Wave Equation and Properties of Waves

Wave Equation and Frequency

• The wave equation relates wave speed (v), wavelength (λ), and frequency (f): v = fλ.
• The equation can be rewritten as v = 1/T, where T is the period of the wave, the time for one complete oscillation.

Determining Wave Frequency

• An oscilloscope can be used to measure the frequency of a wave.
• The timebase of the oscilloscope represents time on the x-axis and amplitude on the y-axis.
• By measuring the time for one complete oscillation, the frequency can be calculated.

Reflection, Refraction, Diffraction, and Polarization

• Waves can be reflected, refracted, and diffracted.
• Reflection: A wave changes direction at a boundary between two media, remaining in the original medium.
• Refraction: A wave changes direction as it changes speed when entering a new medium.
• Diffraction: A wave spreads out as it passes through a gap.
• Polarization: A property of transverse waves where the oscillation is restricted to one plane.

Demonstrating Wave Effects

• Ripple tank: Demonstrates reflection, refraction, and diffraction of water waves.
• Polarizing filters: Demonstrate polarization of light.
• Metal grille: Demonstrates polarization of microwaves.

Intensity of a Progressive Wave

• Intensity is the power passing through a surface per unit area (P/A).
• Units are watts per square meter (Wm⁻²).
• Intensity is inversely proportional to the square of the radius for a point source.
• Intensity is proportional to the square of the amplitude of the wave.

Electromagnetic Waves

• Electromagnetic waves are transverse progressive waves with oscillating electric and magnetic fields.
• They travel through a vacuum at a speed of 3.0 x 10⁸ m/s.
• Visible light is a small part of the electromagnetic spectrum.

Refraction of Light

• When light travels from one medium to another, it refracts due to a change in speed.
• Refractive index (n) is the ratio of the speed of light in a vacuum to the speed of light in the medium.
• Snell's Law relates the angles of incidence and refraction: n₁sinθ₁ = n₂sinθ₂.

Total Internal Reflection

• Occurs when light travels from a medium with a higher refractive index to a medium with a lower refractive index.
• The critical angle (C) is the angle of incidence above which total internal reflection occurs.
• The formula for the critical angle is sinC = 1/n, where n is the refractive index of the material the light is entering.

Superposition

• The principle of superposition states that when waves meet, the resultant displacement is the sum of the individual wave displacements.
• Constructive interference: Displacements in the same direction add together.
• Destructive interference: Displacements in opposite directions cancel each other out.

Interference and Coherence

• Two waves are coherent if they emit with a constant and unchanging phase difference.
• Interference occurs when coherent waves superpose.
• The maximum resultant displacement occurs when the phase difference is an even multiple of π.
• The minimum resultant displacement occurs when the phase difference is an odd multiple of π.

Investigating Superposition and Wavelength

• Audio signal generators: Investigate superposition in sound waves by creating an interference pattern.
• Young's double-slit experiment: Investigate superposition and determine the wavelength of light.
• Diffraction grating: Investigate the wavelength of light.

Stationary Waves

• Formed when two progressive waves with the same frequency traveling in opposite directions superpose.
• Characterized by nodes (zero amplitude) and antinodes (maximum displacement).
• Frequency is the same at all points except at the nodes.

Producing Stationary Waves

• Stretched string: A vibrating generator creates waves that reflect, forming a stationary wave.
• Microwaves: A transmitter and a reflecting surface create a stationary wave.
• Sound in a tube: A tuning fork creates sound waves that interfere, forming a stationary wave depending on the tube's length and end conditions.
• The fundamental frequency is the lowest frequency of vibration for a given arrangement.

Charge

• Charge is a physical quantity that can be positive or negative.
• Charge is measured in Coulombs (C) where 1 Coulomb is the amount of charge that flows in 1 second when the current is 1 Ampere.
• Like charges repel, while opposite charges attract.
• The charge of ions and atomic components is quantised.
• A proton has a charge of +1 and an electron has a charge of -1.
• These charges are multiples of the elementary charge (e) which is 1.6 x 10⁻¹⁹ C.
• The net charge of a particle is due to the gain or loss of electrons.
• A neutral atom has an equal number of protons and electrons.
• Increasing the number of electrons creates a negative ion.
• Removing electrons creates a positive ion.
• The net charge on a particle (Q) is given by Q = ± ne, where n is the number of electrons added or removed and e is the elementary charge.

Electric Current

• Electric current (I) is the rate of flow of charge.
• Current is measured in Amperes (A).
• Current is calculated as I = dQ/dt.
• An ammeter is used to measure current and is placed in series in a circuit.

Charge Carriers

• Electric current is the rate of flow of charge, but charge can be carried in different ways depending on the material.
• In metals, current is carried by electrons.
• Metals have a lattice of positive ions surrounded by free electrons.
• The positive ions are fixed, but the electrons can move.
• When a potential difference is applied across a metal, electrons are attracted to the positive terminal and create a current.
• Some liquids can conduct charge, these are called electrolytes and are typically ionic solutions.
• Electrolytes contain positive and negative ions.
• When electrodes are placed in an electrolyte, cations are attracted to the cathode and anions are attracted to the anode, leading to an electric current.

Conventional Current

• Conventional current is the flow of charge from the positive to the negative terminal.
• This is the standard way current is treated, regardless of the actual direction of charge carriers.
• In metals, electrons flow from negative to positive, so the electron flow is opposite to the conventional current.

Kirchhoff’s 1st Law

• Kirchhoff's 1st law states that at any point in an electrical circuit, the sum of the currents into that point is equal to the sum of the currents out of that point.
• This is a consequence of the conservation of charge.
• Charge cannot be created or destroyed, only transferred.

Mean Drift Velocity

• Electrons in a metal move randomly due to collisions with positive ions.
• When a potential difference is applied, the electrons are attracted to the positive terminal, but still collide with positive ions.
• Mean drift velocity (v) is the average velocity of electrons as they travel through the wire.
• The number density (n) of a material is the number of free electrons per unit volume.
• Conductors have high number densities (around 10²⁸ per m³).
• Insulators have much lower number densities and semi-conductors have values in-between.
• Lower number densities mean electrons must travel faster to carry the same current.
• Current can be calculated as I = nAve, where A is the cross-sectional area of the wire and e is the elementary charge.

Explore the fundamentals of wave behavior through this quiz on the wave equation, frequency, reflection, refraction, diffraction, and polarization. Test your understanding of how these concepts interact and their practical applications using tools like oscilloscopes.