6.4.2 Fundamental particles

Okay so this section is all about how we group different particles into different...well....into different groups. We have matter and antimatter and leptons and and baryons and kaons and pions and mesons and hadrons and quarks and so on...

Antimatter
(This is the prediction that) every particle has a corresponding antiparticle and they completely destroy each other in annihilation if they meet where the masses of both particles are converted into a high-energy pair of photons. An antiparticle has the opposite charge to its corresponding particle and the exact same rest mass:

• electron - positron
• proton - antiproton
• neutron - antineutron
• neutrino - antineutrino

Fundamental particles
A fundamental particle is a particle with no internal structure (therefore it cannot be divided into smaller bits). 


Hadrons and leptons
Subatomic particles are classified into two families, hadrons and leptons:

• Hadrons
• particles and antiparticles affected by the strong nuclear force
• examples include protons/neutrons/mesons
• They also experience the electromagnetic force if charged
• They decay by the weak nuclear force
• All hadrons are made up of quarks
• Leptons
• Particles and antiparticles not affected by the strong nuclear force
• examples include electrons, neutrinos, and muons
• leptons experience the electromagnetic force if charged
• They are fundamental particles so they do not decay

Quarks

All hadrons are made up of quarks. Quarks and leptons are the building blocks of matter (they are the fundamental particles). Any particle that contains a quark is a hadron. The standard model of elementary particles requires 6 quarks (up, down, charm, strange, top, and bottom) and 6 antiquarks. all quarks have a charge Q that is a fraction of e. For example, an up quark has a charge of 2e/3. 

All hadrons experience the strong nuclear force. Individual quarks are bound together within the particle by the attractive strong nuclear force.

A proton consists of 3 quarks, up up down (uud). The total charge must equal e. A neutron also consists of 3 quarks , up down down (udd). The total charge is 0.


Mesons and baryons
Baryons are any hadrons made with a combination of 3 quarks. Mesons are the hadrons made with a combination of a quark and an anti-quark.


Neutrinos
Neutrinos are quite mysterious fundamental particles that carry no charge and may have a tiny mass. Its existence was predicted in order to explain beta decay in terms of conservation laws. Each neutrino has a corresponding antineutrino. There are 3 types of neutrino:

• the electron nuetrino
• the muon neutrino
• the tau neutrino

Beta decay
Beta radiation is the emission of either electrons (β⁻) or positrons (β⁺). The force responsible for beta decay is the weak nuclear force.

• Beta minus decay
• a neutron in an unstable nucleus decays into a proton, an electron, and an electron antineutrino:
• n --> p + e +  ̅ν_e
• Quark transformation: udd --> uud ∴ d --> u + e +  ̅ν_e
• It is important to realise that nucleon and proton number and conserved (as is total charge)
• Beta plus decay
• A proton decays into a neutron, a positron, and an electron neutrino
• p --> n + e + ν_e
• Quark transformation : uud --> udd ∴ u --> d + e +  ν_e
• Charge is conserved and so is the proton/nucleon number

6.4.1 The nuclear atom

Okay so here we need to know about the alpha particle scattering experiment etc and how this lead to out current understanding of atoms and what our current nuclear model is.

In Rutherford's alpha-scattering experiment a narrow beam of alpha particles of the same kinetic energy from a radioactive source were targeted at a thin piece of gold foil which was only a few atomic layers thick. The alpha particles were scattered by the foil and detected on the zinc sulphide screen that was mounted in front of a microscope. Each alpha particle hitting the fluorescent screen produced a tiny speck of light. The microscope was moved in order to count the number of alpha particles scattered through the different values of angle θ per minute.

The results led to the following significant observations (which rued out the Thomson plum-pudding):

• Most of the alpha particles passed straight through the thin gold foil with little scattering (1/2000 was scattered).
• This means that most of the atom was empty space with most of the mass concentrated in a small region (the nucleus).
• Very few (1/10000) were deflected through angles greater than 90°
• This concluded that the nucleus had a positive charge (as it repelled few positive alpha particles). The charge on the nucleus is quantised to +Ze (where Z is the atomic number of the element)

The scattering of the alpha particles can be modelled in terms of Coulomb's law (any two point charges exert an electrostatic (electrical) force on each other that is directly proportional to the product of their charges and inversely proportional to the square of the distance between them).

Rutherford predicted that a fraction of alpha particles would be scattered through an angle θ. He found that more energetic alpha particles managed to get much closer to the nucleus. He concluded that the nucleus has a radius of about 10^-14 m. In one experiment he used alpha particles of Ek 1.2 x 10^-12 J. The distance (d) of the closest approach between and alpha particle and the gold nucleus can be calculated using the idea of conservation of energy. At this distance the alpha particle momentarily stops meaning that initial Ek = electric potential energy at d:

E = Qq/4πε0r = 1.2 x 10^-12 = Qq/4πε0d

(NOTE: Q = Ze = 79e, q = 2e)

1.2 x 10^-12 = (79 x 2 x e) / (4πε0d)

d = 3.0 x 10^-14 ≈ 10^-14 m

This gives an upper limit for the radius of the gold nucleus. More energetic alpha particles might get closer. The order of magnitude for the value of the radius of a nucleus is ~ 10^-15 m. The radius of most atoms is 10^-10 m so the nucleus is 10⁵ times smaller than the atom.

The nuclear model of the atom

The nucleus of an atom contain positive protons and uncharged neutrons (a proton and a neutron approximately have the same mass). Isotopes are nuclei of the same element that have the same number of protons but different numbers of neutrons. Isotopes of an element undergo the same chemical reactions.

The masses of atoms and nuclear particles are often expressed in atomic mass units (u). 1 u is one twelfth the mass of a neutral carbon-12 atom. The experimental value of 1 u is about 1.661 x 10^-27 kg.

The radius of a nucleus depends on the nucleon number (A) of the nucleus. Fast moving electrons have a de Broglie wavelength of ~ 10^-15 m. Diffraction of these electrons can be used to determine the radii of isotopes. The radius (R) of a nucleus is given by the equation:

R = r0 A^1/3

r0 has an approximate value of 1.2fm. All nuclei have a density of about 10¹⁷ kg m^-3.


Regarding the strong nuclear force, lets take a helium-r nucleus as an example. The two protons are separated by a distance of 10^-15 m and exert a large electrostatic force on each other. According to Coulomb's law:

F = Qq/4πε0r²

e²/ 4πε0(10^-15)² ≈ 230N

(THIS IS NOT THE STRONG NUCLEAR FORCE - THIS IS THE REPULSIVE FORCE)

This is extremely large but the protons do not fly apart. Basically, the attractive gravitational force between the protons is very small (10^-34 N) so there must be another force acting on the protons. This is the strong nuclear force. The strong nuclear force acts between all nucleons and is a very short range force (effective over only a few femtometres). The force is attractive to about 3 fm and repulsive below 0.5 fm.

6.5.3 Using ultrasound

Humans can hear sound within the range of 20-20,000Hz. Ultrasound is longitudinal sound wave with a frequency greater than 20kHz (we can't hear this). It can be used to form images of the internal structures of the body and it is good because it is non-ionising (harmless) and non-invasive (no risk of infection) and quick. Medical imaging ultrasound has a frequency in the range of 1-15MHz. It can be refracted at a boundary between two substances and also diffracted. The wavelength of ultrasound in the human body is ,1mm so it can be used to identify features as small as a few mm. An ultrasound transducer is used to generate and to receive ultrasound. It can change electrical energy into sound and sound into electrical energy by means of the piezoelectric effect.

The piezoelectric effect
Crystals such as quartz produce an e.m.f (the energy transferred from chemical to electrical per unit charge) when compressed/stretched/twisted/distorted. This is a reversible process. When an external p.d. is applied across the opposite faces of the crystal the electric field can either compress/stretch the crystal.

To generate ultrasound a high-frequency (e.g. 5MHz) alternating p.d. is applied across the opposite faces of a crystal repeatedly compressing and expanding the crystal. The frequency chosen is the same as the natural frequency of oscillation of the crystal and the result is that the crystal resonates producing an intense ultrasound signal. As ultrasound transducer emits pulses of ultrasound (about 5,000 per second). The same transducer is used to detect ultrasound - any ultrasound incident on the crystal will make it vibrate so the crystal is compressed and expanded by tiny amounts. This generates an alternating e.m.f across the ends of the crystal which can be detected by electronic circuits. Modern ultrasound transducers use lead zirconate titanate or polyvinylidene fluorine rather than quartz.

A and B scans
A scans
This is the simplest type of ultrasound scan. A single transducer is sued to record along a straight line through the patient. Tis can be used to determine the thickness of bone/distance between the lens and retina of the eye (for example).

Each pulse sent by the transducer into the body of a patient will be partly reflected and partly transmitted at the boundary between two different tissues. The reflected pulse (known as the echo pulse) will be received at the transducer. It will have less energy than the original pulse due to energy losses within the body and because some of the energy of the original pulse is transmitted through the body. The pulsed voltage at the transducer is displayed on an oscilloscope screen/computer screen as a voltage-time graph. The amplitudes of the voltage signals are attenuated by absorption and reflection losses. The time interval is the time taken for the ultrasound pulse to travel from the (front of the) transducer to the retina and back to the transducer therefore the total distance travelled by the ultrasound pulse is 2L where L can be calculated provided the average speed of the ultrasound in the eye is known.

B scans
B scans produce a 2D image on a screen. The transducer is moved over the patients skin and the output of the transducer is connected to a high-speed computer. For each position of the transducer the computer produces a row of dots on the screen (each dot corresponds to the boundary between two tissues). The brightness of the dot is proportional to the intensity of the reflected ultrasound pulse.


Acoustic impedance
The fraction of ultrasound intensity reflected at the boundary depends on the acoustic impedance of both media. The acoustic impedance (Z) of a substance is defined as the produce of the density of the substance and the speed of the ultrasound in the substance. It has the SI unit kg m^-2 s^-1:

Z = ρc

The reflected intensity of ultrasound depends on the values of Z1 and Z2 (the acoustic impedances of two substances). For normal incidence, when the angle of incidence is 0°, the ratio of reflected intensity (Ir) to incident intensity (Io) is given by the following equation:

Ir/Io = (Z2-Z1)²/(Z2+Z1)²

Ir/Io = ((Z2-Z1)/(Z2+Z1))²

The ration fo Ir/Io is known as the intensity reflection coefficient. There is more reflection when the values of acoustic impedances are very different (e.g there will be a greater reflection at a one/muscle boundary than a blood-muscle boundary. The acoustic impedance of bone is much different to the rest of the body (the rest of the body is pretty similar) so bone is easily distinguishable in an ultrasound scan.

Coupling gel is very important. When an ultrasound transducer is placed on the skin of a patient air pockets will be trapped between the transducer and the skin. The air-skin boundary means that about 99.9% of the incident ultrasound will be reflected before it even enters the patient. To overcome this coupling gel is used. Coupling gel has a similar acoustic impedance to the skin and fills the air gaps between the transducer and the skin ensuring that almost all the ultrasound enters the patient's body. Here, we can use the term impedance/acoustic matching. This is when two substances have similar values of acoustic impedance and negligible reflection occurs at their boundary as a result.


Doppler imaging
The frequency of ultrasound changes when it is reflected off a moving object (this is known as the Doppler effect). Doppler ultrasound is a non-invasive technique that uses the reflection of ultrasound (from iron-rich blood cells) to help doctors evaluate blood flow through major arteries. It can be used to reveal blood clots/atheroma and evaluate the amount of blood flow to transplanted organs.

During Doppler ultrasound the transducer is pressed lightly over the skin above the blood vessel. It sends pulses of ultrasound and receives the reflected pulses from inside the patient.  Ultrasound reflected off tissues will return with the same frequency and wavelength but ultrasound reflected of moving objects (blood cells) will have a changed frequency. The frequency increases when blood is moving towards the transducer and decreases when blood is moving away from the transducer. The transducer is connected to a computer that produces a colour-coded image to show the direction and speed of the blood flow.

As we know ultrasound scans have a frequency of 5-15MHz. In blood flow analysis this can give a Doppler shift up to 3kHz. The frequency shift (Doppler shift in frequency) (Δf) is directly proportional to the speed (v) of approach/recession of the blood. 

The axis of the probe (ultrasound transducer) must be held at an angle θ to the blood vessel. This is because holding perpendicular would give no observed change in the frequency as cos90 = 0. The usual θ is 60°. The change in observed frequency (Δf) is given by the following equation:

Δf = (2 f v cosθ) / c

F: original ultrasound frequency
v: speed of moving blood cells
c: speed of ultrasound in the blood

6.5.2 Diagnostic methods in medicine

Radioactive isotopes have to be placed inside the patient and their radiation is detected from the outside. Gamma emitters are ideal sources as gamma photons are the least ionising and can also penetrate through the patient and be detected externally. Radioisotopes that are used for medical imaging must have a short half-life to ensure high enough activity from the source so only a small amount is needed for the image to form (this is also important as the patient is not subjected to a high dosage of radiation after the procedure). Radioisotopes such as fluorine-18 (used in PET scans) are produced artificially on-site (as they have a short half-life). Technetium-99m (produced by the natural radioactive decay of molybdenum-99) can be used to monitor the function of major organs such as the heart, liver, lungs, kidneys, and brain.

Radioisotopes are chemically combined with elements that will target the desired tissue to make a radiopharmaceutical (this is a medical tracer). E.g Tc-99m can be combined with sodium and oxygen to produce NaTcO4 (this will target the brain once injected). Its progress through the body can be traced using a gamma camera as the Tc-99m emits gamma photons. The concentration of Tc-99m can identify irregularities in the function of the body.


Fluorine-18 is a radiopharmaceutical used in PET scans (positron emission tomography). It has a half life of ~110 minutes and it will decay into a nucleus of oxygen-18, a positron, a neutrino, and a gamma photon. It has to be made in a laboratory near the hospital or with a particle accelerator (e.g high speed protons collide with oxygen-18 nuclei to produce fluorine-18 nuclei and neutrons.


The gamma camera
The gamma camera is a diagnostic tool that detects gamma photons emitted from radioactive nuclei injected into the patient, and an image is constructed which indicates the concentration of the tracer in the body. The gamma photons travel towards the collimator, any arriving at an angle are absorbed by the tubes so only those travelling along the axis of the tubes reach the scintillator. The scintillator is usually sodium iodide. A single photon striking the scintillator produces thousands of visible light photons (but not all the gamma photons produce these flashes as there is only a 1/10 chance a gamma photon will interact with the scintillation. The photons of visible light travel through the light guide into photomultiplier tubes. These are arranged in a hexagonal pattern and a single photon is converted into an electrical pulse. The outputs of each photomultiplier tube is connected to a computer and the electrical impulses are processed to locate the impacts of the photons on the scintillator. These impact positions construct a high quality image showing the concentrations of the tracer in the patient and the final image is displayed on a screen.

Gamma cameras produce an image that shows the function and processes of the body rather than the anatomy (like an x-ray).


Positron emission tomography
PET scans can be used to construct a detailed 3D image with gamma radiation (instead of X-rays). More often than not the radiopharmaceutical fluorodeoxyglucose (FDG) is used as it is similar to naturally occurring glucose but tagged with a radioactive fluorine-18 atom in place of an oxygen atom. Our bodies treat FDG as normal glucose and incorporate it into tissues with a high rate of respiration. Its activity can be monitored using gamma detectors. Carbon monoxide (with the carbon-11 isotope) can also be used as a radiopharmaceutical for PET scans. This emits a positron and has a half life of ~20 minutes. It attaches to haemoglobin in red blood cells (meaning it can be transported in the blood).

Pet scanners work as follows:

• the patient lies on a horizontal table surrounded by gamma detectors
• each detector consists of a photomultiplier tube and sodium iodide scintillator and produces a voltage pulse/signal for every gamma photon incident at its scintillator
• the detectors are connected to a computer
• the patient is injected with FDG
• the pet scanner detects the gamma photons emitted when positrons (from decaying Fl-18) annihilate with electrons inside the patient
• photons detected by the scanner come from the annihilation of these positrons, not from the gamma photons emitted by Fl-18 decaying. The annihilation of a positron and electron produces 2 gamma photons that are travelling in opposite directions (momentum is conserved)
• the computer determines the point of annihilation from the difference in arrival times of the photons at two diametrically opposite detectors and the speed of the photons (3x10^8)
• the voltage signals from the detectors are fed into the computer which analyses and manipulates the signals to form an image on a display screen
• different concentrations of the tracer show up as areas of different colours and brightnesses

Okay so we need to know the issues raised when equipping a hospital with an expensive scanner, such as a PET scanner. The advantages and disadvantages are as follows:

• Advantages
• non-invasive technique
• help diagnose different types of cancers/plan complex heart surgery/observe function of the brain
• help doctors to identify the onset of certain disorders of the brain (e.g. Alzheimer's)
• can be used to assess the effect of new medicines and drugs on organs
• Disadvantages
• very expensive due to the facilities required to produce the medical tracers
• only found at larger hospitals
• only patients with complex health problems are recommended for a PET scan.

6.5.1 Using X-rays

X-ray photons have 10-10,000 times more energy than a photon of visible light (depending on their wavelength). They are harmful to living cells and can kill them (useful for treating cancer).  X-ray photons are produced when fast-moving electrons are decelerated by interactions with atoms of a metal (e.g tungsten). The kinetic energy of the electrons is transformed into X-ray photons.

X-ray machines contain an X-ray tube that produces X-ray photons that pass through the patient to the detection plate (below the thing being X-rayed). An X-ray tube consists of an evacuated tube containing two electrodes (it is evacuated so electrons can pass through the tube without interacting with gas atoms). A large p.d. is created between the electrodes by an external power supply. The negative electrode (cathode) is a heater which produces electrons by thermionic emission. These electrons accelerate towards the positive electrode (anode). It is important to note that the anode is made from a metal known as the target metal (e.g tungsten). This must have a high melting point. This is because the X-ray photons are produced when the electrons are decelerated by hitting the anode, the energy output fo the X-rays is less than 1% of the kinetic energy of the incident electrons and the remainder of the energy is transformed into thermal energy of the anode (hence a high melting point). In many X-ray tubes oil is circulated to cool the anode/the anode is rotated to spread the heat over a larger surface area. The anode is shaped so the X-rays are emitted in a desired direction with lead to shield the radiographer from sporadic X-rays emitted in a different direction...

An electron accelerated through a p.d. 'V' gains the kinetic energy eV (from W = V Q). It is important to realise that one electron releases one X-ray photon. From the principle of conservation of energy we can determine that the maximum energy of a photon from an X-ray tube must equal the maximum kinetic energy of a single electron. The energy of a photon (eV) is equal to the Planck constant x the freqyency (maximum frequency is the speed divided by the minimum wavelength):

hf = eV

hc/λ = eV

λ = hc/eV

This means that the wavelength from an X-ray tube is inversely proportional to the accelerating p.d. (therefore increasing the current will increase the intensity of X-rays).


The term attenuation is used to describe the decrease in the intensity of electromagnetic radiation as it passes through matter (e.g bone attenuates X-rays more than tissue would). There are four attenuation mechanisms by which X-ray photons interact with atoms (each reduces the intensity of the collimated/parallel/ beam in the original direction of travel):

• Simple scatter
• the X-ray photon is scattered elastically (kinetic energy is conserved) by an electron
• This mechanism is important for X-ray photons with energy in the range of 1-20keV
• The X-ray photon interacts with an electron in the atom but has less energy than the energy required to remove the electron (work function) so the X-ray photon bounces off without a change to its energy
• Photoelectric effect
• the X-ray photon disappears and removes an electron from the atom
• This is significant for X-ray photons with energy less than 100keV
• the X-ray photon is absorbed by one of the electrons in the atom, the electron uses this energy to escape from the atom
• Attenuation of X-ray photons by this type of mechanism is dominant wen an X-ray image is taken as hospital X-ray machines use 30-100kV supplies
• Compton scattering
• the X-ray photon is scattered by an electron, it's energy is reduced, and the electron is ejected from the atom
• This is significant for X-ray photons with energy in the range of 0.5-5MeV
• The X-ray photon interacts with an electron in the atom and is ejected from the atom with reduced energy
• Pair production
• the X-ray photon disappears to produce an electron-positron pair
• This only occurs when X-ray photons have energy equal or greater than 1.02MeV
• An X-ray photon interacts with the nucleus of the atom, it disappears and the electromagnetic energy fo the photon is used to create an electron and a positron

The transmitted intensity of X-rays depends on the energy of the photons and on the thickness and type of the substance (e.g bones will attenuate X-rays more than soft tissue will). For a given substance and energy of photons the intensity falls with the thickness of the substance. Transmitted intensity can be calculated using the following equation:

I = I₀e^-xμ

(NOTE: the μ is meant to be like ^μ  (as in superscript) but for some reason it wouldn't format properly)

I₀: the initial intensity before absorption
x: the thickness of the substance
μ: the attenuation/absorption coefficient (the larger the μ the better absorber the substance is). SI unit: m^-1.


Soft tissues have a low absorption coefficient so a contrast medium is sue to improve the visibility of their internal structures in X-ray images. The most common are barium sulphate and iodine. They have relatively large atomic number (Z) which is good as μ ∝ Z³:

• iodine is used as a contrast medium in liquids (e.g to view blood flow). An organic compound of iodine is injected into blood vessels so that doctors can diagnose blockages in the blood vessels and structure of organs (e.g the hears) from X-ray images.
• Barium sulphate is used for digestive systems. It is given to the patient in the form of a white liquid mixture which is swallowed.

CAT scans

A CAT scanner records a large number of X-ray images from different angles and assembles them into a 3D image (with sophisticated software). In essence, a patient lies horizontally on their back and slide in/out of a gantry/large vertical ring. The gantry houses an X-ray tube on one side and an array of electronic X-ray detectors on the other side. These rotate within the gantry. The X-ray tube produces a fan-shaped beam of X-rays (~1-10mm thick). This thin beam irradiates a thin 'slice' of the patient, the X-rays are attenuated by different amounts by different tissues. The intensity of the transmitted X-rays is recorded by the detectors which send electrical signals to a computer. A 2D slice is acquired each time the X-ray tube and detectors complete one full rotation. The slices can be manipulated to produce a 3D image of the patient.

Advantages:

• CAT scans can be used to create 3D images - this can help doctors to assess the shape/size/position of disorders (e.g cancers)
• CAT scans can distinguish between soft tissues of similar attenuation coefficients

Okay so the spec doesn't say we need to know the disadvantages but they'll probably be useful to know:

• A traditional X-ray is quicker and cheaper
• The X-rays are harmful as they are ionising radiation - some CAT scans can be quite prolonged so expose the patients to a radiation dose equivalent to several years of background radiation
• Patients have to remain very still during the scanning process (any movement will blur the slice) - super tricky with young patients

4.4.4 stationary waves

Stationary waves are also known as standing waves. They can be created by longitudinal and transverse waves and form when two progressive waves with the same frequency (and ideally the same amplitude) travelling in opposite directions superpose. At points in antiphase the waves cancel out forming a node. At points in phase the waves cancel out forming an antinode. Where displacement is zero, amplitude and therefore intensity are also zero. The separation between adjacent nodes (or antinodes, for that matter) equates to half the wavelength of the original progressive waves. It is important to realise that there is no net transfer of energy as the two progressive waves are travelling in opposite directions so they sort of cancel each other out.

All the particles between adjacent nodes are oscillating in phase with one another. This is because, although they have different amplitude, they all reach their maximum positive displacements at the same time. On different sides of a node the particles are in antiphase; the particles to the left of a node reach their maximum positive displacement at the same time the particles on the right reach their maximum negative displacement.

This set of graphs nicely demonstrates the motion of a stationary wave:

We need to know how to demonstrate stationary waves using microwaves, stretched strings, and air columns:

Microwaves
We can form a stationary wave by reflecting microwaves off a metal sheet so that two microwaves of the same frequency are travelling in opposite directions. Using a microwave receiver we can detect the changes in intensity between nodes (low/no intensity) and antinodes (maximum intensity). The distance between the transmitter and the metal sheet must be adjusted until the receiver detects a series of notes/antinodes. As we are aware already, the distance between adjacent nodes or antinodes equates to half the wavelength of the microwaves from the transmitter.

Stretched strings
Each string has a fundamental mode of vibration. The frequency of this vibration is the fundamental frequency (f₀). This depends on factors such as the strings mass, tension, and length. When a string is stretched between two points the two points act as nodes. If the string is plucked a progressive wave travels along the string and reflects off its ends which creates two progressive waves travelling in opposite directions and these superpose and a stationary wave is formed. When plucked, the string vibrates in its fundamental mode of frequency, the wavelength of the progressive wave is double the length of the string.

The fundamental frequency (f₀) is the minimum frequency of a stationary wave for a string. However, it is possible to form other stationary waves known as harmonics at higher frequencies. For a given string at a fixed tension the speed of the progressive wave is constant. From v = fλ we can see that as frequency increases λ decreases proportionally. E.d at a frequency of 2f₀ the wavelength is half what it was at f₀. This table (from the kerboodle OCR A Physics textbook) nicely demonstrates this:

Air columns
Most woodwind instruments (sound is longitudinal) produce notes from blowing over the top of a tube creating a standing wave inside. This produces a note at a particular frequency (the length of the tube determines the wavelength of the note it produces). Sound waves reflected off a surface can produce a stationary wave. The original wave and the reflected wave travel in opposite directions and superpose. Stationary sound waves can also be made in tubes by making the air column inside the tube vibrate at frequencies related to the length of the tube. The stationary wave formed depends on whether the ends of the tube are open or closed.

Closed at one end:
In order for a stationary wave to form in a tube closed at one end there must be an antinode at the open end and a node at the closed end. The air at the closed end cannot move so it must form a node whilst at the open end the oscillations of the air are at their greatest amplitude so it must be an antinode. The fundamental mode of vibration has a node at the base and antinode at the top, the wavelength is 4 times the length of the tube.

In a tube closed at one end it is not possible to form harmonics at 2f₀, 4f₀, 6f₀, etc. This is because the open end must be an antinode. The frequencies of the harmonics in tubes closed at one end are always an odd multiple of f₀, 3f₀, 5f₀, 7f₀, etc as demonstrated in this diagram:

Open at both ends:
A tube open at both ends will have an antinode at both ends and a node in the centre (if vibrating at f₀). Harmonics at all integer multiples of the fundamental frequency are possible. This diagram nicely shows this:

4.4.3 Superposition

The principle of superposition of waves
When two waves (of the same type) meet they sort of pass through each other/overlap. This is known as superposition and a single wave is produced whose instantaneous displacement is the sum of the two former waves (the principle of superposition reads: 'when two waves meet at a point the resultant displacement at that point is equal to the sum of the displacements of the individual waves'). It is important to realise that, since displacement is a vector quantity, this resultant displacement can be bigger or smaller than the displacements of the previous waves. This effect is known as interference. If two waves are in phase the maximum positive displacements will line up causing constructive interference. This results in an increase in amplitude since intensity ∝ (amplitude)². If the two waves are in antiphase the maximum positive displacement of one will line up with the maximum negative displacement of another - this results in destructive interference. If the amplitudes of both waves are the same, the resultant will have zero amplitude:

Interference patterns can be seen when, for example, raindrops fall on a pond. As the waves ripple outward they overlap with waves caused by other drops. At this point they superpose and can interfere constructively (if in phase) or destructively (if out of phase). If is important to realise that this does not produce a stable interference pattern but rather one that changes all the time, for a stable pattern the waves must be coherent. Coherence refers to waves emitted from two sources that have a constant phase difference. For example, two filament lamps cannot produce stable interference patterns as they emit light of a range of different frequencies and changing phase difference between different waves (in other words, they do not produce coherent light).

Interference patterns contain a series of fringes known as maxima (louder/brighter etc, where constructive interference occurs) and minima (quieter/dimmer etc, where destructive interference occurs). Maxima and minima are the result of two waves that have travelled different distances from their sources. This difference is known as the path difference. If the path difference to a point is a whole number (or 0) the two waves will arrive in phase (producing constructive interference) producing a point that has maximum amplitude. If the path difference to a point is an odd number of half wavelengths (n+0.5 where n is an integer) the two waves will arrive in antiphase (producing destructive interference) producing a point that has minimum amplitude.

At the central maxima the path difference is zero so the phase difference (the difference between displacements of particles along a wave) is zero. At the first order maxima the path difference is one wavelength, so the phase difference is 360° (the peaks from the first wave line up with the peaks from the second wave so constructive interference occurs). At the first order minima the path difference is half a wavelength so the phase difference is 180° (the peaks from the first wave line up with the troughs from the second waves which results in destructive interference).

Experiments:
Okay so there are a series of experiments we need to know regarding superposition/two-source interference. We need to know techniques and procedures to investigate superposition experiments using sound, light and microwaves.

Light
One way to observe the interference patterns of light is to observe the pattern of coloured light on thin oil films on water. Basically, light reflecting off the bottom surface of the oil interferes with the light reflected off the top surface. If the thickness of the oil results in a path difference that is a non-integer half number of wavelengths of light the two sets of light wavs are out of phase and destructive interference occurs and the waves cancel out. The colours result from the different wavelengths in white light and the differences in the thickness of the oil layer. The distance the light travels through the oil before reflecting off the surface differs. Different wavelengths of light are cancelled out by different thicknesses of oil. The wavelengths are not cancelled out from the colours we observe.

Credit: Kerboodle OCR Physics A textbook

Sound

• Connect two loudspeakers to the same signal generator (this means they will emit coherent sound waves)
• the sound waves will travel out from each loudspeaker and overlap forming an interference pattern
• this interference pattern comprises a series of maxima (louder areas) and minima (quieter areas)
• the positions of maxima and minima can be detected with a microphone (or your ears, but a microphone is more accurate)

Microwaves

• Introduce a pair of slits in front of a single microwave source
• the microwaves will diffract and overlap forming an interference pattern
• the interference pattern can be detected using a microwave receiver connected to a voltmeter or an oscilloscope
• if the receiver were to me moved in an arc around the double slit the maxima and minima created as part of the interference pattern can be detected
• the positions of each maxima and minima can be marked on a piece of paper situated below the apparatus

The Young double-slit experiment

As we know from above we need two coherent waves to form an interference pattern. Young used a monochromatic light source (by using a light filter). This means only light of a specific frequency can pass through a narrow slit that follows to diffract the light. Light diffracting from this slit then arrives at a double slit in phase. It diffracts again at the double slit. Each slit acts as a source of coherent waves which spread from each slit overlapping and forming an interference pattern that can be seen on a screen as fringes (alternating bright and dark regions). This experiment demonstrates the wave nature of light - it can also determine the wavelengths of various different colours of visible light.

Okay so now for some maths. The separation between the double slits is denoted as 'a'. The interference pattern is observed on a screen at distance 'D'from the slits (D>>a). A bright fringe (maxima) is seen on the screen at position 'Y' and the next an position 'X', the distance between 'Y' and 'X' is x. The path difference S1P is one wavelength and the angles θ1 and θ2 are almost the same (they are very VERY small). We can use trig to show that:

sinθ1 ≈ sinθ2 ≈ tanθ2

where sinθ1 = wavelength/a and tanθ2 = x/D

This means that wavelength/a ≈ x/D. We can roughly express this as:

λ = ax/D

Provided D>>a.

Isaac Newton had a theory in which light was made up of tiny particles. Christiaan Huygens (a Dutch physicist) believed that light was made up of waves vibrating up and down perpendicular to the direction of the light travels. From this he formulated a way of visualising wave propagation (known as 'Huygens' Principle').  Huygens theory was the successful theory of light showing wave motion in three dimensions. He suggested that in a vacuum, or other uniform mediums, the light waves are spherical, and these wave surfaces advance or spread out as they travel at the speed of light. This theory explains why light shining through a pin hole or slit will spread out rather than going in a straight line (diffraction). Huygens theory better describes early experiments. Huygens' principle lets you predict where a given wavefront will be in the future, if you have the knowledge of where the given wavefront is in the present.

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.