Waves

Periodic Motion• We are surrounded by oscillations – motions that

repeat themselves• Understanding periodic motion is essential for the

study of waves, sound, alternating electric currents, light, etc.– How many of you play an instrument?

• An object in periodic motion experiences restoring forces that bring it back toward an equilibrium position

• Those same forces cause the object to “overshoot” the equilibrium position

• Think of a block oscillating on a spring or a pendulum swinging back and forth past its equilibrium position Demonstrate

Definitions of a Waves A wave is a traveling disturbance that

carries energy through space and matter without transferring mass. Transverse Wave: A wave in which the

disturbance occurs perpendicular to the direction of travel (Light).

Longitudinal Wave: A wave in which the disturbance occurs parallel to the line of travel of the wave (Sound).

Surface Wave: A wave that has charact-eristics of both transverse and longitudinal waves (Ocean Waves).

Wave types

How does a wave vary in position and velocity?

- Full body Demonstrate

- PVA graphs

Types of Waves

Mechanical Waves: Require a material medium* such as air, water, steel of a spring or the fabric of a rope.

Electromagnetic Waves: Light and radio waves that can travel in the absence of a medium.

* Medium = the material through which the wave travels.

Wave Motion• The wave is another basic model used to describe

the physical world (the particle is another example)• Any wave is characterized as some sort of

“disturbance” that travels away from its source• In many cases, waves are result of oscillations

– For example, sound waves produced by vibrating string

• For now, we will concentrate on mechanical waves traveling through a material medium– For example: water, sound, seismic waves– The wave is the propagation of the disturbance: they do

not carry the medium with it

• Electromagnetic waves do not require a medium• All waves carry momentum and energy

Types of Waves• In solids, both transverse and longitudinal waves can

exist– Transverse waves result from shear disturbance– Longitudinal waves result from compressional disturbance

• Only longitudinal waves propagate in fluids (they can be compressed but do not sustain shear stresses)– Transverse waves can travel along surface of liquid,

though (due to gravity or surface tension)

• Sound waves are longitudinal– Each small volume of air vibrates back and forth along

direction of travel of the wave

• Earthquakes generate both longitudinal (4–8 km/s P waves) and transverse (2–5 km/s S waves) seismic waves– Also surface waves which have both components

Transverse Wave Characteristics Crest: The high point of a wave. Trough: The low point of a wave. Amplitude: Maximum displacement from its

position of equilibrium (undisturbed position).

John Wiley & Sons

Transverse Wave Characteristics (cont.) Frequency(f): The number of oscillations the

wave makes in one second (Hertz = 1/seconds).

Wavelength(): The minimum distance at which the wave repeats the same pattern (= 1 cycle). Measured in meters.

Velocity (v): speed of the wave (m/s).

v = f Period (T): Time it takes for the wave to

complete one cycle (seconds).

T = 1/f

The Inverse Relationshipsv = f

The speed of a wave is determined by the medium in which it travels. Since velocity is constant for a given medium,

the frequency and wavelength must be inversely proportional.

• As one increases, the other decreases

Wavelength

Fre

que

ncy

The Inverse RelationshipsT = 1/f

Similar to the inverse relationship for frequency and wavelength, a similar relationship exists for frequency and the period.

Period

Fre

que

ncy

Waves at Fixed Boundaries A wave incident upon a

fixed boundary will have its energy reflected back in the opposite direction. Note that the wave pulse is inverted after reflecting off the boundary.

Example of Waves at Fixed Boundaries

www.electron4.phys.utk.edu

Start Per 5/6 here

Interference

Interference occurs whenever two waves occupy the same space at the same time. Law of Linear Superposition: When two or

more waves are present at the same time at the same place, the resultant disturbance is equal to the sum of the disturbances from the individual waves.

Constructive Wave Interference

www.electron4.phys.utk.edu

Constructive Interference – Process by which two waves meet producing a net larger amplitude.

Destructive Wave Interference

Destructive Interference – Process by which two waves meet canceling out each other.

Standing Waves Standing Wave: An interference pattern resulting

from two or more waves moving in opposite directions with the same frequency and amplitude such that they develop a consistent repeating pattern of constructive and destructive interference. Node: The part of a standing wave where interference

is destructive at all times (180o out of phase) . Antinode: The part of the wave where interference is

maximized constructively. Standing Wave

Continuous Waves When a wave impacts a boundary, some of the

energy is reflected, while some passes through. The wave that passes through is called a

transmitted wave. A wave that is transmitted through a boundary

will lose some of its energy. Electromagnetic radiation will both slow down and have

a shorter wavelength when going into a denser media. Sound will increase in speed when transitioning into a

denser media. Speed of Light in different mediums

Incident + Reflected Wave

Higher speed

Longer wavelength

Lower speed

Shorter wavelength

Transmitted Wave

Continuous Waves – Higher Speed to Lower Speed Note the differences in wavelength and amplitude between

of the wave in the two different mediums

Displacement

Boundary

v1 v2-v1

Note: This phenomena is seen with light traveling from air to water.

Waves at Boundaries Examples of Waves at Boundaries Wave Types (Cutnell & Johnson) Waves - Colorado.edu Other Examples

Key Ideas Waves transfer energy without transferring

matter. Longitudinal waves like that of sound require a

medium. Transverse waves such as electro-magnetic

radiation do not require a medium. In transverse waves, displacement is

perpendicular to the direction of the wave while in longitudinal waves, the displacement is in the same direction.

Key Ideas Waves travel at different speeds in

different mediums. Light slows down when going from air

to a liquid or solid. Sound speeds up when going from air

to a liquid or solid. Waves can interfere with one

another resulting in constructive or destructive interference.

Continuous Waves – Lower Speed to Higher Speed Note the differences in wavelength and amplitude between

of the wave in the two different mediums

Incident + Reflected Wave

Lower speed

Shorter wavelength

Higher speed

Longer wavelength

Transmitted Wave

-v1

v2

v1

Displacement

Boundary

Review of Springs• Classic example of periodic motion:

– Spring exerts restoring force on block:

– k = spring constant (a measure of spring stiffness)

– “Slinky” has k = 1 N/m; auto suspensions have k = 105 N/m

– Movie of vertical spring:

• Elastic potential energy stored in spring:– Uel = 0 when x = 0 (spring relaxed)

– Uel is > 0 always

– We do not have freedom to pick where x = 0

– Uel conserves mechanical energy

kxFs (Hooke’s Law)

2el 2

1kxU

Shock Absorbers• Shock absorbers provide a

damping of the oscillations– A piston moves through a viscous

fluid like oil– The piston has holes in it, which

creates a (reduced) viscous force on the piston, regardless of the direction it moves (up or down)

– Viscous force reduces amplitude of oscillations smoothly after car hits bump in road

– When oil leaks out of the shock absorber, the damping is insufficient to prevent oscillations

• Shock absorber is example of an underdamped oscillator (see also critically damped and overdamped)

Properties of Waves• Superposition principle: The overlap of 2 or more

waves (having small amplitude) results in a wave that is a point-by-point summation of each individual wave

(constructive interference) (destructive interference)

Properties of Waves• Traveling waves can both reflect and transmit

across a boundary between 2 media– Reflected wave pulse is inverted (not inverted) if wave

reaches a boundary that is fixed (free to move)