WavesWaves can transfer energy and information without a net motion of the medium through which they travel.

They involve vibrations (oscillations) of some sort.

Wave frontsWave fronts highlight the part of a wave that is moving together (in phase).= wavefrontRipples formed by a stone falling in water

Rays Rays highlight the direction of energy transfer.

Transverse wavesThe oscillations are perpendicular to the direction of energy transfer.Direction of energy transferoscillation

Transverse wavespeaktrough

Transverse wavesWater ripples

Light

On a rope/slinky

Earthquake S

Longitudinal wavesThe oscillations are parallel to the direction of energy transfer.

Direction of energy transferoscillation

Longitudinal wavescompressionrarefraction

Longitudinal wavesSound

Slinky

Earthquake P

Displacement - xThis measures the change that has taken place as a result of a wave passing a particular point. Zero displacement refers to the average position.= displacement

Amplitude - AThe maximum displacement from the mean position.amplitude

Period - TThe time taken (in seconds) for one complete oscillation. It is also the time taken for a complete wave to pass a given point.One complete wave

Frequency - fThe number of oscillations in one second. Measured in Hertz.

50 Hz = 50 vibrations/waves/oscillations in one second.

Wavelength - The shortest distance between points that are in phase (points moving together or in step). wavelength

Wave speed - vThe speed at which the wave fronts pass a stationary observer.330 m.s-1

Period and frequencyPeriod and frequency are reciprocals of each other

f = 1/TT = 1/f

The Wave EquationThe time taken for one complete oscillation is the period T. In this time, the wave will have moved one wavelength .

The speed of the wave therefore is distance/time

v = /T = fLets try some questions

A water wave has a frequency of 2Hz and a wavelength of 0.3m. How fast is it moving?A water wave travels through a pond with a speed of 1m/s and a frequency of 5Hz. What is the wavelength of the waves?The speed of sound is 330m/s (in air). When Dave hears this sound his ear vibrates 660 times a second. What was the wavelength of the sound?Purple light has a wavelength of around 6x10-7m and a frequency of 5x1014Hz. What is the speed of purple light?Some example wave equation questions0.2m0.5m0.6m/s3x108m/s

Representing wavesThere are two ways we can represent a wave in a graph;

Displacement/time graphThis looks at the movement of one point of the wave over a period of time1

Displacement/time graphThis looks at the movement of one point of the wave over a period of time1PERIODIMPORTANT NOTE: This wave could be either transverse or longitudnal

Displacement/distance graphThis is a snapshot of the wave at a particular moment1Distance cm-1-20.40.81.21.6displacement cm

Displacement/distance graphThis is a snapshot of the wave at a particular moment1Distance cm-1-20.40.81.21.6displacement cmWAVELENGTHIMPORTANT NOTE: This wave could also be either transverse or longitudnal

Wave intensityThis is defined as the amount of energy per unit time flowing through unit area

It is normally measured in W.m-2

Wave intensityFor example, imagine a window with an area of 1m2. If one joule of light energy flows through that window every second we say the light intensity is 1 W.m-2.

Intensity at a distance from a light sourceI = P/4d2

where d is the distance from the light source (in m) and P is the power of the light source(in W)

Intensity at a distance from a light sourceI = P/4d2

d

Intensity and amplitudeThe intensity of a wave is proportional to the square of its amplitude

I a2

(or I = ka2)

Intensity and amplitudeThis means if you double the amplitude of a wave, its intensity quadruples!

I = ka2

If amplitude = 2a, new intensity = k(2a)2 new intensity = 4ka2

Electromagnetic spectrum 700 - 420 nm 10-7 - 10-8 m 10-9 - 10-11 m 10-12 - 10-14 m 10-4 - 10-6 m 10-2 - 10-3 m 10-1 - 103 m

What do they all have in common? 700 - 420 nm 10-7 - 10-8 m 10-9 - 10-11 m 10-12 - 10-14 m 10-4 - 10-6 m 10-2 - 10-3 m 10-1 - 103 m

What do they all have in common?They can travel in a vacuumThey travel at 3 x 108m.s-1 in a vacuum (the speed of light)They are transverseThey are electromagnetic waves (electric and magnetic fields at right angles to each oscillating perpendicularly to the direction of energy transfer)

RefractionWhen a wave changes speed (normally when entering another medium) it may refract (change direction)

Water wavesWater waves travel slower in shallow water

Sound wavesSound travels faster in warmer air

Light wavesLight slows down as it goes from air to glass/water

Snells lawThere is a relationship between the speed of the wave in the two media and the angles of incidence and refractionirRay, NOT wavefronts

Snells lawspeed in substance 1 sin1speed in substance 2 sin2

=

Snells lawIn the case of light only, we usually define a quantity called the index of refraction for a given medium asn = c = sin1/sin2cmwhere c is the speed of light in a vacuum and cm is the speed of light in the medium

vacuumccm

Snells lawThus for two different media

sin1/sin2 = c1/c2 = n2/n1

Refraction a few notesThe wavelength changes, the speed changes, but the frequency stays the same

Refraction a few notesWhen the wave enters at 90, no change of direction takes place.

DiffractionWaves spread as they pass an obstacle or through an opening

DiffractionDiffraction is most when the opening or obstacle is similar in size to the wavelength of the wave

Diffraction patterns HL later!

DiffractionDiffraction is most when the opening or obstacle is similar in size to the wavelength of the wave

DiffractionThats why we can hear people around a wall but not see them!

Diffraction of radio waves

Superposition

Principle of superpositionWhen two or more waves meet, the resultant displacement is the sum of the individual displacements

Constructive and destructive interferenceWhen two waves of the same frequency superimpose, we can get constructive interference or destructive interference.

+=+=

SuperpositionIn general, the displacements of two (or more) waves can be added to produce a resultant wave. (Note, displacements can be negative)

Interference patternsRipple Tank Simulation

If we pass a wave through a pair of slits, an interference pattern is produced

Path differenceWhether there is constructive or destructive interference observed at a particular point depends on the path difference of the two waves

Constructive interference if path difference is a whole number of wavelengths

Constructive interference if path difference is a whole number of wavelengthsantinode

Destructive interference if path difference is a half number of wavelengths

Destructive interference if path difference is a half number of wavelengthsnode

Phase differenceis the time difference or phase angle by which one wave/oscillation leads or lags another. 180 or radians

Phase differenceis the time difference or phase angle by which one wave/oscillation leads or lags another. 90 or /2 radians

Simple harmonic motion (SHM)periodic motion in which the restoring force is proportional and in the opposite direction to the displacement

Simple harmonic motion (SHM)periodic motion in which the restoring force is in the opposite durection and proportional to the displacement

F = -kx

Graph of motionA graph of the motion will have this form

Graph of motionA graph of the motion will have this formAmplitude x0Period T

Graph of motionNotice the similarity with a sine curveangle

2 radians/23/22

Graph of motionNotice the similarity with a sine curveangle

2 radians/23/22Amplitude x0x = x0sin

Graph of motionAmplitude x0Period T

Graph of motionAmplitude x0Period Tx = x0sint

where = 2/T = 2f = (angular frequency in rad.s-1)

When x = 0 at t = 0Amplitude x0Period Tx = x0sint

where = 2/T = 2f = (angular frequency in rad.s-1)

When x = x0 at t = 0Time displacement

Amplitude x0Period Tx = x0cost

where = 2/T = 2f = (angular frequency in rad.s-1)

When x = 0 at t = 0Amplitude x0Period Tx = x0sint v = v0cost

where = 2/T = 2f = (angular frequency in rad.s-1)

When x = x0 at t = 0Time displacement

Amplitude x0Period Tx = x0costv = -v0sint

where = 2/T = 2f = (angular frequency in rad.s-1)

To summarise!When x = 0 at t = 0 x = x0sint and v = v0cost

When x = x0 at t = 0 x = x0cost and v = -v0sint

It can also be shown that v = (x02 x2) and a = -2x

where = 2/T = 2f = (angular frequency in rad.s-1)

Maximum velocity?When x = 0

At this point the acceleration is zero (no resultant force at the equilibrium position).

Maximum acceleration?When x = +/ x0

Here the velocity is zero

amax = -2x0

Oscillating springWe know that F = -kx and that for SHM, a = -2x (so F = -m2x)

So -kx = -m2xk = m2 = (k/m)Remembering that = 2/TT = 2(m/k)

S.H.M.Where is the kinetic energy maxiumum?

Where is the potential energy maximum?

It can be shown that.Ek = m2(xo2 x2)ET = m2xo2Ep = m2x2

where = 2f = 2/T

DampingIn most real oscillating systems energy is lost through friction.

The amplitude of oscillations gradually decreases until they reach zero. This is called damping

UnderdampedThe system makes several oscillations before coming to rest

Overdamped The system takes a long time to reach equilibrium

Critical dampingEquilibrium is reached in the quickest time

Natural frequencyAll objects have a natural frequency that they prefer to vibrate at.

Forced vibrationsIf a force is applied at a different frequency to the natural frequency we get forced vibrations

ResonanceIf the frequency of the external force is equal to the natural frequency we get resonanceYouTube - Ground Resonance - Side ViewYouTube - breaking a wine glass using resonancehttp://www.youtube.com/watch?v=6ai2QFxStxo&feature=relmfu

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