Oscillations and Waves

What is a wave?

How do the particles move?

Some definitions…

1) Amplitude – this is “how high” the wave is:

2) Wavelength (λ) – this is the distance between two corresponding points on the wave and is measured in metres:

3) Frequency – this is how many waves pass by every second and is measured in Hertz (Hz)

Define the terms displacement, amplitude, frequency and period.

Describing waves

T

T

Time

Displacement

ω = 2π/T

f = 1/T

ω = 2πf

ω = Angular frequency

f = Frequency

T = Time period

2π

1 cycle is described by 2π radians of phase

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What are radians?

Hyperlink and scroll down

Phase and angle

Examples of phase difference

Oscillations

http://www.acoustics.salford.ac.uk/feschools/waves/shm.htm#motion

Simple harmonic motion

Any motion that repeats itself after a certain period is known as a periodic motion, and since such a motion can be represented in terms of sines and cosines it is called a harmonic motion.

The following are examples of simple harmonic motion:

a test-tube bobbing up and down in water (Figure 1)�a simple pendulum�a compound pendulum�a vibrating spring�atoms vibrating in a crystal lattice�a vibrating cantilever�a trolley fixed between two springs�a marble on a concave surface�a torsional pendulum�liquid oscillating in a U-tube�a small magnet suspended over a horseshoe magnet�an inertia balance

Data loggers

• Use the data loggers to find the variation of displacement with time for an oscillating mass on a spring.
• Process this data to find velocity and acceleration with time.
• Now use the data to obtain a graph of Acceleration versus displacement.

Analysing your graphs

• From the graph, find the
• Time period
• Angular frequency
• Amplitude
• The value of the velocity at maximum displacement
• The value of the acceleration at zero displacement
• The relationship between displacement and acceleration.

SHM definition

• Find the gradient of your graph of acceleration and displacement.
• Use this to calculate ω
• Calculate T from the graph.
• How the graph fit in with the definition of SHM?

SHM

Free body diagram for SHM

• Draw the free body diagram for the mass when it is
• in the centre of the motion
• At the top of the motion
• Between the bottom and the middle, down
• Between the bottom and the middle, heading upwards.

http://www.acoustics.salford.ac.uk/feschools/waves/shm2.htm

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Spring pendulum

Spring Pendulum

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Oscillations to wave motion

Restoring forces

Simple Harmonic Motion

Consider a pendulum bob:

Let’s draw a graph of displacement against time:

Pendulum

Simple Pendulum

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SHM Graphs

Time

Displacement

Velocity

Acceleration

Time

Time

T

Definition of SHM

Acceleration

Displacement

Now write your OWN definition of SHM

The Maths of SHM

Therefore we can describe the motion mathematically as:

x = x₀cosωt

v = -x₀ωsinωt

a = -x₀ω²cosωt

a = -ω²x

Students are expected to understand the

significance of the negative sign in the equation

and to recall the connection between ω and T.

ω = 2π/T

SHM questions

5

2

1. Calculate the gradient of this graph
2. Use it to work out the value of ω
3. Use this to work out the time period for the oscillations

4. Ewan sets up a pendulum and lets it swing 10 times. He records a time of 20 seconds for the 10 oscillations. Calculate the period and the angular speed ω.
5. The maximum displacement of the pendulum is 3cm. Sketch a graph of a against x and indicate the maximum acceleration.

a

x

Questions

• Q’s 1 – 9 from the worksheet
• Using the equations
• V = V₀cosωt
• V = V₀sinωt
• x = x₀cosωt
• x = x₀sinωt
• V = ± ω√(x₀²-x²)

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When do you use cos or sin?

4.2 Energy changes during simple harmonic motion (SHM)

Energy

Displacement (x)

-x₀

x₀

At which points are

-max displacement?

-max velocity?

-max acceleration?

- max E_k

-max E_p

-max total energy?

Total energy

SHM: Energy change

Energy

Time

GPE

K.E.

Energy formulae

Energy

Displacement (x)

-x₀

x₀

Total energy

E_k = ½ mω²(x₀² – x²)

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E_p = ½ mω²x²

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E_total = ½ mω²x₀²

Questions

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Answers

4.3 Forced oscillations and resonance

4.3.1 State what is meant by damping.

“It is sufficient for students to know that damping

involves a force that is always in the opposite

direction to the direction of motion of the

oscillating particle and that the force is a dissipative force.”

Free and Forced oscillations

Forcing frequency too slow

Forcing frequency too fast

Forcing frequency equals natural frequency

Resonance

Resonance and frequency

Hyperlink

Resonance and frequency

The width of the curve (Q value) is determined by the damping in the system. The value of the resonant frequency depends factors such as the size of the object…..

Tacoma Narrows

Useful resonance

• Musical instruments
• Microwave ovens
• Electrical resonance when tuning a radio

Damping

Damped oscillations

Damping

Low damping

High damping

Damping

How much damping is best?

Critical damping

Wave characteristics

The wave pulse transfers energy

If the source continues to oscillate, then a continuous progressive wave is produced.

Students should be able to distinguish between oscillations and wave motion, and appreciate that in many examples, the oscillations of the particles are simple harmonic.

Travelling Waves

Definition: A travelling wave (or “progressive wave”) is one which travels out from the source that made it and transfers energy from one point to another.

Energy dissipation

Clearly, a wave will get weaker the further it travels. Assuming the wave comes from a point source and travels out equally in all directions we can say:

Example questions

1. Darryl likes doing his homework. His work is 2m from a 100W light bulb. Calculate the energy flux arriving at his book.
2. If his book has a surface area of 0.1m² calculate the total amount of energy on it per second (what assumption did you make?).
3. Matti doesn’t like the dark. He switches on a light and stands 3m away from it. If he is receiving a flux of 2.2Wm^-2 what was the power of the bulb?
4. Matti walks 3m further away. What affect does this have on the amount of flux on him?

State that progressive (travelling)�waves transfer energy.

Students should understand that there is no net motion of the medium through which the wave travels.

Transverse vs. longitudinal waves

Transverse waves are when the displacement is at right angles to the direction of the wave…

Longitudinal waves are when the displacement is parallel to the direction of the wave…

Transverse wave

Transverse waves

Students should describe the waves in terms of the direction of oscillation of particles in the wave relative to the direction of transfer of energy by the wave. Students should know that light waves and water waves are transverse and that water waves cannot be propagated in gases or liquids.

Longitudinal waves

Sound waves and earthquake P-waves are longitudinal

Longitudinal slinky

Loudspeaker

Describe waves in two dimensions,�including the concepts of wavefronts�and of rays.

Energy is transferred in 2 dimensions

Watch the wavefront(s) propagate

Wavefronts and rays

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Wavefronts and rays

Rays show the direction of travel of the energy. The wavefronts are where the crests of the waves are. The rays are always at 90 deg to the wavefronts.

rays

Wavefronts

Longitudinal waves

Compressions and rarefactions

Transverse waves

Crests

Troughs

Displacement graphs

Define the terms displacement, amplitude, frequency, period,�wavelength, wave speed and intensity

WAVELENGTH

- the distance from one crest to another or one trough to another. (In fact generally from any point on the wave to the next exactly similar point i.e. 2 consecutive points in phase)

FREQUENCY

- the number of vibrations of any part of the wave per second. The bigger the frequency the higher the pitch of the note or the bluer the light��AMPLITUDE

- the maximum distance that any point on the wave moves from its mean position. The bigger the amplitude the louder the sound, the rougher the sea, or the brighter the light

Period (T)

The time it takes for one complete cycle of the wave.

Displacement (x)

How far the “particle” has travelled from its mean position.

Wave speed (v)

The speed at which the wavefronts pass a stationary observer

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Intensity (I)

The power per unit area that is received by an observer. Students should know that intensity α amplitude²

Derive and apply the relationship between wave speed, wavelength and frequency.

Speed = Dist/time

For 1 cycle of the wave, dist = λ and time =T

Speed = λ/T f = 1/T

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Therefore V=fxλ

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The Wave Equation

The wave equation relates the speed of the wave to its frequency and wavelength:

Wave speed (v) = frequency (f) x wavelength (λ)

in m/s in Hz in m

V

λ

f

Some example wave equation questions

1. A water wave has a frequency of 2Hz and a wavelength of 0.3m. How fast is it moving?
2. 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?
3. 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?
4. Purple light has a wavelength of around 6x10^-7m and a frequency of 5x10¹⁴Hz. What is the speed of purple light?

0.2m

0.5m

0.6m/s

3x10⁸m/s

Electromagnetic waves

Click to play

4.5 Wave properties

Wave diagrams

1) Reflection

4) Diffraction

3) Refraction

2) Refraction

• Describe the reflection and transmission of waves at a boundary between two media.

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• This should include the sketching of incident, reflected and transmitted waves.

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The amount of transmission and reflection depends upon the difference in the “density” of the 2 media. i.e the bigger the difference, the greater the amount of reflection.

Refraction through a glass block:

Wave slows down but is not bent, due to entering along the normal

Refraction of Light applet

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Finding the Critical Angle…

1) Ray gets refracted

4) Ray gets internally reflected

3) Ray still gets refracted (just!)

2) Ray still gets refracted

Optical fibres

Uses of Total Internal Reflection

Optical fibres:

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An optical fibre is a long, thin, _______ rod made of glass or plastic. Light is _______ reflected from one end to the other, making it possible to send ____ chunks of information

Optical fibres can be used for _________ by sending electrical signals through the cable. The main advantage of this is a reduced ______ loss.

Words – communications, internally, large, transparent, signal

Other uses of total internal reflection

1) Endoscopes (a medical device used to see inside the body):

2) Binoculars and periscopes (using “reflecting prisms”)

Huygen’s principle

1. Velocity decreases
2. Wavelength decreases
3. Frequency same

Snell’s law

Questions

• 10,11,12
• Practice Q 2

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• n_air = 1.00
• n_water = 1.33
• n_diamond = 2.42
• n_glass= 1.50

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Diffraction

More diffraction if the size of the gap is similar to the wavelength

More diffraction if wavelength is increased (or frequency decreased)

Diffraction

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Sound can also be diffracted…

The explosion can’t be seen over the hill, but it can be heard. We know sound travels as waves because sound can be refracted, reflected (echo) and diffracted.

Diffraction depends on frequency…

A high frequency (short wavelength) wave doesn’t get diffracted much – the house won’t be able to receive it…

Diffraction depends on frequency…

A low frequency (long wavelength) wave will get diffracted more, so the house can receive it…

Superposition

Superposition is seen when two waves of the same type cross. It is defined as “the vector sum of the two displacements of each wave”:

Superposition

Interference of 2 pulses

Click to play

Constructive interference i.e. Loud or bright. Waves are in phase

Destructive interference i.e. dark or quiet. Waves are π rads out of phase.

Interference of sound waves

Where are the positions of constructive and destructive interference?

Interference of 2 point sources

Click to play

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Superposition patterns

Consider two point sources (e.g. two dippers or a barrier with two holes):

Superposition of Sound Waves

Path Difference

Constructive interference

Destructive interference

2^nd Max