Waves
Waves
Overview of Waves
Definition of a Wave:
“A wave is a vibration or oscillation that transfers energy without net transfer of matter..”
Important to note: A wave does NOT cause any net movement of matter.
Types of Waves
Longitudinal Waves
Definition:
A longitudinal wave consists of particles that oscillate parallel to the direction of propagation of the wave.
Characteristics:
Mechanical waves require a medium for transmission and arise due to vibrations.
Example:
Sound waves are a primary example of longitudinal waves.
Transverse Waves
Definition:
A transverse wave consists of particles that oscillate perpendicular to the direction of propagation of the wave.
Example:
Light waves are a common example of transverse waves.
Waveforms
Sinusoidal Motion
Displacement-time graph:
Illustrates how vibrations occur over time.
Key Terms:
Amplitude (A): The magnitude of the maximum displacement (from the mean position) reached by an oscillation in the wave (measured in metres, m).
Period (T): The time taken for one complete oscillation at one point on the wave (measured in seconds, s).
Frequency (f): The number of complete wave cycles per second (measured in hertz, Hz).
Important: Always pay attention to the x-axis label on graphs.
Frequency and Time Period
Explanation:
If an oscillator has a frequency of 10 Hz, it completes ten oscillations every second, leading to each oscillation taking 0.1 s.
Relationship:
The relationship between frequency and period is essential for understanding wave behavior.
Displacement-Distance Graphs
General Characteristics
While similar to the displacement-time graph, it represents all particle positions along that wave section instead of a single particle's motion.
Key Concept:
The distance between two adjacent positions that are in phase is defined as the wavelength, λ, of the wave.
Longitudinal Wave Profile
Displacement Graph:
For longitudinal waves, illustrating particle displacement can be challenging as it runs parallel to the motion.
A wave profile may show:
Positive displacements to the right of the mean positions.
Negative displacements to the left, leading to particle displacement in opposite directions, which results in increased pressure.
Pulse-Echo Measurements
Overview:
An ultrasound transducer transmits a pulse of ultrasound into a medium and records the time taken for the pulse to return.
Mechanics:
The pulse reflects at boundaries between different materials/media/densities.
A data logger connected to the transducer creates a graph of the echo.
Important Note:
The time taken for the echo to return should not overlap with the original sound duration.
Amplitude Significance:
The amplitude of the returning echo indicates the energy of the sound waves, which diminishes as it travels through the medium, resulting in a lower amplitude upon return.
Wave Equation
Speed of a Travelling Wave:
The speed of all waves can be calculated using a specific equation.
Fundamental Equation:
A wave will travel one wavelength, λ, in the time taken to complete one cycle, T.
Relationship:
The relationship can be summarized as:
\text{Wave speed} = \text{frequency} \times \text{wavelength}
Phase
Definition:
The phase of an oscillation relates to the position within a cycle that a particle occupies relative to its starting point.
Important Concept:
Oscillations that are half a cycle out of step are considered to be in antiphase.
For sinusoidal waveform calculations:
1 cycle = 360° = 2π radians.
Wavefronts
Definition of Wavefront:
A wavefront is defined as a line or surface that connects points in a wave that are in phase.
Wavefront Concept:
The distance between wavefronts is equivalent to the wavelength, λ.
Superposition of Waves
General Definition:
Superposition occurs when two or more waves meet at a point, resulting in the resultant displacement being the vector sum of individual displacements.
Important Cases:
Constructive Superposition: When two waves are in phase, they add to create a maximum amplitude.
Destructive Superposition: When two waves are in antiphase, they combine to create zero amplitude (𝝅 radians out of phase).
Standing Waves
Formation and Characteristics
Definition:
Standing waves (stationary waves) are formed by the superposition of two progressive waves of equal speed and frequency moving in opposite directions, retaining a constant phase relationship.
Mechanism:
Typically achieved using a travelling wave and its reflection, ensuring the frequencies match exactly.
Nodes and Antinodes:
Points of zero amplitude within a standing wave are termed nodes, while maxima are referred to as antinodes.
Phase and Energy Distribution
Phase Relationships:
All points between one node and the next remain in the same phase (these sections of the wave oscillate together).
Energy Considerations:
Although each point has varying amplitude, they all oscillate as a unit; energy does not pass along the standing wave due to its stationary nature.
Harmonics
Resonance in Stringed Instruments:
Instruments like guitars and violins produce standing waves on strings fixed at both ends.
Simplest Standing Wave Characteristics:
The simplest standing wave will contain one antinode between two nodes, resulting in a string length of half a wavelength (λ).
Speed of Transverse Waves on a String/Wire
Formula for Speed:
The speed of a transverse wave moving along a stretched wire is given by:
v = \sqrt{\frac{T}{\mu}}Where:
T is the tension (measured in N).
μ is the mass per unit length (measured in kg m⁻¹).
Diffraction
Definition:
Diffraction is the spreading out of waves as they navigate through a narrow slit or around an obstacle.
Conditions:
For optimal diffraction, the slit size must be comparable to the wavelength of the passing wave.
Strategies to Increase Diffraction:
Reduce the size of the gap.
Increase the wavelength of the wave.
Huygens’ Construction
Concept:
Huygens stated that each point on a single slit acts as a source of secondary wavefronts.
Visualization:
This model illustrates how waves spread out, where the new wavefront is the surface tangential to all secondary wavelets.
Single Slit Diffraction
Diffraction Pattern Overview:
When light passes through a narrow slit, a diffraction pattern is observed consisting of:
A bright central maximum (fringe).
Dark areas flanking the central maximum.
Bright maxima of decreasing intensity following the dark areas.
Observational Notes:
A narrower slit results in a widened central maximum alongside the further maxima and minima.
Diffraction Grating
Definition:
A diffraction grating consists of numerous closely spaced, narrow slits.
Mechanism:
When illuminated by monochromatic coherent light, a diffraction pattern is generated which features a series of sharp rays dispersing from the grating.
Interference Relationship:
A greater number of slits results in enhanced interference effects.
Diffraction Grating Patterns
Equation for Patterns:
Patterns produced follow specific equations to relate to wavelengths and angles of diffraction.
General Conclusions from Diffraction
If λ increases, then sin(θ) increases, resulting in larger θ (more dispersion).
If d increases, then sin(θ) decreases; a smaller d results in a coarser grating leading to more spreading of the pattern.
Values of sin(θ) greater than 1 are impossible; if the calculation results in such a value for n, the order cannot exist.
Practice Questions
Calculate the angle to the normal for a yellow laser light of wavelength 6.00 x 10⁻⁷ m through a diffraction grating of 4.0 x 10⁵ lines per meter:
a) For first and second-order bright lines (4 marks).
b) State whether there is a fifth order line and provide an explanation (1 mark).
Two Source Interference
Definition:
Interference occurs when waves from two or more sources superpose, forming patterns of maxima (antinodes) and minima (nodes).
Conditions for Stable Patterns:
The waves must be from the same type (both light waves, for example).
The sources must be coherent (same wavelength and frequency maintaining a constant phase relationship).
The amplitudes should be similar at the interference point.
Two Source Interference with Sound
Setup Overview:
Two loudspeakers connected to a signal generator produce identical, coherent sound waves.
Position Outcomes:
Position X results in maximum amplitude (antinodes) producing loud sounds.
Position Y leads to zero amplitude (nodes) producing silence.
Path Difference Concept
Definition:
The difference in distance travelled by two waves from their sources to the point where they meet is known as the path difference, which can lead to constructive or destructive interference depending on whether the path difference is an integer multiple of the wavelength.
Outcomes:
Maximum amplitude occurs when path difference is zero or a whole number of wavelengths (constructive interference).
Minimum amplitude occurs when path difference is an odd half wavelength (destructive interference).
Phase Difference and Path Difference Relationships
Key Relationships:
Path Difference for Constructive Interference: nλ and for Destructive Interference: (n + ½)λ
Phase Difference for Constructive Interference: 2πn and for Destructive Interference: (2n + 1)π
Young’s Double Slit Experiment
Experiment Overview:
Uses either a monochromatic laser source or an incoherent source combined with a single slit.
Setup Details:
A crest/trough simultaneously passes through each slit creating a central maximum signal received due to zero path difference.
Relationship Calculation:
The wavelength of the source can be deduced using the equation relating wavelength and path differences across the two slits.
Young’s Fringe Spacing
Setup Description:
Consider two slits S₁ and S₂, with distances defined and a point P on a screen.
Bright Spot Production:
For bright spots at point P, the path difference must equate to a whole number of wavelengths leading to constructive interference.
Path Difference Equation:
S{1}P - S{2}P = nλ
Refraction of Light
Definition:
Refraction occurs due to changes in wave speed of light across various media.
Angle Reference:
All angles are measured from the normal to the light beam (the perpendicular at the interface).
Bending Behavior:
Light bends away from the normal when moving from denser to less dense medium (speeds up), and towards the normal when moving from less dense to dense (slows down).
Refraction of Wavefronts
Overview:
When wavefronts encounter a denser medium, they slow down and change direction.
In shallow water, wavefront spacing varies, exhibiting longer wavelengths in lighter mediums compared to denser ones.
Frequency Consistency:
The frequency remains unchanged during refraction; thus, changes in speed produce modifications in wavelength.
Ray Behavior:
Any ray crossing the interface along the normal remains unchanged in direction.
Refractive Index, n
Definition:
Refractive index measures the extent of refraction caused by media, represented as a ratio:
n = \frac{c}{v}
where c is the speed of light in vacuum (or air), and v is the speed in the material.
Expression:
It may also be represented as the ratio of angles of incidence and refraction:
n = \frac{\sin i}{\sin r}
Significance of Higher Index:
A higher refractive index indicates that light travels slower in that medium (denser).
Snell’s Law
Conceptual Importance:
This law captures the relationship between the direction of light and the refractive index, yielding a practical measure of refractive properties.
Snell’s Law Formula:
n1 \sin(θ1) = n2 \sin(θ2)
Application:
Useful for measuring changes in speed based on changes in angle when transitioning across different media.
Dispersion
Definition:
Dispersion involves the splitting of white light into a spectrum of colors using a prism, due to its composition of multiple wavelengths.
Wavelength Effects:
Refraction affects wavelengths differently; shorter wavelengths (e.g., violet) experience greater angles of refraction in air.
Dispersive Effects:
This property arises because the speed of light within glass varies by wavelength, thus varying refractive indices for different colors.
Total Internal Reflection
Conditional Requirements:
The incident substance must possess a larger refractive index than the adjacent substance (moving from dense to less dense material).
The angle of incidence must exceed the critical angle.
Critical Angle, θc:
Defines the threshold angle above which total internal reflection occurs, with certain calculations derivable through Snell’s Law.
Critical Angle Calculations
Using Snell’s Law:
To derive the critical angle for a material:
For medium 1 (optically denser) and θ₂ as 90° at the critical angle, where θc equals the critical angle.
Plane Polarisation of Electromagnetic Waves
Definition of Polarisation:
Polarisation restricts oscillations within a wave to occur in a single plane.
Characteristics of EM Waves:
Composed of perpendicular electric (blue) and magnetic (red) fields, which may oscillate in distinctly defined planes; the electric field plane determines the wave's plane of polarisation.
Polarisation Overview
Nature of Polarised Light:
Polarised light contains only one plane of polarisation.
Interaction with Polaroids:
When unpolarised light encounters a polarising filter, it becomes polarised.
Cross Polarisation:
Two filters at right angles to each other (cross polarisers) will block all light from passing through.
Analysers and Intensity of Light
Filter Behaviour:
When aligned, all light passing through the first filter will also pass through the second; however, varying amounts will pass upon rotating the second filter.
Intensity Changes:
Less light gets through as the vertical component of the second filter's transmission axis decreases, impacting overall intensity.
Measurements:
At angles of 45° and 90°, intensity will exhibit specific values (half and zero respectively).
Polarisation by Reflection
Reflection Dynamics:
Light striking a surface such that the angle between the reflected and refracted rays equals 90° will result in linearly polarised reflected light.
Polarisation Direction:
The direction of polarisation corresponds parallel to the plane of the reflecting medium surface, leading to glare.
Polarising Sunglasses Utility:
These glasses efficiently absorb horizontally oriented glare from surfaces (like water or roads), reducing visual discomfort.
Polarisation by Refraction
Interaction at Surface:
Light striking a surface capable of refraction will result in partial reflection of horizontally polarised light and partial transmission of vertically polarised light into the new medium.
Quantum Physics: Wave-Particle Duality
Fundamental Overview:
Evidence supports light's dual behavior, acting as both a wave and a particle under specific circumstances, described through distinct phenomena.
Evidence Summary:
Waves: Diffraction, interference, polarisation.
Particles: Photoelectric effect, electron diffraction, ionisation.
Huygen’s Principle
Principle Description:
Huygens formulated this principle to elucidate wave diffraction, correctly predicting wavefront movements across various scenarios and propagation aspects.
Important Argument:
Particles are incapable of producing a standing wave pattern featuring nodes and antinodes, unlike wave fronts.
Light as a Particle
Max Planck's Contribution:
Introduced the notion of light as quantised packets (photons); the energy of these particles can be computed by the equation:
E = hf
Contextual Importance:
The energy quantisation explains energy transfer processes in photon interactions.
Are Electrons Particles or Waves?
Evidence for Electron Behavior:
As particles, electrons possess defined charge and mass.
Experiments demonstrating electron diffraction patterns validate their wave-like characteristics.
Evidence further supports electrons' ability to produce interference patterns like light does, reinforcing the wave-particle duality concept.
The Photoelectric Effect
Definition:
The photoelectric effect refers to photons of UV light causing electron release (photoelectrons) from a negatively charged metal surface.
Work Function:
The minimum energy required for an electron to escape the metal surface is termed the work function (Φ).
Critique of Classical Wave Theory:
The photoelectric effect presents challenges classical physics cannot address succinctly.
Wave Model Limitations
Key Characteristics that Classical Theory Cannot Explain:
Lack of lag time for electron emission.
Kinetic energy independence from incident radiation intensity.
Necessity of a cut-off frequency for photoelectron emission.
Photoelectric Effect: Wave vs. Particle Theory
Comparisons:
Wave Theory: Suggests energy is correlated to amplitude, predicting photoemission based on wave brightness regardless of frequency.
Particle Theory: Clarifies the instantaneity of photon-to-electron energy transfer, requiring threshold frequency for electron release, irrespective of wave intensity.
The Photoelectric Equation
Equation Summary:
During the photoelectric effect, photons impart energy to electrons needing a threshold amount equal to the work function for emission.
Conservation of Energy Perspective:
Any surplus energy post emission is converted to kinetic energy, described mathematically.
The Photoelectric Cell Experiment
Experiment Setup:
In a vacuum, the metal acts as an anode with a cathode gap. Knowledgeable frequency light triggers photoelectron emission, causing current registration.
Stopping Voltage Importance:
The stopping voltage (Vs) quantifies maximum kinetic energy of emitted photoelectrons.
Application of Results:
Varying light frequencies and analyzing stopping voltage produce data useful for calculating Planck’s constant and work function.
Electron Diffraction
Historical Experimentation:
Conducted by Davisson and Germer, underlying electron wave behavior against crystal surfaces, showcasing diffraction patterns correlated with atomic spacing.
Validation of Wave Nature:
Electrons’ small wavelengths necessitate appropriate diffraction obstacles, utilizing atomic features for observation.
Two-Slit Interference of Electrons
Experimental Evidence for Electron Waves:
Electrons exhibited two-slit interference patterns mirroring light behavior when input through similar arrangements.
Latest Discoveries:
Recent research illustrated pattern establishment through individual electron movements in the experimental setup, supporting wave-particle duality.
Electron Microscopy
Application Relevance:
Electron wave properties are crucial for high-resolution imaging at minuscule scales, facilitated by controlling electron wavelengths through applied voltages.
Minimum Imaging Resolution:
The smallest detectable object size aligns closely with the wavelength being utilized for observation.
Atomic Electron Energies
Energy Levels Overview:
In free atoms, energy states for electrons are quantized, being confined to discrete values known as energy levels.
Ground State Definition:
An electron in its ground state is at its lowest energy level, with energy values generally represented as negative (requiring energy input for excitation).
Excitation Processes
Conditions for Excitation:
An atom may become excited through:
Collisions with other particles.
Absorbing a photon possessing the precise energy requisite for jumping to a higher energy level.
Photons as Energy Transitions:
After absorption, the photon ceases to exist, transferring its energy to the electron involved.
Excitation and Absorption Spectra
Interaction with Photons:
If a photon lacks enough energy for electron excitation, it fails to interact with the atom, thereby not being absorbed.
Spectra Appearance:
Gas atoms illuminated by varied frequencies exhibit absorption spectra, distinguished by missing color lines in the emitted spectrum due to absorbed wavelengths.
Ionisation Energies
Definition:
The minimum energy required to liberate an electron from its ground state is termed the principal ionisation energy, illustrating energy level interactions.
Energy Level Diagram Explanation:
The relationship between ground state energy levels and ionisation is illustrated, where less energy is needed to extract electrons at higher energy levels.
Electron Volt (eV)
Definition of Electron Volt:
An electronvolt (eV) represents the kinetic energy gained by an electron accelerated through a potential difference of 1 volt.
Calculative Context:
Using the equation E = QV , the parameters lead to the determination of 1 eV values in joules.
De-excitation Mechanism
Explanation:
An excited electron, after a random timeframe, will de-excite to lower energy levels:
Returning to the ground state.
Shifting to an intermediate energy level.
Photon Emission:
Due to energy conservation, emitted photons upon de-excitation display energy equivalent to the difference between energy levels.
De-excitation and Emission Spectra
Spectra Generation:
A collection of excited gas atoms emits light across various frequencies, forming emission spectra characterized by distinct bright lines against darkness.
Unique Fingerprints:
Each element has a distinguishable emission pattern allowing for precise element identification through diffraction grating techniques.
Intensity Measurement of Light
Measurement Concept:
Light intensity signifies energy transported per unit area per unit of time (complex intensity assessment).
Power Formula:
Power as the rate of energy transfer specified by P = \frac{E}{t}
Solar Emission Consideration:
Consideration of emitted light from the Sun is dispersed spherically, making it crucial to utilize spherical surface area for energy calculation as a source of emission.