4.4.1 Wave motion

A progressive wave is an oscillation that transfers energy, but not matter, from one place to another. The particles of matter do not move in the direction of the wave. Instead they move from their equilibrium position to a new position and back. The particles exert forces on each other - a displaced particle experiences a restoring force meaning it is pulled back to its equilibrium position. 

There are two types of progressive wave, transverse waves, and longitudinal waves.

Transverse waves
In transverse waves, oscillations/vibrations are perpendicular to the direction of energy transfer. They can be in any orientation - up and down, side to side, etc - provided that they occur at right angles to the direction of energy transfer. The peak/trough is where the oscillating particles have maximum displacement from their equilibrium positions. Examples of transverse waves include water waves/electromagnetic waves/waves on a stretched string (e.g a guitar string)/S- waves (produced in earthquakes).

Longitudinal waves
In longitudinal waves, oscillations are parallel to the direction of energy transfer. When they travel through a medium they crease a series of compressions and rarefactions. Examples include sound waves and P-waves (produced in earthquakes). Since the displacement of particles occurs in the same plane as the direction of energy transfer you may be wondering how the restoring forces work. Well lets take sound for an example. Air particles are displaced and bounce off their neighbors - this provides the restoring force. As the wave moves region of higher pressure (compressions) and regions of lower pressure (rarefactions). again, the particles are still oscillating around their equilibrium positions.

Key terms
Okay so there are quite a few key terms we have to commit to memory for this topic - but we will use them loads so i'm sure you'll remember them soon enough:

• Displacement - the distance from the equilibrium position in a particular direction
• Amplitude - the maximum displacement from the equilibrium position
• Wavelength - minimum distance between two points in phase on adjacent waves
• Period (of oscillation) - the time taken for one oscillation/the time taken for a wave to move one whole wavelength past a given point
• Frequency - the number of wavelengths passing a given point per unit time
• Wave speed - the distance travelled by the wave per unit time

NOTE: wavespeed has the unit v, but if we're talking about electromagnetic waves then it has the unit c (for the speed of light, 3 x 10^8 ms^-1).

The wave equations

We can see from the definition above that the frequency of a wave and its period of oscillation are reciprocals of eachother. From this we can form an equation that relates the frequency of a wave to its period:

f = 1 / T

We also know that if a wave has a frequency of say 10Hz, then there are 10 complete oscillations each second. Say we have a wavelength of 1m, this means that the wave has travelled 10m in each second meaning its speed must be 10ms^-1. this means that for a certain frequency the wave has trvelled a distance of f x λ (frequency x wavelength) per second which is equal to the wavelength. From this information we can form another important equation...

V = f λ

Graphical representations
So, like in forces and motion, we can show the displacement of the particles of a wave against the distance along the wave on a displacement-distance graph (this can be called a wave profile). The wave profile can be used to determine the wavelength and amplitude of both longitudinal and transverse waves. The wave profile of a transverse and longitudinal wave will look the same (well, the same shape anyway (sinusoidal), not necessarily the same numbers) because it is a measure of the displacement and distance of the wave/particles NOT how the wave looks.

Phase difference describes the difference in displacements of particles along a wave (or on different waves). One complete cycle is 360° (2π radians). If particles reach their maximum positive (or negative)displacements at the same time they are in phase and their phase difference is zero. Similarly, if one particle reaches its maximum positive displacement at the same time another reaches its maximum negative displacement the particles are in antiphase and their phase difference is 180° (π radians). There is an equation that we can use to determine phase difference:

ϕ = (x/λ) x 360°

We can also use displacement-time graphs to show how the displacement of a given particle varies with time (duh). They look the same for transverse and longitudinal waves. These types of wave can be used to determine the period (and therefore frequency) of a wave.

Oscilloscope experiment

So we need to know techniques and procedures used to use an oscilloscope to determine frequency. Basically, using the set up below we can see that using a microphone produces a trace on the oscilloscope screen. Each horizontal square on the screen represents a certain time interval known as the timebase. This is set to a certain ms cm^-1 (e.g 10 ms cm^-1) - this means that each square represents a time interval of 10 mc cm^-1. The up/down squares represent the y sensitivity which is measured in V cm^-1. E.g a setting of 10 V cm^-1 means that each square will represent a pd. of 10V. Using the timebase we can do f = 1/T to determine the frequency.

Reflection, refraction, polarisation, and diffraction.

Reflection: this occurs when a wave changes direction at a boundary between two different media but remains in he original medium. The law of reflection states that whenever waves are reflected the angle of incidence is equal to the angle of reflection. When waves are reflected their frequency and wavelength do not change.

Refraction: this occurs when a wave changes direction as ti changes speed when it passes from one medium to another. There is always some refection off the surface (partial reflection).If a wave slows down as it enters the medium it will refract toward the normal, if it speeds up it will refract away from the normal. Sound waves speed up when they enter a denser medium whereas electromagnetic waves usually slow down. Since the speed of the waves changes and frequency is constant, this means that wavelength also changes as V = f λ. water waves can also be refracted - when a water wave enters a shallower bit of water is slows down and it's wavelength gets shorter.

Diffraction: this is the spreading out of a wave as it passes through a gap/travels around an obstacle. ALL waves can be diffracted and the speed, wavelength, and frequency are all constant (they do not change). The effects of diffraction are most significant when the gap the wave travels through is the same as the waves wavelength.

Polarisation: this means that the particles oscillate in one plane only. We cannot polarise longitudinal waves as their oscillations already act in one plane only (the direction of energy transfer). If a wave is plane polarised its oscillations occur in one plane only (e.g some sunglasses contain polarising filters so you can only see in one plan only). Partial polarisation can also occur (this happens when  transverse waves reflect off a surface). This means that more waves oscillate in one particular plane compared to others/another but they wave is not completely plane polarised.

Most naturally occurring electromagnetic waves are unpolarised. we can polarised them using polarising filters (each filter only allows waves with a particular orientation of oscillations through). We need to know how to observe polarising effects with microwaves and light:

• Unpolarised microwaves can be polarised by placing a metal grille in front of the transmitter (in between the transmitter and the receiver).
• If you take two pieces of polaroid filter and place them together (at right angled orientations to each other) you can nicely see the effect of polarisation. Unpolarised light travels through the first filter and becomes plane polarised. It cannot pass through the second filter as the second filter is not in the same plane as the first (it is 90° sideways). This means that the intensity of the light transmitted drops - no light is in fact transmitted through the second filter and the intensity falls to zero.

NOTE: think we need to know what wave fronts are, they are just lines joining all the points on a wave that are in phase

Intensity

This nicely leads me on to intensity. The intensity of a progressive wave is the radiant power passing through a surface per unit area. It has the units W m^-2 and is calculates with the following equation:

I = P/A

where A is the cross sectional area of the surface, P is the radiant power passing through the surface, and I is the intensity of the wave at the surface.

for a point source the radiant power will spread out uniformly in all directions (e.g over the surface of a sphere). This makes the equation I = P / (4πr²). From this we can see that intensity has an inverse square relationship with the distance from the source. 

Intensity drops as the energy becomes more spread out and the wave height (amplitude) decreases. Decreased amplitude means a reduced average speed  of the oscillating particles. For example, if you were to half th amplitude you would have particles that oscillate with half the speed which means a quarter of the kinetic energy and energy is proportional to intensity so intensity is proportional to amplitude squared...

intensity ∝ (amplitude)²

A ripple tank can be set up with a camera above it to take photos of the wavefronts. The frequency is changed and the images allow the wavelength to be measured. This allows the wave equation to be investigated and also shows that the wave speed does not depend on frequency.

3.5.2 Collisions

The principle of conservation of momentum states that for a closed system the total momentum in a specified direction remains constant as long as no external forces act on the system.

It is important to recognise that both momentum and total energy are conserved when objects collide. There are two types of collision:

• perfectly elastic:
• momentum is conserved, total kinetic energy is conserved.
• inelastic:
• momentum is conserved, total kinetic energy is not conserved.

As we know, linear momentum must always be conserved. We can resolve the different directions of momenta into 'x' 9horizontal) and 'y' (vertical) directions and ensure that each is constant.

3.5.1 Newton’s laws of motion

okay so Newton (v clever chap) has 3 laws of motion:

• First law: An object will remain at rest or continue to move with constant velocity unless acted upon by a resultant force.
• Second law: The net force acting on an object is directly proportional to the rate of change of momentum.
• Third law: When two objects interact they exert an equal and opposite force on each other.

The linear momentum of an object depends on its mass and velocity. We can calculate it using the equation:

p = m v

It has the unit kg m s^-1. It is a vector as it is a product of mass (scalar) and velocity (vector).

Newton's second law we can formulate the equation:

F = Δp/Δt

NOTE: The constant (k) in this equation can be made equal to 1 by defining the unit of force (the Newton) as the force required to give a 1kg mass an acceleration of 1m s^-2. It is also important to recognise that F = ma is a special case of this equation.

If we rearrange this equation we can get F Δt = Δp. The impulse of a force is defined as the product of force and the time for which this force acts on an object. It follows that the area under a force-time graph is equal to impulse.

2.2.1 Measurements and uncertainties

Okay so this is not exactly content in itself but is sort of like good information to know.

A true value is the value that would be obtained in an ideal measurement. A measurement error is the difference between a measured value and the true value for the quantity being measured. Mistakes are not counted as errors. If you make a mistake you can repeat an experiment without the mistake. However, an experiment will still contain errors no matter if you repeat it. In science, an error is the difference between the result you get and the correct result. They are usually caused by measuring devices even if they are used correctly. They can also be caused by the design of the experiment itself.

Random errors can happen when any measurement is being made. They are measurement errors in which measurements vary unpredictable. They cannot be corrected but we can reduce their effects by making more measurements and finding the mean value. Reasons for this include:

• factors that are not controlled in the experiment
• difficulty in deciding on the reading given by a measuring device.

Systematic errors are measurement errors in which the measurements differ from the true values by a consistent amount each time a measurement is made. Unlike random errors it is possible to correct systematic errors e.g by changing equipment. Reasons for this include:

• the way in which measurements are taken
• faulty measuring devices

Examples include poor contact between a thermometer and the object whose temperature is being measured, a faulty measuring device (e.g calibrated incorrectly/zero errors).

When we talk about obtaining measurements it is vital that we know the difference between precision and accuracy:

• precision regards how close repeated measurements are to each other (the closer together the more precise)
• accuracy regards how close a measurement result is to the true value (the closer the more accurate)

Uncertainties

Okay so because of random and systematic errors basically it is very hard to obtain the same value for a particular measurement. A mean value can be calculated by adding all the values and dividing it by the number of values. The range is the difference between the smallest and largest readings. The uncertainty in the measurement is an interval within which the true value can be expected to lie.

The absolute uncertainty in the mean value of a measurement can be approximated as half the range. It is expressed as a ± value. When you have a single measurement/repeat measurements are equal you approximate the absolute uncertainty to be equal to the resolution of the measuring instrument.

The percentage uncertainty can be calculated from its absolute uncertainty and mean values:

% uncertainty = 100 x absolute uncertainty/mean value

The final uncertainty depends on how quantities are combined:

• adding or subtracting quantities: add the absolute uncertainties for each value
• multiplying or dividing quantities: add the percentage uncertainties for each value
• raising to a power: increase the percentage uncertainty by the magnitude of the power

I'm sure you will have completed this in PAGs, but we can also use error bars/lines of best and worst fit. The absolute uncertainty in the gradient is the positive difference between the gradient of the line of best fit and the gradient of the line of worst fit. The percentage uncertainty can then be calculated using the following equation:

% uncertainty = 100 x absolute uncertainty/gradient of best fit line

3.3.3 Power

Power is the rate of work done (or the rate of energy transfer, since work done = energy transferred). We can determine power using the equation:

P = W / t

Power has the unit J s^-1 or W (watts). One watt is equal to one joule per second.

In lots of situations a constant force must exerted to maintain a constant speed. For example, the rate of work done (power) by the forward force provided by  cars engine is equal to the rate of work done against the frictional forces acting on the cat (so the net force on the car is 0). We can depict this as:

A constant force F moves the car a distance x in a time t

work done by the force W = F x

P = W/t = Fx/t

The speed of the car (v) is the rate of change of distance: v = x/t

P = Fv


Efficiency
Not all machines/processes convert all their energy into useful work. Lots of energy often is outputted as thermal energy (heat). We can calculate the efficiency of something using the following equation:

efficiency = 100% x useful output energy / total input energy

3.3.2 Kinetic and potential energies

Kinetic energy
Kinetic energy is the energy associated with the motion of an object. Kinetic energy (Ek) can be calculated from mass and speed:

Ek = 0.5 m v²

This means that, for a constant speed, Ek is directly proportional to mass. For a given mass, Ek is directly proportional to the square of the objects speed.

We need to  know how to derive this:

Take an object that initially starts at rest (u = 0). We can determine it's velocity at a certain distance using v² = u² + 2as:

s = (v²-u²)/2a = v²/2a

This means that the work done by the force moving the object is entirely transferred to Ek:

Work done = Ek = F x = F s

F = ma

Ek = ma s = ma (v²/2a)

Ek = mv²/2 = 0.5 m v²


Gravitational potential energy
Gravitational potential energy os the capacity for doing work as a result of an object's position in a gravitational field. You can calculate change in GPE from it's mass and height:

Ep = m g h

This one is a lot simpler to derive: When you lift something through a height h at a constant speed (no change in Ek) you have applied a force that equates to mg. The work done is transferred into Ep (GPE)

Ep = W = force x distance moved in the direction of the force

Ep = (mg) h

Ep = m g h


Also (when we cover gravitational fields):
The gravitational potential energy 'E' of any object with the mass m within a gravitational field is defined as the work done to move the mass from infinity to a point in a gravitational field:

E = m Vg

In a uniform gravitational field, in order to change the gravitational potential energy of an object its height above the surface must be changed as this results in a change in gravitational potential.

In a radial field (remember Vg = -GM/r) the gravitational potential energy can be written as:

E = m Vg = -GMm/r

Energy exchanges
Often, kinetic energy and gravitational energy will be exchanged. For example, if you drop a book it's gravitational energy will decrease whilst its terminal velocity increases:

0.5 m v² = m g h

0.5 v² = g h

v² = 2 g h

v = √(2 g h)

3.3.1 Work and conservation of energy

Work done is defined as the product of force and distance moved in the direction of the force. We can use the following equation to determine work done:

W = F x

(at an angle: W = F x cosθ)

It has the unit Nm. 1 Nm = 1J. One Joule is defined as the work done when a force of 1 N moves its point of application 1 m in the direction of the force.

It is important to understand that work done = energy transferred (this is because energy is defined as the capacity to do work). Energy is a scalar quantity with the unit Joule. The energy of all systems with mass can be classified as kinetic energy (energy due to the movement of an object) and gravitational potential energy (the energy due to the position of an object in the Earth's gravitational field). 

It is imperative to understand that the total final energy is always equal to the total initial energy. This can be represented in the principle of conservation of energy: the total energy of a closed system remains constant, energy can never be created nor destroyed but can be transferred from one form to another.