Physics (9702) | Waves

👋 Welcome to the World of Waves! (Physics 9702 - Chapter 7 & 8)

Hello future Physicists! This chapter is all about how energy moves through space and matter—from the sound waves carrying music to your ears, to the radio waves bringing you data. Waves are fundamental to almost every area of physics, so mastering these concepts is crucial. Don't worry if terminology seems new; we’ll break down every complex idea using simple analogies!

Key Concepts You Must Master:

• The properties and equations of progressive waves.
• Distinguishing between transverse and longitudinal waves (including polarisation).
• The Doppler effect for sound.
• The principles of superposition, interference, and diffraction.

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7.1 Progressive Waves: Carrying Energy

A progressive wave is a transfer of energy through a medium (or free space) without the net transfer of matter. Think of a stadium wave: the energy (the 'wave') moves around the stadium, but the people (the 'matter') just move up and down in place.

Wave Terminology (The Essentials)

Understanding these terms is half the battle won:

• Displacement ($x$): The distance moved by a point on the wave from its equilibrium (rest) position. It can be positive or negative.
• Amplitude ($A$): The maximum displacement from the equilibrium position. This is always positive and relates directly to the energy carried.
• Wavelength ($\lambda$): The shortest distance between two points on a wave that are oscillating in phase (e.g., crest to crest, or trough to trough). Unit: metres (m).
• Period ($T$): The time taken for one complete oscillation (or one complete wave to pass a point). Unit: seconds (s).
• Frequency ($f$): The number of complete oscillations per unit time. Unit: hertz (Hz). It is the reciprocal of the period: \(f = \frac{1}{T}\).
• Wave Speed ($v$): The distance travelled by the wave energy per unit time. Unit: m/s.

Phase Difference ($\phi$)

Phase difference describes how far one point on a wave is through a cycle compared to another point.

It is measured in degrees or radians.

• One full cycle (one wavelength $\lambda$) corresponds to $360^{\circ}$ or $2\pi$ radians.
• Points moving together at the same time are in phase (phase difference = 0 or $2\pi$).
• Points moving in exactly opposite directions (e.g., crest and trough) are in anti-phase (phase difference = $180^{\circ}$ or $\pi$ radians).

Memory Aid: If two points are separated by a distance $d$, the phase difference $\phi$ is calculated by: \[\phi = \frac{d}{\lambda} \times 2\pi \quad (\text{in radians})\]

The Wave Equation

Since speed is distance over time, one full wavelength ($\lambda$) travels in one period ($T$). \[v = \frac{\text{distance}}{\text{time}} = \frac{\lambda}{T}\] Since \(f = \frac{1}{T}\), we derive the essential wave equation: \[v = f\lambda\]

Energy and Intensity (7.1.6 & 7.1.7)

A progressive wave transfers energy. The amount of energy transferred is related to its amplitude and frequency.

Intensity ($I$) is defined as the power transmitted per unit area, perpendicular to the direction of energy transfer:

\[I = \frac{\text{Power}}{\text{Area}}\]

For any progressive wave, the intensity is proportional to the square of its amplitude:

\[I \propto A^2\]

Why is this important? If you double the amplitude of a sound wave, the loudness (intensity) increases by a factor of four ($2^2=4$).

Using the Cathode-Ray Oscilloscope (CRO)

The CRO is used to measure the properties of electrical signals (which often represent sound waves or alternating voltages).

• Y-Gain (Vertical Axis): Controls the scale of displacement (or voltage). Used to determine the amplitude of the wave.
• Time-Base (Horizontal Axis): Controls the scale of time. Used to determine the period ($T$).

Step-by-step to find Frequency:

1. Measure the time taken for one complete oscillation (Period, $T$) from the CRO screen, using the time-base setting.
2. Calculate frequency using $f = 1/T$.

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7.2 Transverse and Longitudinal Waves

Waves are classified based on the direction of oscillation of the particles (or field) relative to the direction of energy transfer.

1. Transverse Waves (7.2.1)

In a transverse wave, the oscillation of the particles (or fields) is perpendicular ($90^{\circ}$) to the direction of wave propagation (energy transfer).

• Examples: Light (all Electromagnetic waves), ripples on water, waves on a stretched string.
• They consist of alternating peaks (crests) and dips (troughs).

2. Longitudinal Waves (7.2.1)

In a longitudinal wave, the oscillation of the particles is parallel (along the same line) to the direction of wave propagation.

• Example: Sound waves (in air or liquids), compression waves in a spring.
• They consist of regions where particles are crowded together (compressions) and regions where they are spread out (rarefactions).

Graphical Representations (7.2.2)

When analyzing wave graphs:

For Transverse waves: The displacement is clearly perpendicular to the direction of travel, making the sine or cosine shape easily visible.

For Longitudinal waves: We plot the displacement of the medium particles from their equilibrium position.

• A compression occurs where the displacement is zero, but the gradient is steepest (particles are moving towards each other).
• A rarefaction occurs where the displacement is zero, but the gradient is steepest in the opposite direction (particles are moving away from each other).

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7.3 Doppler Effect for Sound Waves

The Doppler effect is the change in observed frequency ($f_o$) of a wave when the source of the waves moves relative to the observer (7.3.1).

Real-world example: Think of an ambulance siren. As it approaches you, the pitch (frequency) sounds higher; as it moves away, the pitch sounds lower.

When the source moves, the distance between successive wavefronts changes, causing a shift in wavelength and therefore a shift in frequency.

Doppler Equation for Sound

For a source of sound waves moving relative to a stationary observer (7.3.2):

\[f_o = f_s \frac{v}{v \pm v_s}\]

Where:

• $f_o$: Observed frequency.
• $f_s$: Source frequency.
• $v$: Speed of sound in the medium.
• $v_s$: Speed of the source.

The crucial sign convention:

• Use \(v - v_s\) (minus in the denominator) when the source is approaching the observer. This makes the observed frequency ($f_o$) higher.
• Use \(v + v_s\) (plus in the denominator) when the source is receding (moving away) from the observer. This makes the observed frequency ($f_o$) lower.

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7.4 Electromagnetic Spectrum

The electromagnetic (EM) spectrum consists of waves produced by oscillating electric and magnetic fields.

Key Facts (7.4.1)

• All EM waves are transverse waves.
• All EM waves travel at the same speed ($c$) in a vacuum (free space).
• The speed of light $c \approx 3.00 \times 10^8 \text{ m/s}$.

Spectrum Order (7.4.2)

It's vital to recall the order of the principal regions, generally listed from longest wavelength / lowest frequency to shortest wavelength / highest frequency:

1. Radio waves
2. Microwaves
3. Infrared (IR)
4. Visible light (400 nm to 700 nm) (7.4.3)
5. Ultraviolet (UV)
6. X-rays
7. Gamma ($\gamma$)-rays

Memory Aid: Real Men In Violet Underwear X-ray Girls.

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7.5 Polarisation

Polarisation is a key phenomenon that proves a wave is transverse (7.5.1).

An unpolarised transverse wave (like regular light) vibrates in many planes perpendicular to the direction of travel. A polarising filter (or polariser) acts like a fence, only allowing oscillations parallel to the transmission axis to pass through. The resulting light is plane-polarised.

Malus's Law (7.5.2)

If plane-polarised light of intensity $I_0$ passes through a second polarising filter (called the analyser), the intensity ($I$) transmitted depends on the angle ($\theta$) between the polarisation plane of the light and the axis of the filter:

\[I = I_0 \cos^2 \theta\]

Where:

• $I_0$: Intensity of the incident plane-polarised wave.
• $\theta$: Angle between the plane of polarisation and the axis of the analyser.

Key point: Maximum intensity ($I_0$) is transmitted when $\theta = 0^{\circ}$ ($\cos 0^{\circ} = 1$). Zero intensity (darkness) is transmitted when $\theta = 90^{\circ}$ ($\cos 90^{\circ} = 0$).

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8.1 Stationary Waves (Standing Waves)

When two progressive waves of the same frequency and amplitude travel in opposite directions and overlap, they superimpose to form a stationary wave (8.1.3).

Principle of Superposition (8.1.1)

When two or more waves meet at a point, the resultant displacement at that point is the algebraic sum of the displacements of the individual waves.

If Wave 1 has displacement $x_1$ and Wave 2 has displacement $x_2$, the resultant displacement $x$ is: \[x = x_1 + x_2\]

Formation and Features (8.1.3)

A stationary wave does not transfer energy. The energy is stored within the system.

• Nodes (N): Points where the displacement is always zero. These points are in permanent destructive interference.
• Antinodes (A): Points where the amplitude is maximum. These points undergo maximum oscillation due to constructive interference.

Wavelength Determination (8.1.4):

The distance between adjacent Nodes (N to N) is \(\frac{1}{2}\lambda\).
The distance between an adjacent Node and Antinode (N to A) is \(\frac{1}{4}\lambda\).

Demonstrating Stationary Waves (8.1.2)

Stationary waves can be demonstrated using:

• Stretched strings: (e.g., using a vibration generator). Different vibration modes (harmonics) can be generated.
• Air columns: (e.g., resonance tube experiment). Sound waves reflect off the closed/open ends.
• Microwaves: Using a metal plate as a reflector to create the second progressive wave.

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8.2 Diffraction

Diffraction is the phenomenon where waves spread out as they pass through an aperture (a gap) or around the edge of an obstacle (8.2.1).

Qualitative Effect (8.2.2)

The amount of diffraction depends on the relationship between the wavelength ($\lambda$) of the wave and the size of the aperture ($a$).

• Maximum Diffraction: Occurs when the gap width ($a$) is approximately equal to the wavelength ($\lambda$) ($a \approx \lambda$). The waves spread out in semicircles.
• Less Diffraction: Occurs when the gap width ($a$) is much larger than the wavelength ($\lambda$). The waves mostly continue straight, with only slight spreading at the edges.

Example: If you listen to someone talking around a corner, the sound waves (long wavelength) diffract easily. If you look around the corner, you can't see them easily because light waves (very short wavelength) diffract very little.

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8.3 Interference

Interference occurs when two or more waves overlap, resulting in a change in the resultant amplitude at points in space (8.3.1).

Coherence (8.3.1 & 8.3.3)

To observe a clear, stable (non-moving) interference pattern (fringes), the sources must be coherent. This requires two conditions:

1. The sources must have the same frequency ($f$) (and thus wavelength $\lambda$).
2. The sources must maintain a constant phase difference ($\phi$).

Note: Light sources must usually be derived from a single source (like splitting a laser beam) to ensure coherence.

Types of Interference

Interference patterns show alternating regions of constructive and destructive interference:

• Constructive Interference: Occurs when waves meet in phase ($\phi = 0, 2\pi, 4\pi, ...$). Crest meets crest, or trough meets trough. Result: Maximum intensity (bright fringe / loud sound / large ripple).
• Destructive Interference: Occurs when waves meet in anti-phase ($\phi = \pi, 3\pi, 5\pi, ...$). Crest meets trough. Result: Minimum or zero intensity (dark fringe / quiet sound / no ripple).

Double-Slit Interference using Light (Young's Fringes) (8.3.4)

When monochromatic light passes through two narrow slits (separation $a$) and the interference pattern is observed on a screen a distance $D$ away, the spacing between the fringes ($x$) is given by the formula:

\[\lambda = \frac{ax}{D}\]

Where:

• $\lambda$: Wavelength of light (m).
• $a$: Separation of the slits (m).
• $x$: Fringe separation (distance between adjacent bright or dark lines) (m).
• $D$: Distance from the slits to the screen (m).

Tip: To find the most accurate $x$, measure the distance across many fringes (e.g., 10 fringes) and divide the total distance by the number of fringe spacings.

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8.4 The Diffraction Grating

A diffraction grating is a plate containing a large number of equally spaced parallel lines (slits) very close together. Gratings produce much sharper, brighter interference patterns than double slits.

Grating Equation (8.4.1)

For monochromatic light incident normally on a grating, the positions of the bright maxima (lines) are given by the equation:

\[d \sin \theta = n\lambda\]

Where:

• $d$: The grating spacing (distance between adjacent slits) (m).
• $\theta$: The angle of diffraction for the maximum (measured from the normal).
• $n$: The order number ($n=0$ is the central maximum, $n=1$ is the first order, etc.).
• $\lambda$: Wavelength of the light (m).

Calculating Grating Spacing ($d$):

If a grating has $N$ lines per metre, the spacing $d$ is: \[d = \frac{1}{N}\]

Using the Grating to Determine Wavelength (8.4.2)

By shining a known light source through the grating, measuring the angle $\theta$ for a specific order $n$, and knowing the grating spacing $d$, you can calculate the wavelength $\lambda$ using the formula $d \sin \theta = n\lambda$. This method is highly accurate for measuring wavelengths.

You have successfully navigated the challenging topic of Waves and Superposition! Keep practicing those equations and visualizations.