Determining the wavelength from the positions of nodes or antinodes is a fundamental concept in the study of stationary waves. This topic is pivotal for students pursuing AS & A Level Physics (9702), as it enhances understanding of wave behavior, resonance phenomena, and various practical applications in physics and engineering.

Key Concepts

Stationary Waves

Stationary waves, also known as standing waves, are formed by the superposition of two waves of identical frequency and amplitude traveling in opposite directions. Unlike traveling waves, stationary waves remain confined to a particular region and do not exhibit net propagation. Instead, they exhibit points of constant amplitude called nodes and points of maximum amplitude called antinodes.

Nodes and Antinodes

In a stationary wave, nodes are specific points where the medium does not oscillate, resulting in zero displacement. Antinodes, on the other hand, are points where the medium experiences maximum displacement. The distance between two consecutive nodes or two consecutive antinodes is half the wavelength ($\lambda/2$).

Wavelength ($\lambda$)

The wavelength is the distance between two successive points that are in phase, such as two consecutive nodes or two consecutive antinodes. It is a key parameter that describes the spatial periodicity of the wave and is related to both the frequency ($f$) and the speed of the wave ($v$) through the equation:

$$\lambda = \frac{v}{f}$$

Wave Speed ($v$)

The speed of a wave traveling through a medium is determined by the medium's properties and is given by:

$$v = \lambda f$$

Where:

$\lambda$ = Wavelength
$f$ = Frequency

Determining Wavelength from Nodes

To determine the wavelength using the positions of nodes, follow these steps:

Identify Consecutive Nodes: Locate two adjacent nodes in the stationary wave pattern.
Measure Distance ($d$): Measure the distance between these two nodes.
Calculate Wavelength: Since the distance between consecutive nodes is half the wavelength, multiply the measured distance by two to obtain the wavelength.

Formula: $$\lambda = 2d$$

Example: If the distance between two consecutive nodes is 0.4 meters, the wavelength is:

$$\lambda = 2 \times 0.4 \, \text{m} = 0.8 \, \text{m}$$

Determining Wavelength from Antinodes

The process to determine the wavelength using antinodes is similar to using nodes:

Identify Consecutive Antinodes: Locate two adjacent antinodes in the stationary wave pattern.
Measure Distance ($d$): Measure the distance between these two antinodes.
Calculate Wavelength: Multiply the measured distance by two to obtain the wavelength.

Formula: $$\lambda = 2d$$

Example: If the distance between two consecutive antinodes is 0.6 meters, the wavelength is:

$$\lambda = 2 \times 0.6 \, \text{m} = 1.2 \, \text{m}$$

Mathematical Representation of Stationary Waves

The general equation for a stationary wave on a string fixed at both ends is represented as:

$$y(x, t) = 2A \sin(kx) \cos(\omega t)$$

Where:

$A$ = Amplitude of the traveling waves
$k$ = Wave number, $k = \frac{2\pi}{\lambda}$
$\omega$ = Angular frequency, $\omega = 2\pi f$

This equation shows that the stationary wave is a product of a spatial component ($\sin(kx)$) and a temporal component ($\cos(\omega t)$), indicating that the spatial pattern remains fixed while the wave oscillates in time.

Resonance and Standing Wave Patterns

Resonance occurs when the frequency of an external force matches the natural frequency of the system, leading to the formation of standing waves with specific numbers of nodes and antinodes. For a string fixed at both ends, the allowed wavelengths (resonant wavelengths) are given by:

$$L = n \frac{\lambda}{2}$$

Where:

$L$ = Length of the string
$n$ = Harmonic number (integer)

Solving for $\lambda$ provides the wavelengths that form standing waves:

$$\lambda_n = \frac{2L}{n}$$

Example: For a string of length 1 meter supporting the second harmonic ($n=2$), the wavelength is:

$$\lambda_2 = \frac{2 \times 1}{2} = 1 \, \text{m}$$

Wave Speed on a String

The speed of a wave traveling on a string is influenced by the tension ($T$) in the string and its linear mass density ($\mu$). The wave speed is determined by:

$$v = \sqrt{\frac{T}{\mu}}$$

Where:

$T$ = Tension in the string (Newtons)
$\mu$ = Linear mass density (kg/m)

This relationship is crucial for understanding how physical properties of the string affect wave propagation.

Applications of Stationary Waves

Stationary waves have numerous applications across various domains:

Musical Instruments: Strings on instruments like guitars and violins produce standing waves to create musical notes.
Microwave Ovens: Use standing electromagnetic waves to heat food uniformly.
Engineering: Analysis of vibrations in structures to prevent resonance-related failures.
Optics: Lasers utilize standing light waves within optical cavities to produce coherent light beams.

Determining Frequency from Wavelength

Once the wavelength ($\lambda$) is known, the frequency ($f$) of the wave can be calculated using the wave speed ($v$) with the relation:

$$f = \frac{v}{\lambda}$$

Example: If a wave has a wavelength of 1.5 meters and travels at a speed of 3 m/s, the frequency is:

$$f = \frac{3}{1.5} = 2 \, \text{Hz}$$

Experimental Determination of Wavelength

In laboratory settings, wavelengths can be determined by measuring the distances between nodes or antinodes using rulers, calipers, or laser measurement tools. Precision in measurement is essential for accurate calculation of wave parameters.

Limitations and Considerations

While determining wavelength from nodes or antinodes is straightforward, several factors must be considered:

Measurement Accuracy: Precise measurement tools are required to accurately determine distances between nodes or antinodes.
Harmonic Overlap: In complex systems with multiple harmonics, identifying specific nodes or antinodes can be challenging.
Boundary Conditions: The fixed or free ends of the medium affect the standing wave pattern and must be appropriately considered.

Real-World Applications

Understanding wavelength determination is essential in various real-world applications, including:

Acoustics: Designing concert halls and auditoriums to optimize sound quality through controlled standing wave patterns.
Telecommunications: Antenna design relies on standing wave principles to ensure efficient signal transmission and reception.
Medical Imaging: Technologies like ultrasound utilize wave properties to create images of internal body structures.
Engineering: Vibration analysis in buildings and bridges to prevent resonance-induced structural failures.

Advanced Concepts

Mathematical Derivation of Standing Waves

To derive the equation for standing waves, consider two identical traveling waves moving in opposite directions along a string:

$$y_1(x, t) = A \sin(kx - \omega t)$$ $$y_2(x, t) = A \sin(kx + \omega t)$$

Superimposing these waves gives the stationary wave:

$$y(x, t) = y_1 + y_2$$ $$y(x, t) = A \sin(kx - \omega t) + A \sin(kx + \omega t)$$

Using the trigonometric identity:

$$\sin a + \sin b = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right)$$

We obtain:

$$y(x, t) = 2A \sin(kx) \cos(\omega t)$$

This equation represents a standing wave with a spatial component ($\sin(kx)$) and a temporal oscillation component ($\cos(\omega t)$). Nodes occur where $\sin(kx) = 0$, i.e., $kx = n\pi$ for integer $n$, leading to positions of zero displacement.

Quantization of Modes in Resonating Systems

In resonating systems such as strings or air columns, only discrete frequencies and wavelengths are allowed. These quantized modes correspond to specific harmonic numbers ($n$) and are determined by the system's boundary conditions. For instance, in a string fixed at both ends, the allowed wavelengths are:

$$\lambda_n = \frac{2L}{n}$$

Where:

$L$ = Length of the string
$n$ = Harmonic number (integer)

This quantization ensures that standing wave patterns fit perfectly within the physical constraints of the system.

Energy Distribution in Standing Waves

In stationary waves, energy is localized at antinodes and absent at nodes. The energy alternates between kinetic and potential forms, maintaining a constant total energy within the standing wave pattern. This distribution is crucial for understanding wave interference and resonance phenomena.

Interference of Waves

Standing waves result from the interference of two waves traveling in opposite directions. Constructive interference occurs at antinodes where wave amplitudes add up, while destructive interference occurs at nodes where wave amplitudes cancel out. This interference pattern is fundamental to the formation of stationary waves.

Mode Shapes and Patterns

Each harmonic mode of a standing wave has a distinct shape characterized by a specific number of nodes and antinodes. The fundamental mode (first harmonic) has the fewest nodes, while higher harmonics exhibit more complex patterns with additional nodes and antinodes. Understanding these mode shapes is essential for applications in musical instrument design and structural engineering.

Advanced Problem Solving

Problem: A string of length 3 meters is fixed at both ends and supports the fourth harmonic. If the wave speed on the string is 240 m/s, determine the wavelength and frequency of the fourth harmonic.

Solution:

Determine Wavelength ($\lambda$):
Calculate Frequency ($f$):

Answer: The wavelength is 1.5 meters and the frequency is 160 Hz.

Interdisciplinary Connections

Understanding stationary waves and wavelength determination is crucial across various disciplines:

Engineering: Design of structures to withstand vibrational modes and prevent resonance-induced failures.
Medicine: Ultrasound imaging relies on wave propagation and standing wave principles to create detailed images of internal body structures.
Music: Tuning musical instruments involves precise control of standing wave patterns to produce desired pitches.
Telecommunications: Antenna design utilizes standing wave principles to optimize signal transmission and reception.

Experimental Techniques

Advanced experimental techniques enable precise study of stationary waves:

Laser Doppler Vibrometry: Measures vibrations with high precision, allowing detailed analysis of wave properties.
High-speed Photography: Captures rapid oscillations of waves, providing visual insights into wave behavior.
Interferometry: Uses interference patterns to study wave interactions and measure physical quantities with high accuracy.

Nonlinear Effects in Standing Waves

At high amplitudes, stationary waves may exhibit nonlinear effects such as harmonic generation, wave distortion, and soliton formation. These phenomena complicate the simple linear model of standing waves and require advanced theoretical approaches to understand and predict wave behavior in nonlinear media.

Quantum Analogues of Standing Waves

In quantum mechanics, the concept of standing waves is analogous to the wavefunctions of particles in confined systems, such as electrons in an atom. These standing wave patterns define the allowed energy levels and shapes of atomic orbitals, bridging the gap between classical wave theory and quantum physics.

Advanced Mathematical Models

Advanced models of standing waves incorporate factors like damping, variable tension, and heterogeneous media, providing a more accurate and comprehensive description of wave behavior in real-world systems. These models are essential for applications requiring precise control and prediction of wave dynamics.

Comparison Table

Aspect	Nodes	Antinodes
Definition	Points of zero or minimal displacement.	Points of maximum displacement.
Displacement	No movement.	Maximum movement.
Energy Oscillation	None or minimal.	Maximum.
Distance Between Consecutive Points	$\lambda/2$	$\lambda/2$
Identification	Identified by points of no vibration.	Identified by points of greatest vibration.
Applications	Used to determine wavelength by measuring between nodes.	Used to determine wavelength by measuring between antinodes.

Summary and Key Takeaways

Wavelength can be determined by measuring the distance between consecutive nodes or antinodes and multiplying by two.
Stationary waves are formed by the superposition of two traveling waves with the same frequency and amplitude.
Understanding nodes and antinodes is essential for analyzing wave behavior and applying wave principles in various scientific and engineering contexts.

Examiner Tip

Tips

- **Visualize the Wave Pattern:** Draw diagrams of standing waves to clearly identify nodes and antinodes, aiding in accurate measurements and understanding.
- **Use Mnemonics:** Remember "N-A-N-A" to recall the alternating pattern of Nodes and Antinodes.
- **Practice with Real Objects:** Use strings or slinkies to create standing waves in a practical setting, reinforcing theoretical concepts through hands-on experience.
- **Check Units:** Always ensure that measurements are in consistent units when calculating wavelength and frequency to avoid errors.

Did You Know

1. **Resonance in Bridges:** The famous Tacoma Narrows Bridge collapse in 1940 was a result of resonance, where wind-induced standing waves amplified the bridge's oscillations, leading to its dramatic failure.
2. **Laser Technology:** Lasers rely on standing wave patterns within optical cavities to produce highly coherent and focused light beams, essential for applications ranging from medical surgeries to barcode scanners.
3. **Quantum Physics:** Electrons in atoms form standing wave patterns around the nucleus, explaining the discrete energy levels observed in atomic spectra.

Common Mistakes

1. **Confusing Nodes and Antinodes:** Students often mix up nodes (points of no displacement) and antinodes (points of maximum displacement). Remember, nodes are always stationary, while antinodes vibrate with maximum amplitude.
Incorrect Approach: Assuming the distance between a node and an antinode is equal to the wavelength.
Correct Approach: Recognizing that the distance between consecutive nodes or antinodes is half the wavelength ($\lambda/2$).

2. **Incorrect Wavelength Calculation:** Forgetting to multiply the measured distance between nodes or antinodes by two to find the wavelength.
Incorrect Formula: $\lambda = d$
Correct Formula: $\lambda = 2d$

FAQ

1. What is the difference between a node and an antinode in a standing wave?

Nodes are points where there is no displacement in the medium, while antinodes are points of maximum displacement.

2. How do you calculate the wavelength from the distance between two consecutive nodes?

Multiply the distance between two consecutive nodes by two. If the distance is $d$, then $\lambda = 2d$.

3. Can standing waves form in mediums other than strings?

Yes, standing waves can form in various mediums, including air columns in pipes, electromagnetic fields in cavities, and even electromagnetic waves in optical fibers.

4. Why are only specific wavelengths allowed in a standing wave system?

Only wavelengths that fit an integer number of half-wavelengths within the system's length satisfy the boundary conditions, leading to resonance and the formation of standing waves.

5. How does tension in a string affect the wavelength of standing waves?

Increased tension in the string leads to a higher wave speed, which, for a given frequency, results in a longer wavelength.

6. What is the relationship between frequency and wavelength in standing waves?

Frequency and wavelength are inversely related through the wave speed: $v = \lambda f$. For a given wave speed, an increase in frequency results in a decrease in wavelength.

1. Capacitance

1.1 Energy Stored in a Capacitor

1.1.1 Determine electric potential energy stored in a capacitor from the area under the potential–charge g

1.1.2 Recall and use W = ½QV = ½CV²

1.2 Discharging a Capacitor

1.2.1 Analyze graphs of potential difference, charge, and current during capacitor discharge

1.2.2 Recall and use τ = RC for the time constant during discharge

1.2.3 Use equations of the form x = x₀e^(-t / RC) for current, charge, or potential during discharge

1.3 Capacitors and Capacitance

1.3.1 Define capacitance for isolated spherical conductors and parallel plate capacitors

1.3.2 Recall and use C = Q / V

1.3.3 Derive formulae for combined capacitance of capacitors in series and parallel

1.3.4 Use capacitance formulae for capacitors in series and parallel

2. Nuclear Physics

2.1 Mass Defect and Nuclear Binding Energy

2.1.1 Sketch the variation of binding energy per nucleon with nucleon number

2.1.2 Explain nuclear fusion and nuclear fission

2.1.3 Calculate the energy released in nuclear reactions using E = c²∆m

2.1.4 Understand the equivalence between energy and mass (E = mc²)

2.1.5 Define and use the terms mass defect and binding energy

2.2 Radioactive Decay

2.2.1 Understand that fluctuations in count rate provide evidence for random decay

2.2.2 Understand that radioactive decay is spontaneous and random

2.2.3 Define activity and decay constant, and recall A = λN

2.2.4 Define half-life

2.2.5 Use λ = 0.693 / t₁/₂ for the half-life relationship

2.2.6 Understand exponential decay and use the relationship x = x₀e^(-λt)

3. Kinematics

3.1 Equations of Motion

3.1.1 Define and use distance, displacement, speed, velocity, acceleration

3.1.2 Use graphical methods to represent motion

3.1.3 Determine displacement from velocity–time graph

3.1.4 Determine velocity using gradient of displacement–time graph

3.1.5 Determine acceleration using gradient of velocity–time graph

3.1.6 Derive equations for uniformly accelerated motion

3.1.7 Solve problems of uniformly accelerated motion

3.1.8 Describe experiment to determine acceleration due to gravity

3.1.9 Explain motion with uniform velocity and perpendicular acceleration

4. Deformation of Solids

4.1 Stress and Strain

4.1.1 Understand and use the terms load, extension, compression, and limit of proportionality

4.1.2 Recall and use Hooke’s law

4.1.3 Recall and use the formula for spring constant k = F / x

4.1.4 Define and use terms stress, strain, and Young's modulus

4.1.5 Describe an experiment to determine Young’s modulus

4.1.6 Understand deformation caused by tensile or compressive forces

4.2 Elastic and Plastic Behaviour

4.2.1 Understand elastic and plastic deformation and elastic limit

4.2.2 Understand area under the force–extension graph as work done

4.2.3 Calculate elastic potential energy using EP = ½kx²

5. Gravitational Fields

5.1 Gravitational Field of a Point Mass

5.1.1 Understand that g is approximately constant for small height changes near Earth's surface

5.1.2 Recall and use g = GM / r²

5.1.3 Derive and use g = GM / r² for gravitational field strength

5.2 Gravitational Potential

5.2.1 Define gravitational potential as work done per unit mass in bringing a test mass from infinity

5.2.2 Use ϕ = -GM / r for gravitational potential due to a point mass

5.2.3 Use EP = -GMm / r for gravitational potential energy between two point masses

5.3 Gravitational Field

5.3.1 Define gravitational field as force per unit mass

5.3.2 Represent a gravitational field using field lines

5.4 Gravitational Force Between Point Masses

5.4.1 For point outside uniform sphere, treat mass as a point mass at its center

5.4.2 Analyze circular orbits in gravitational fields

5.4.3 Recall and use Newton's law of gravitation F = Gm₁m₂ / r²

6. Electricity

6.1 Resistance and Resistivity

6.1.1 Understand that the resistance of a thermistor decreases as temperature increases

6.1.2 Define resistance and recall and use V = IR

6.1.3 Sketch I–V characteristics for metallic conductor, semiconductor diode, and filament lamp

6.1.4 Explain that the resistance of a filament lamp increases as current increases

6.1.5 State Ohm’s law and recall and use R = ρL / A

6.1.6 Understand that the resistance of an LDR decreases as light intensity increases

6.2 Electric Current

6.2.1 Understand that an electric current is a flow of charge carriers

6.2.2 Understand that the charge on charge carriers is quantised

6.2.3 Recall and use Q = It

6.2.4 Use I = Anvq for current-carrying conductors

6.3 Potential Difference and Power

6.3.1 Define potential difference as energy transferred per unit charge

6.3.2 Recall and use V = W / Q

6.3.3 Recall and use P = VI, P = I²R, P = V² / R

7. D.C. Circuits

7.1 Practical Circuits

7.1.1 Recall and use the circuit symbols in the syllabus

7.1.2 Draw and interpret circuit diagrams using circuit symbols

7.1.3 Define electromotive force (e.m.f.) of a source as energy transferred per unit charge

7.1.4 Distinguish between e.m.f. and potential difference (p.d.)

7.1.5 Understand the effects of internal resistance on terminal potential difference

7.2 Kirchhoff’s Laws

7.2.1 Recall Kirchhoff’s first law: conservation of charge

7.2.2 Recall Kirchhoff’s second law: conservation of energy

7.2.3 Derive the formula for combined resistance of resistors in series using Kirchhoff’s laws

7.2.4 Use the formula for combined resistance of resistors in series

7.2.5 Derive the formula for combined resistance of resistors in parallel using Kirchhoff’s laws

7.2.6 Use the formula for combined resistance of resistors in parallel

7.2.7 Use Kirchhoff’s laws to solve simple circuit problems

7.3 Potential Dividers

7.3.1 Understand the principle of a potential divider circuit

7.3.2 Recall and use the principle of the potentiometer for comparing potential differences

7.3.3 Understand the use of a galvanometer in null methods

7.3.4 Explain the use of thermistors and LDRs in potential dividers to provide a potential dependent on te

8. Thermal Equilibrium

8.1 Thermal Energy Transfer

8.1.1 Understand that thermal energy transfers from higher to lower temperature

8.1.2 Understand that regions of equal temperature are in thermal equilibrium

9. Temperature Scales

9.1 Temperature Measurement

9.1.1 Understand physical properties that vary with temperature (e.g., density, gas volume, resistance)

9.1.2 Convert temperatures between Kelvin and Celsius, and recall T(K) = θ(°C) + 273.15

10. Magnetic Fields

10.1 Concept of a Magnetic Field

10.1.1 Understand that a magnetic field is a field of force produced by moving charges or permanent magnets

10.1.2 Represent a magnetic field using field lines

10.2 Force on a Current-Carrying Conductor

10.2.1 Understand that a force acts on a current-carrying conductor in a magnetic field

10.2.2 Recall and use F = BIL sin θ for the force on a current-carrying conductor in a magnetic field

10.2.3 Define magnetic flux density as the force per unit current per unit length on a wire at right angles

10.3 Force on a Moving Charge

10.3.1 Determine the direction of the force on a charge moving in a magnetic field

10.3.2 Recall and use F = BQv sin θ for the force on a moving charge in a magnetic field

10.3.3 Understand the origin of the Hall voltage and use the expression VH = BI / (ntq) for the Hall voltag

10.3.4 Understand the use of a Hall probe to measure magnetic flux density

10.4 Magnetic Fields Due to Currents

10.4.1 Sketch magnetic field patterns due to currents in a straight wire, flat coil, and solenoid

10.4.2 Understand that magnetic field in a solenoid is increased by a ferrous core

10.4.3 Explain the origin of forces between current-carrying conductors and determine the direction of the

10.5 Electromagnetic Induction

10.5.1 Define magnetic flux as the product of magnetic flux density and area perpendicular to flux directio

10.5.2 Recall and use Φ = BA for magnetic flux

10.5.3 Understand and use the concept of magnetic flux linkage

10.5.4 Explain experiments that show changing magnetic flux induces an e.m.f.

10.5.5 Recall and use Faraday’s and Lenz’s laws of electromagnetic induction

11. Specific Heat Capacity and Latent Heat

11.1 Specific Heat Capacity

11.1.1 Define and use specific heat capacity

11.2 Latent Heat

11.2.1 Define and use specific latent heat; distinguish between fusion and vaporisation

12. Dynamics

12.1 Momentum and Newton’s Laws of Motion

12.1.1 Mass resists change in motion

12.1.2 Recall F = ma and solve related problems

12.1.3 Define and use linear momentum

12.1.4 Force as rate of change of momentum

12.1.5 State and apply Newton’s laws

12.1.6 Describe weight as effect of gravitational field

12.1.7 Weight of an object is mass × gravitational acceleration

12.2 Non-uniform Motion

12.2.1 Understand frictional and drag forces

12.2.2 Describe motion in uniform gravitational field with air resistance

12.2.3 Objects moving against resistive force may reach terminal velocity

12.3 Linear Momentum and its Conservation

12.3.1 State conservation of momentum

12.3.2 Apply conservation of momentum to solve problems

12.3.3 For elastic collision, kinetic energy and relative speed are conserved

12.3.4 Momentum is conserved, but some kinetic energy may change

13. Medical Physics

13.1 Production and Use of Ultrasound

13.1.1 Understand that a piezo-electric crystal changes shape when a p.d. is applied and generates an e.m.f

13.1.2 Understand how ultrasound waves are generated and detected by a piezoelectric transducer

13.1.3 Understand how ultrasound reflection at boundaries provides diagnostic information about internal st

13.1.4 Define specific acoustic impedance as Z = ρc

13.1.5 Use the equation IR / I₀ = (Z₁ – Z₂)² / (Z₁ + Z₂)² for the intensity reflection coefficient

13.2 Production and Use of X-rays

13.2.1 Explain that X-rays are produced by electron bombardment of a metal target and calculate the minimum

13.2.2 Understand the use of X-rays in imaging internal body structures and the term 'contrast' in X-ray im

13.2.3 Recall and use I = I₀e^(-μx) for the attenuation of X-rays in matter

13.2.4 Understand how computed tomography (CT) produces 3D images by combining multiple 2D X-ray images

13.3 PET Scanning

13.3.1 Understand that a tracer contains radioactive nuclei and is introduced into the body for imaging

13.3.2 Understand that β+ decay is used in positron emission tomography (PET)

13.3.3 Understand that annihilation occurs when a particle interacts with its antiparticle, conserving mass

13.3.4 Explain how gamma-ray photons from annihilation are detected to create an image of tracer concentrat

14. Waves

14.1 Progressive Waves

14.1.1 Describe wave motion in ropes, springs, and ripple tanks

14.1.2 Understand and use the terms displacement, amplitude, phase difference, period, frequency, wavelengt

14.1.3 Use time-base and y-gain on a cathode-ray oscilloscope (CRO) to determine frequency and amplitude

14.1.4 Derive and use v = f λ for the wave equation

14.1.5 Recall and use v = f λ

14.1.6 Understand energy transfer by a progressive wave

14.1.7 Recall and use intensity = power / area, intensity ∝ (amplitude)²

14.2 Transverse and Longitudinal Waves

14.2.1 Compare transverse and longitudinal waves

14.2.2 Analyze and interpret graphical representations of transverse and longitudinal waves

14.3 Doppler Effect for Sound Waves

14.3.1 Understand the change in frequency when a sound source moves relative to a stationary observer

14.3.2 Use the expression f₀ = fₛ(v / (v ± vₛ)) for the observed frequency

14.4 Electromagnetic Spectrum

14.4.1 State all electromagnetic waves are transverse waves traveling at the same speed in free space

14.4.2 Recall the range of wavelengths from radio waves to γ-rays

14.4.3 Recall that wavelengths 400–700 nm are visible to the human eye

14.5 Polarisation

14.5.1 Understand that polarisation is a phenomenon associated with transverse waves

14.5.2 Recall and use Malus's law (I = I₀ cos²θ) to calculate intensity of a plane-polarised wave after pas

15. The Mole

15.1 The Mole

15.1.1 Understand that amount of substance is an SI base quantity with the unit mol

15.1.2 Use molar quantities where one mole contains Avogadro's constant of particles

16. Equation of State

16.1 Ideal Gas Law

16.1.1 Understand that a gas obeying pV ∝ T is an ideal gas

16.1.2 Recall and use the equation pV = nRT for an ideal gas

16.1.3 Recall pV = NkT, where N = number of molecules and k is Boltzmann constant

16.1.4 Use the relationship k = R / NA

17. Kinetic Theory of Gases

17.1 Kinetic Theory

17.1.1 State the basic assumptions of the kinetic theory of gases

17.1.2 Explain pressure exerted by a gas due to molecular movement

17.1.3 Derive and use pV = (3/2)NmkT for pressure of a gas

17.1.4 Compare pV = (3/2)NmkT with pV = NkT to deduce average kinetic energy of a molecule

18. Particle Physics

18.1 Atoms, Nuclei and Radiation

18.1.1 Infer the existence and small size of the nucleus from α-particle scattering

18.1.2 Describe the nuclear atom model: protons, neutrons, and electrons

18.1.3 Distinguish between nucleon number and proton number

18.1.4 Understand isotopes as forms of the same element with different neutrons

18.1.5 Use the notation AZX for nuclides representation

18.1.6 Understand conservation of nucleon number and charge in nuclear processes

18.1.7 Describe the composition, mass, and charge of α-, β-, and γ-radiations

18.1.8 Understand antiparticles and positrons as antiparticles of electrons

18.1.9 State that (electron) antineutrinos are produced during β- decay and neutrinos during β+ decay

18.1.10 Represent α- and β-decay using radioactive decay equations

18.1.11 Use unified atomic mass unit (u) for mass

18.2 Fundamental Particles

18.2.1 Understand quarks as fundamental particles with six flavours

18.2.2 Understand hadrons as baryons (three quarks) or mesons (one quark and one antiquark)

18.2.3 Describe changes in quark composition during β- and β+ decay

18.2.4 Describe protons and neutrons as composed of quarks

18.2.5 Recall and use the charge of each quark and its respective antiquark

18.2.6 Recall electrons and neutrinos as fundamental particles (leptons)

19. Internal Energy

19.1 Internal Energy

19.1.1 Understand that internal energy is the sum of kinetic and potential energies of molecules in a syste

19.1.2 Relate rise in temperature to an increase in internal energy

20. Forces, Density and Pressure

20.1 Turning Effects of Forces

20.1.1 Weight acts at a single point: centre of gravity

20.1.2 Define and apply the moment of a force

20.1.3 Understand the concept of a couple

20.1.4 Define and apply torque of a couple

20.2 Equilibrium of Forces

20.2.1 State and apply the principle of moments

20.2.2 System in equilibrium has no resultant force or torque

20.2.3 Use a vector triangle to represent coplanar forces in equilibrium

20.3 Density and Pressure

20.3.1 Define and use density

20.3.2 Define and use pressure

20.3.3 Derive equation for hydrostatic pressure ∆p = ρg∆h

20.3.4 Use the equation ∆p = ρg∆h

20.3.5 Understand upthrust as the effect of hydrostatic pressure

20.3.6 Calculate upthrust using F = ρgV (Archimedes' principle)

21. Astronomy and Cosmology

21.1 Standard Candles

21.1.1 Understand luminosity as the total power radiated by a star

21.1.2 Recall and use the inverse square law F = L / (4πd²) for radiant flux intensity

21.1.3 Understand that an object with known luminosity is a standard candle

21.1.4 Use standard candles to determine distances to galaxies

21.2 Stellar Radii

21.2.1 Recall and use Wien’s displacement law λmax ∝ 1 / T to estimate the surface temperature of a star

21.2.2 Use Stefan–Boltzmann law L = 4πσr²T⁴

21.2.3 Use Wien’s law and Stefan–Boltzmann law to estimate the radius of a star

21.3 Hubble’s Law and Big Bang Theory

21.3.1 Understand redshift and the increase in wavelength of light from distant objects

21.3.2 Use ∆λ / λ = ∆f / f = v / c for redshift

21.3.3 Explain why redshift suggests the Universe is expanding

21.3.4 Recall and use Hubble’s law v = H₀d to explain the Big Bang theory

22. First Law of Thermodynamics

22.1 First Law

22.1.1 Recall and use W = p∆V for work done when the volume of a gas changes at constant pressure

22.1.2 Understand the difference between work done by a gas and work done on a gas

22.1.3 Recall and use the first law of thermodynamics: ∆U = q + W

23. Alternating Currents

23.1 Characteristics of Alternating Currents

23.1.1 Understand and use terms period, frequency, and peak value for alternating current or voltage

23.1.2 Use x = x₀ sin(ωt) for sinusoidal alternating current or voltage

23.1.3 Recall that the mean power in a resistive load is half the maximum power for sinusoidal current

23.1.4 Distinguish between root-mean-square (r.m.s.) and peak values and recall I₀r.m.s = I₀ / √2 and V₀r.m

23.2 Rectification and Smoothing

23.2.1 Distinguish graphically between half-wave and full-wave rectification

23.2.2 Explain the use of a single diode for half-wave rectification

23.2.3 Explain the use of four diodes (bridge rectifier) for full-wave rectification

23.2.4 Analyze the effect of a capacitor in smoothing, including the effect of capacitance and load resista

24. Superposition

24.1 Stationary Waves

24.1.1 Explain and use the principle of superposition

24.1.2 Understand experiments demonstrating stationary waves using microwaves, stretched strings, and air c

24.1.3 Explain the formation of stationary waves using graphical methods and identify nodes and antinodes

24.1.4 Determine wavelength from the positions of nodes or antinodes

24.2 Diffraction

24.2.1 Explain diffraction and show understanding of related experiments

24.2.2 Understand the effect of gap width relative to wavelength in diffraction experiments

24.3 Interference

24.3.1 Understand the terms interference and coherence

24.3.2 Demonstrate two-source interference using water waves, sound, light, and microwaves

24.3.3 Understand the conditions required for two-source interference fringes

24.3.4 Recall and use λ = ax / D for double-slit interference with light

24.4 Diffraction Grating

24.4.1 Recall and use d sin θ = nλ for diffraction grating calculations

24.4.2 Describe the use of a diffraction grating to determine the wavelength of light

25. Simple Harmonic Oscillations

25.1 Simple Harmonic Motion

25.1.1 Understand and use terms displacement, amplitude, period, frequency, angular frequency, and phase di

25.1.2 Understand that simple harmonic motion occurs when acceleration is proportional to displacement from

25.1.3 Use a = -ω²x and recall x = x₀ sin(ωt) as the solution

25.1.4 Use v = v₀ cos(ωt) and v = ±ω√(x₀² - x²) for velocity

26. Energy in Simple Harmonic Motion

26.1 Energy in SHM

26.1.1 Describe the interchange between kinetic and potential energy during SHM

26.1.2 Recall and use E = ½mω²x₀² for the total energy of a system undergoing SHM

27. Quantum Physics

27.1 Energy and Momentum of a Photon

27.1.1 Understand that electromagnetic radiation has a particulate nature

27.1.2 Recall and use E = hf for the energy of a photon

27.1.3 Use the electronvolt (eV) as a unit of energy

27.1.4 Understand that a photon has momentum and use p = E / c for photon momentum

27.2 Photoelectric Effect

27.2.1 Understand that photoelectrons are emitted from a metal surface when illuminated by electromagnetic

27.2.2 Understand and use the terms threshold frequency and threshold wavelength

27.2.3 Explain photoelectric emission in terms of photon energy and work function

27.2.4 Recall and use hf = Φ + ½mv² for the maximum kinetic energy of photoelectrons

27.2.5 Explain why the maximum kinetic energy of photoelectrons is independent of intensity, while current

27.3 Wave-Particle Duality

27.3.1 Understand that the photoelectric effect shows the particulate nature of light, while diffraction sh

27.3.2 Describe and interpret electron diffraction as evidence for the wave nature of particles

27.3.3 Understand the de Broglie wavelength as the wavelength associated with a moving particle

27.3.4 Recall and use λ = h / p for de Broglie wavelength

27.4 Energy Levels in Atoms and Line Spectra

27.4.1 Understand discrete electron energy levels in atoms (e.g., atomic hydrogen)

27.4.2 Understand the appearance and formation of emission and absorption line spectra

27.4.3 Recall and use hf = E₁ – E₂ for the energy difference between electron energy levels

28. Damped and Forced Oscillations, Resonance

28.1 Damped Oscillations

28.1.1 Understand that damping occurs due to resistive forces

28.1.2 Understand the types of damping: light, critical, and heavy damping

28.2 Resonance

28.2.1 Understand that resonance occurs when an oscillating system is forced to oscillate at its natural fr

29. Work, Energy and Power

29.1 Energy Conservation

29.1.1 Understand the concept of work, and recall W = F × s

29.1.2 Apply the principle of conservation of energy

29.1.3 Understand efficiency as useful energy output / total energy input

29.1.4 Use efficiency concept to solve problems

29.1.5 Define power as work done per unit time

29.1.6 Solve problems using P = W / t

29.1.7 Derive P = Fv and solve problems

29.2 Gravitational Potential Energy and Kinetic Energy

29.2.1 Derive ∆EP = mg∆h for gravitational potential energy changes

29.2.2 Recall and use ∆EP = mg∆h for gravitational potential energy changes

29.2.3 Derive EK = ½mv² for kinetic energy

29.2.4 Recall and use EK = ½mv² for kinetic energy

30. Electric Fields

30.1 Electric Fields and Field Lines

30.1.1 Understand that an electric field is a field of force and define it as force per unit positive charg

30.1.2 Represent an electric field using field lines

30.2 Uniform Electric Fields

30.2.1 Recall and use E = ∆V / ∆d to calculate electric field strength between charged parallel plates

30.2.2 Describe the effect of a uniform electric field on the motion of charged particles

30.3 Electric Force Between Point Charges

30.3.1 Understand that, for a point outside a spherical conductor, the charge may be considered a point mas

30.3.2 Recall and use Coulomb’s law F = Q₁Q₂ / (4πε₀r²) for the force between two point charges

30.4 Electric Field of a Point Charge

30.4.1 Recall and use E = Q / (4πε₀r²) for the electric field strength due to a point charge

30.5 Electric Potential

30.5.1 Define electric potential as work done per unit positive charge in bringing a test charge from infin

30.5.2 Recall and use V = Q / (4πε₀r) for electric potential due to a point charge

30.5.3 Understand how electric potential leads to electric potential energy of two point charges and use EP

31. Physical Quantities and Units

31.1 Physical Quantities

31.1.1 Physical quantities consist of magnitude and unit

31.1.2 Estimate physical quantities from the syllabus

31.2 SI Units

31.2.1 SI base units: mass, length, time, current, temperature

31.2.2 Express derived units from SI base units

31.2.3 Check homogeneity of physical equations

31.2.4 Use prefixes (pico, nano, micro, kilo, etc.)

31.3 Errors and Uncertainties

31.3.1 Understand systematic and random errors

31.3.2 Assess uncertainty in derived quantities

31.4 uniform Electric Fields

31.4.1 Distinguish between precision and accuracy

31.5 Scalars and Vectors

31.5.1 Difference between scalar and vector quantities

31.5.2 Add and subtract coplanar vectors

31.5.3 Represent a vector as components

32. Motion in a Circle

32.1 Centripetal Acceleration

32.1.1 Understand centripetal acceleration and its role in circular motion

32.1.2 Recall and use a = rω² and a = v² / r

32.1.3 Recall and use F = mrω² and F = mv² / r

32.1.4 Understand force causing centripetal acceleration is always perpendicular to motion

32.2 Kinematics of Uniform Circular Motion

32.2.1 Recall and use ω = 2π / T and v = rω

32.2.2 Understand and use angular speed

32.2.3 Define and express angular displacement in radians

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Determine Wavelength from the Positions of Nodes or Antinodes

Introduction

Key Concepts

Stationary Waves

Nodes and Antinodes

Wavelength ($\lambda$)

Wave Speed ($v$)

Determining Wavelength from Nodes

Determining Wavelength from Antinodes

Mathematical Representation of Stationary Waves

Resonance and Standing Wave Patterns

Wave Speed on a String

Applications of Stationary Waves

Determining Frequency from Wavelength

Experimental Determination of Wavelength

Limitations and Considerations

Real-World Applications

Advanced Concepts

Mathematical Derivation of Standing Waves

Quantization of Modes in Resonating Systems

Energy Distribution in Standing Waves

Interference of Waves

Mode Shapes and Patterns

Advanced Problem Solving

Interdisciplinary Connections

Experimental Techniques

Nonlinear Effects in Standing Waves

Quantum Analogues of Standing Waves

Advanced Mathematical Models

Comparison Table

Summary and Key Takeaways

Coming Soon!

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