Module 3: Waves and Thermodynamics
Wave Properties
- Vibrating objects transfer energy through waves
- Waves are classified by what they move through
- There are two types: mechanical and electromagnetic
Mechanical Waves
- Mechanical waves transfer energy through vibrations in a medium
- Examples include water, sound and wind
- A wave can be a single pulse, or continuous
Waves only transfer energy from one point to another. THEY DO NOT TRANSFER MATTER! However, matter may move as the wave passes through it.
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Mechanical waves can be either transverse or longitudinal.
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In a transverse wave, particles oscillate (move back and forth) perpendicular (at 90°) to the direction of energy transfer.
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In a longitudinal wave, particles move parallel to the direction of energy transfer.
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Measuring Mechanical Waves
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Waves can be represented by displacement-distance graphs and displacement-time graphs.
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The graph takes a sinusoidal shape (like a sine wave)
The Role of the Medium
- Speed of sound waves in a medium depends on 2 main factors:
- Elasticity of the medium
- Density of the medium
- The more rigid a material is, the faster the mechanical waves are transmitted
- The more elastic a material is, the slower a mechanical wave is transmitted
- The higher the temperature a given medium is, the faster a mechanical wave will move through it.
Electromagnetic Waves
- All EM waves propagate through space at $3\times10^{8}m/s$
- EM waves do not require a medium as they self-propagate
- A charged particle produces an electric field.
- A moving charged particle produces a magnetic field.
- An oscillating charged particle produces an EM wave.
- EM waves consist of perpendicular electric and magnetic fields.
Other Properties of EM Waves
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All EM waves are transverse waves.
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All able to undergo:
- Reflection
- Refraction
- Polarisation
- Interference
- Diffraction
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All EM waves travel in straight lines
Displacement-Time Graph of a Transverse Wave
- The displacement time graph of a transverse wave shows how a single particle is displaced from natural resting position as a function of time
Wave Behaviour
Wavefronts and Rays
- Rays refer to direction in which energy moves from initial disturbance which created the wave
- Wavefront essentially marks points at which energy of particles are equal
Reflection
- The law of reflection state that $θ_{i}=θ_{r}\newcommand{orange}{\color{orange}}\newcommand{pink}{\color{pink}}$
- A reflected wave has the same frequency, wavelength, and speed as the incident (original) wave
- Wavefronts are identical before and after reflection
$\newcommand{orangebox}{\bbox[5px, border: 2px solid orange]}\newcommand{pinkbox}{\bbox[5px, border: 2px solid pink]}\newcommand{greenbox}{\bbox[5px, border: 2px solid green]}$
Convex Mirrors
- Rays diverge after reflection
- Used in car mirrors and surveillance systems
Concave Mirrors
- Curves inward and waves converge at the focus
- Used to reflect microwaves or radio waves from collecting dishes to receiving antenna
Refraction
- When an EM wave travels from one medium to another some of the wave will be absorbed, some reflected, and the rest propagates through the new medium
$$\color{orange}{n_{1}\sin\left(\alpha\right)=n_{2}\sin\left(\beta\right)}$$
- Where $\alpha$ and $\beta$ are the angles from the normal (the red line in the image) and n values are the refractive indices of the media
- the refractive index of a material will usually be given in exams, but it is useful to remember that air is $n\approx 1,$ glass is $n\approx 1.5,$ and water is $n\approx \frac{4}{3}$
- Usually, a more dense substance will have a higher refractive index
When light moves from one medium to another:
- Frequency of light remains unchanged
- Wavelength of light changes
- Velocity of wavelength changes
- When a wave slows down, it bends towards the normal
Diffraction
- Refers to the spreading if waves as the waves passes an object or travels through a gap between objects
- Wave diffracts around barrier at edges, leaving a shadow region behind it where the wave does not reach.
- The amount of diffraction is proportional to the wavelength AND the size of the slit:
$$\orange{\text{Diffraction}=\frac{\omega}{\lambda}}$$
Superposition
- Superposition is the overlapping of two waves
- There are two types of superposition: constructive (waves add to make a larger wave) and destructive (waves cancel each other out to make a smaller wave)
- Superposition results in a temporary change of frequency, amplitude and phase
- Once the waves have passed each other, the superposition collapses, and both waves return to their original properties
Standing Waves
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Standing waves are waves which do not appear to be moving along the medium
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Instead, the antinodes (peaks and troughs) seem to switch displacement twice every cycle
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Standing waves are frequently used in music, where:
$$\orange{\mathcal{f}_{n}=n\cdot\mathcal{f}_{1}}$$
$$\orange{\lambda_{n}=\frac{1}{n}\cdot\lambda_{1}}$$
- In other words, the frequency of harmonic n is equal to n multiplied by the frequency of harmonic 1 (usually middle C)
Resonance
- A phenomenon in which a vibrating system or external force drives another system to oscillate with greater amplitude at specific frequencies
- Occurs at frequencies where response amplitude is a maximum is termed “resonant frequency”
Sound Waves
- Sound is a mechanical wave; the wave is caused by a vibrating source.
- Travel as alternating regions of compressions and rarefactions.
- Travel at 343m/s
- Changes depending on air pressure and temperature
- Sound waves are measured in $W/m^2$, and are calculated by:
$$\orange{I=\frac{P}{4πr^{2}}}$$
Frequency and Pitch
- Frequency is the quantitative equivalent of pitch
Amplitude and Volume
- Amplitude and volume are directly proportional (increasing one increases the other)
Echoes:
- Minimum distance the boundary needs to be away (from both the source and the listener) is 17m.
- The sound wave has to travel a total of at least 34m.
- The best echoes come from hard, smooth surfaces.
- For a human to perceive two sounds as separate sounds, they MUST be 0.1 seconds apart.
The Doppler Effect
- When the source of a sound wave approaches an observer, the observer interprets the sound as compressed, and therefore higher pitched
- The opposite is true when the source is travelling away from an observer
- The frequency shift can be calculated by:
$$\orange{f\prime=f\left(\frac{v_{wave}+v_{observer}}{v_{wave}-v_{source}}\right)}$$
Ray Model of Light
Ray Diagrams
Describing Ray Diagrams
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Questions will often ask you to describe the image produced by a ray diagram
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To describe these images, use TOMP:
- Type of image (real/virtual)
- Orientation (upright/inverted)
- Magnification (enlarged/diminished/true size)
- Position (distance from the mirror)
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Magnification can be calculated by:
$$\orange{M=\frac{h_{i}}{h_{o}}\text{, where:}}$$
- $M=$magnification scale
- $h_i=$ image height
- $h_{o}=$ object height
Mirror Formula
$$\orange{\frac{1}{f}=\frac{1}{u}+\frac{1}{v}\text{, where:}}$$
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$f=$ focal length of the mirror (cm)
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$u=$ distance between mirror and object (cm)
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$v=$ distance between mirror and image (cm)
- Now that we have the mirror formula, we can expand the magnification formula:
$$\orange{M=\frac{h_{i}}{h_{o}}=\frac{-v}{u}}$$
Lenses
- A lens is a transparent piece of material which bends light in a specific manner
- Converging lenses bend light rays to meet at a specific focal point
- Diverging lenses bend light away from each other
Converging Lens:
Diverging Lens:
Refractive Index
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The speed of a wave depends on the density of the medium
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The density, when referring to its effect on waves, is referred to as a medium’s refractive index
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A total vacuum has a refractive index of $n=1.00$
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A higher refractive index means light will travel slower in that medium
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Refractive index is calculated by:
$$\orange{n_{material}=\frac{c}{v_{material}}}$$
- Where $c$ is the speed of light in a vacuum $(3\times10^{8}\text{ m/s})$
Thermodynamics
Temperature and Kinetic Energy
- While an object may be at rest, the particles it is composed of are in a constant state of motion
- As temperature increases, the Kinetic energy increases, so the particles move more quickly
- This allows energy to be transferred faster at higher temperatures
Heat is a measure of the transfer of thermal energy between bodies
Thermal Equilibrium
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Thermal energy is always transferred from regions of higher temperature to regions of lower temperature
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The Zeroth Law of thermodynamics states that:
“If two thermodynamics systems are each in equilibrium with a third system, then they are also in equilibrium with each other.”
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The First Law of Thermodynamics, also known as the Law of Conservation of Energy, states that:
“Energy cannot be created, nor destroyed.”
Specific Heat Capacity
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A measure of how much energy 1kg of a substance must absorb to increase in temperature by $\newcommand{deg}{^{\circ}}1\deg C$
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Water has a specific heat capacity of 4186J/kg/K
- In other words, it takes 4186J of energy to increase the temperature of 1kg of water by 1 Kelvin
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Specific heat capacity can be calculated with the formula:
$$\orange{\Delta Q=mc\Delta T\text{, where:}}$$
- $ \Delta Q=$change in energy (Joules)
- $m=$ mass of the substance (kg)
- $c=$ specific heat capacity of the substance (J/kg/K)
- $\Delta T=$ change in temperature (K)
Latent Heat
- Latent heat of fusion is the amount of energy required to change 1kg of a substance from liquid to gas WITHOUT CHANGING ITS TEMPERATURE
$$\orange{Q=mL\text{, where:}}$$
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$Q=$ Energy released (J)
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$m=$ mass of the substance (kg)
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$L=$ Latent heat of fusion (J/kg)