin),(

tkxAtxD sin),(

fT

12

2

k

fTk

v

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Wave Equation• Equivalent to Newton’s 2nd Law for Particles,

equation of motion for waves. In 1-D:

2

2

22

2 ),(1),(

t

txD

vx

txD

tkxAtxD sin),(

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Superposition of Waves

• If you have multiple waves passing through the same area of space, the total displacement is simply the sum of the displacement from all of the waves.

-200 -100 0 100 200

-1.0

-0.5

0.0

0.5

1.0

Inte

nsi

ty

Position (m)

=15

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Superposition of Waves

• If you have multiple waves passing through the same area of space, the total displacement is simply the sum of the displacement from all of the waves.

-200 -100 0 100 200

-1.0

-0.5

0.0

0.5

1.0

Inte

nsi

ty

Position (m)

=15.5

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Superposition of Waves

• If you have multiple waves passing through the same area of space, the total displacement is simply the sum of the displacement from all of the waves.

-200 -100 0 100 200

-1.0

-0.5

0.0

0.5

1.0

Inte

nsity

Position (m)

=16

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Superposition of Waves

• If you have multiple waves passing through the same area of space, the total displacement is simply the sum of the displacement from all of the waves.

-200 -100 0 100 200

-1.0

-0.5

0.0

0.5

1.0In

ten

sity

Position (m)

=20

Department of Physics and Applied Physics95.141, F2010, Lecture 24

Superposition of Waves

• If you have multiple waves passing through the same area of space, the total displacement is simply the sum of the displacement from all of the waves.

-200 -100 0 100 200-15

-10

-5

0

5

10

15 =15, =15.5

....=20

Inte

nsi

ty

Position (m)