Wave Definitions and Properties

Definitions

• Amplitude refers to a wave's maximum displacement from its equilibrium position
• Frequency (f) refers to the quantity of complete oscillations passing a point per second
• Period (T) refers to the time it takes for one complete oscillation
• Speed (v) refers to the distance travelled by the wave per unit time
• Wavelength (λ) refers to the length of one whole oscillation, such as the distance between successive peaks or troughs

Wave Equation

• The speed (v) of a wave equals the frequency multiplied by its wavelength, expressed as: v = fλ

Longitudinal Waves

• In longitudinal waves, particle oscillation is parallel to the direction of energy transfer
• Longitudinal waves are composed of compressions and rarefactions, and cannot travel in a vacuum
• Sound is an example, and can be shown by horizontally pushing a slinky
• In rarefaction stages, pressure decreases and particles separate
• In compression stages, pressure increases and particles move closer

Transverse Waves

• With transverse waves, oscillations of particles or fields occur at right angles to the direction of energy transfer
• All electromagnetic (EM) waves are transverse, and travel at 3 x 10^8 ​ms​⁻¹ in a vacuum
• Transverse waves are demonstrated by vertically shaking a slinky, or waves on a string connected to a signal generator

Graphs of Transverse and Longitudinal Waves

• Displacement-distance graphs illustrate particle displacement relative to wave travel distance, useful to measure wavelength
• Displacement-time graphs illustrate the displacement of a particle over time to measure the period of a wave
• For transverse waves, displacement distance graphs are similar to the actual wave

Standing Wave Representation

• Standing waves can be placed on a displacement-distance graph

Further Definitions

• Phase refers to a position on a wave cycle, in radians, degrees, or fractions of a cycle
• Phase difference refers to the lag of a particle/wave behind another, using radians, degrees, or fractions of a cycle
• Path difference refers to a variance in distance travelled by two waves
• Superposition is the combination of displacements when waves pass, their resultant being the vector sum of each wave’s displacement
• Coherence indicates the light source having the same frequency and wavelength, but with a fixed phase difference

Wavefronts

• A wavefront connects points on a wave with the same phase
• Dropping a rock into a pond creates wavefronts in the form of ripples

Interference Types

• Constructive Interference: Occurs when two waves are in phase, leading to added displacements
• Destructive Interference: Occurs when two waves are completely out of phase, their displacements subtracted

Constructive vs Destructive Interference

• Constructive interference results with one wave while destructive interference impacts the resultant wave

Phase Difference and Path Difference

• Waves in phase are at the same point in their cycle, share frequency and wavelength (coherent), and have an integer multiple of 360° phase difference (2π radians)
• Waves out of phase maintain the same frequency and wavelength (coherent), but their phase difference is an odd integer multiple of 180° (π radians)

Phase Difference Equation

• The phase difference in radians of two waves with the same frequency and path differences relates as: Δx = (λ / 2π) * Δϕ
• Δx is the path difference, λ is the wavelength, and ΔΦ is the phase difference

Example Question

• Two waves have a 6m path difference, both with a 2m wavelength
• To find the phase difference use: Δϕ = 2π * (Δx / λ)
• Substitute known values to get: Δϕ = 2π * (6 / 2) = 6π
• The phase difference is 6π, thus it is a multiple of 2π, meaning the waves are in phase

Stationary Waves

• Stationary waves, also known as standing waves is formed from the superposition of progressive waves, travelling in opposing directions in the same plane, with the same frequency, wavelength and amplitude
• No net energy is transferred

Wave Meeting Points

• In phase results in constructive interference, creating antinodes with max displacement
• Completely out of phase results in destructive interference, creating nodes with no displacement

String Fixed at One End

• A demonstration of stationary waves involves a string fixed at one end connected to a driving oscillator
• A wave travelling from the oscillator reflects from the string's fixed end, causing superposition
• Given that the waves share wavelength, frequency, and amplitude, a stationary wave forms

Representation of Standing Waves

• Standing Waves have antinodes and nodes

Transverse Wave Speed on a String

• The speed (v) of a transverse wave on a string can be calculated as: v = √(T/μ)
• T is the tension in the string and μ is the mass per unit length

Intensity of Radiation

• Intensity refers to the power (energy transferred per unit time) per unit area
• Equation is: I = P/A
• P is the power and A is the area

Refractive Index and Snell’s Law

• Refractive index (n) is how much a material slows down light that passes through it
• Calculated by dividing the speed of light in a vacuum (c) by the speed of light in a substance (v): n = c/v
• Material with a higher refractive index is more optically dense

Refraction and Snell's Law

• Refraction occurs when a wave changes medium, altering its direction relative to the normal
• Snell’s law calculates light refraction: n₁sinθ₁ = n₂sinθ₂ where:
• n₁ and n₂ are refractive indexes of material 1 and 2 respectively
• θ₁ and θ₂ are the angles of incidence and refraction

Light Boundaries

• Light speed changes across boundaries, altering its direction
• As n₂ is more optically dense than n₁, light slows down and bends toward the normal
• As n₂ is less optically dense than n₁, light bends away from the normal

Critical Angle

• As incidence angle increases, so does refraction angle, approaching 90°
• The critical angle (C) is the incidence angle that causes the refraction angle to be exactly 90°, refracting light along the boundary

Using Air

• Knowing that the refractive index of air (n₂) is approximately 1, calculate critical angle (C) using: sin C = 1/n where n > 1

Total Internal Reflection (TIR)

• Total Internal Reflection occurs when the incidence angle is greater than the critical angle, and when the incident refractive index (n₁) is greater than the refractive index of the boundary material (n₂)

Refractive Index Measurement

To measure the refractive index of a solid material:

• Trace the material
• Remove the material and mark a perpendicular normal line
• Radiate, with a protractor, lines at 10° intervals from 10° to 70°
• Return the block
• Use a ray box to shine light along with the 10° to mark where the ray leaves the block
• Connect marked points and measure

Graph of Refractive Index

• To find the value, graph sine of incident angles (sin i) against the sine of refracted angles (sin r)
• The gradient of the line of best fit denotes the refractive index of the material

Plane Polarisation

• A polarized wave oscillates in only one plane, where only transverse waves can be polarized

Polarised Waves

• In vertically polarized waves, the wave passes through the filter without a problem
• In horizontally polarized waves, the wave cannot pass through the filter as it blocks waves which are not in the vertical plane

Polarised Sunglasses

• Polarized sunglasses reduce glare by blocking partially polarised light reflected from water and tarmac, only allowing oscillations in the plane of the filter to pass through, making it easier to see

Diffraction and Huygens’ Construction

• Diffraction is the spreading out of waves when they pass through or around a gap
• Huygens’ construction states that every point on a wavefront is a point source to secondary wavelets, which spread out to form the next wavefront

Huygens' Principle

• Light travelling through a doorway diffracts, meaning each point on the wavefront then acts as a source of wavelets, thus spreading to form further circular wavefronts
• Light passes through a doorway with little diffraction, resulting in straight shadows
• Light waves barely diffract, while more diffraction occurs with sound waves
• Light’s wavelength is much smaller compared against the doorway, whereas sound’s wavelength is much closer
• Greatest amount of diffraction occurs occurs when the gap is the same size as the wavelength

Diffraction Grating Equation

• A diffraction grating is a slide with equally spaced slits close together which creates an interference pattern with light and dark fringes
• The ray of light passing through the grating's center is the zero order line
• On either side of this line is the first order lines, etc.

Diffraction Grating Equation

• dsinθ = nλ
• d = the distance between slits
• θ = the angle to the normal
• n = the order
• λ = the wavelength

Electron Diffraction

• An electron gun accelerates electrons towards a crystal lattice, where they interact with small gaps between atoms and then form an interference pattern on a screen made of fluorescent behind the crystal
• Electrons behave as waves which explains the diffraction pattern

de Broglie Relation

• The de Broglie hypothesis describes all particles with a wave nature and a particle nature
• Any particle’s wavelength can be found using: λ = h/p
• λ = the de Broglie wavelength
• h = Planck's constant
• p = momentum of the particle

Wave Behaviour

• An interface, a boundary between materials where waves can be: Transmitted - passing into the next material and experiencing refraction
• Reflected - Bouncing off the interface without entering the new material

Pulse-Echo Technique

• The pulse-echo technique employs ultrasound waves (over 20 kHz) to image and reflect medical techniques
• A brief ultrasound pulse is directly transmitted
• The pulse traverses to a boundary between two mediums, reflecting it back
• Boundary Density Variance impacts increased reflection
• Reflected waves detected
• Intensities reflect structure and time informs position using the formula s = vt

Resolution of Pulse-Echo

• Longer pulses can cause overlap, diminishing your data captured
• High wavelengths will resolve fewer details of the data, and subsequently, the data is reduced

Wave Model & Photon Model

• The photon model theorizes that EM waves move via discrete packets called photons
• Photon's energy is directly proportional to its frequency, expressed as E = hf
• On the other hand, in the wave model, EM radiation can be expressed with a transverse wave

Models of Light

• At first, light (an EM wave) was understood as tiny particles to explain the reflection and refraction
• As light behaved as a wave, diffraction experiment occurred, and now the scientific understanding changed, and light was then understood as waves
• It was then learned that light is both particle and wave which leads to acceptance of the photon model of light and wave-particle duality

Photon Energy

• Photons transfer energy which is proportional to frequency as calculated by: E = hf
• E = photon energy
• h = Planck’s constant
• f = wave frequency

Photoelectricity

• Photoelectric Effect: Photoelectrons emitting off of metal surfaces that certain frequencies of light shine on (the threshold frequency)
• Electrons collect closer to the surface and when it absorbs more energy, it leaves the surface

Threshold Frequency, Work Function & Photoelectric Equation

• Threshold Frequency: the minimum required light frequency for photoelectrons that varies depending on the metal
• Work Function: a metal’s minimum energy to eject electron measured as Φ
• In mathematical contexts, the photoelectric equation displays a correlation between work function, frequency, and the maximum kinetic energy: E = hf = Φ + E k (max)
• E = photon energy
• Φ= work function
• k(max) = maximum kinetic energy

Electronvolt

• Electronvolt (eV): a quantity of energy to express small energy commonly where 1 eV means kinetic power with electron increasing amount that corresponds an amount equal 1 V or 1.6 x 10⁻¹⁹J

Electronvolt Conversions

• Use the following methods for joules and volts:
• To electron volts: divide by 1.6 x 10⁻¹⁹
• To joules: multiply by 1.6 x 10⁻¹⁹

Explaining the Wave Theory With Experiments

• Wave Theory claims with light experiment there should be emission when any frequency, not explaining existence of threshold frequency
• Photoelectric Impact exists quick and not like the Wave Theory, which requires the source electron obtain with the energy for task to leave metallic surface
• Increase light source should also make the source amount speed which doesn't happen and instead emits sources electrons
• Electrons have kinetic range

Explaining Wave Theory With the Photon Model

• In contrast, photon model displays EM wave emits small packets with particles, explain all points where Wave Theory failed
• Photon and Electron connect: transfer energy and electron needs one, threshold must exist: the wave works to equate work
• To clarify the model, photon carries to electrons, therefore fast emission exists
• Increase also emits increased due photon interacts
• Electrons obtain power from a photon with collision where power exits