TOPIC 1: WAVES

Introduction to Waves

The Concept of Wave

A wave is a disturbance that travels through a medium from one location to another.

Consider a slinky wave as an example. When the slinky is stretched and held at rest, it assumes a natural position called the equilibrium or rest position, where the coils are equally spaced.

To introduce a wave, the first coil is displaced from its equilibrium position—upwards, downwards, forwards, or backwards—and then returned to its original position. This movement creates a disturbance that travels through the slinky.

A single back-and-forth vibration of the first coil produces a slinky pulse, which is a pulse: a single disturbance moving through a medium.

If the first coil is continuously and periodically vibrated, a repeating disturbance called a wave moves through the medium.

A medium is the substance or material that carries the wave. For example, the slinky coils carry the slinky wave, ocean water carries water waves, and air carries sound waves.

The Terms Wavelength, Frequency and Velocity of a Wave

Consider the transverse wave below:

1. A transverse wave has particles displaced perpendicular to the direction of energy transport.
2. The crest is the point of maximum positive displacement from rest.
3. The trough is the point of maximum negative displacement from rest.
4. The amplitude is the maximum displacement from rest to crest or trough.
5. The wavelength is the length of one complete wave cycle, measured from crest to crest or trough to trough.
6. A longitudinal wave has particles displaced parallel to the direction of energy transport, such as a slinky stretched horizontally and vibrated back-and-forth.
7. A compression is a region of maximum density in a longitudinal wave.
8. A rarefaction is a region of minimum density in a longitudinal wave.
9. The frequency (f) is how often particles vibrate as the wave passes, measured in Hertz (Hz), where 1 Hz = 1 cycle/second.
10. The period (T) is the time for one complete vibrational cycle, measured in seconds.
11. The speed of a wave is the distance traveled per unit time, measured in meters per second (m/s).

Wave Equation

The wave equation relates speed, wavelength, and frequency.

Snapshots of a wave in a rope show that in one period, the wave travels one wavelength. Using speed = distance/time, speed = wavelength/period.

Since period T = 1/frequency (f), the wave equation is:

Speed = Wavelength × Frequency, or v = f × λ.

Types of Waves

Waves can be classified by the medium of propagation:

1. Mechanical Waves
2. Electromagnetic Waves

Mechanical Waves

Mechanical waves depend on the elastic properties of the medium. They include transverse, longitudinal, and surface waves.

Transverse waves have particle motion perpendicular to wave direction; longitudinal waves have particle motion parallel to wave direction; surface waves combine both.

Mechanical waves require a medium to travel. Examples include sound waves, water waves, ocean waves, earthquake waves, and seismic waves.

Electromagnetic Waves

Electromagnetic waves do not require a medium and can travel in a vacuum. Examples include light and radio waves.

• Electromagnetic waves travel in vacuum; mechanical waves do not.
• Mechanical waves need a medium like water or air.
• Electromagnetic waves are disturbances; mechanical waves are periodic disturbances.

Behaviour of Waves

Reflection, Refraction, Diffraction and Interference of Waves

Waves exhibit characteristic behaviors:

1. Reflection
2. Refraction
3. Diffraction
4. Interference

These properties define wave behavior.

Reflection

Reflection is the change in direction of a wavefront at an interface, causing it to return into the original medium. Examples include light, sound, and water wave reflections.

The law of reflection

For smooth surfaces (specular reflection):

1. The incident ray, reflected ray, and normal lie in the same plane.
2. The angle of incidence equals the angle of reflection.
3. The reflected and incident rays are on opposite sides of the normal.

Characteristics of Reflection of waves:

1. Obeys the Law of Reflection.
2. Wavelength (λ) of reflected wave equals that of incident wave.
3. Frequency (f) of reflected wave equals that of incident wave.
4. Speed (v) of reflected wave equals that of incident wave.

Types of reflection:

1. Specular: smooth surfaces reflect light at equal angles.
2. Diffused: rough surfaces scatter light in all directions.
3. Spread: surfaces with mixed textures (e.g., varnished paper).

Ripple Tank

A ripple tank is a shallow glass tank used to demonstrate wave properties such as reflection, refraction, interference, and diffraction. It is usually illuminated from above or below.

Ripples are generated by a motor-driven piece of wood touching the water surface.

Refraction

Refraction is the change in wave direction due to a change in transmission medium.

Refraction changes phase velocity but not frequency. It occurs when waves pass between media at angles other than 0° to the normal.

Commonly observed with light, but also with sound and water waves.

Snell’s law

Snell’s law states that for a wave of single frequency passing between two media:

“The ratio of the sines of the angle of incidence θ₁ and angle of refraction θ₂ equals the ratio of phase velocities (v₁ / v₂) or the inverse ratio of refractive indices (n₂ / n₁).”

The refractive index (n) describes how light propagates through a medium and determines bending of light, reflection, critical angle, and Brewster’s angle.

Snell’s law: n₁ sin θ₁ = n₂ sin θ₂.

Interference

Interference occurs when two waves superpose to form a resultant wave with greater or lower amplitude.

It usually involves coherent waves from the same or similar sources.

Interference occurs with light, radio, acoustic, water, and matter waves.

Constructive Interference

Occurs when two waves have displacements in the same direction, resulting in a larger amplitude.

Example: two downward sine pulses each with displacement -1 unit combine to form a pulse with displacement -2 units.

Destructive Interference

Occurs when two waves have displacements in opposite directions, reducing or canceling amplitude.

Example: a +1 unit sine pulse meets a -1 unit sine pulse.

Diffraction

Diffraction is the bending of waves as they pass through an opening or around obstacles comparable in size to their wavelength.

Described by the Huygens–Fresnel principle and superposition of waves.

Diffraction occurs with sound, water, and electromagnetic waves.

Wave propagation can be modeled as secondary spherical waves from points on a wavefront.

Diffraction effects depend on slit width relative to wavelength.

Examples: water waves bending around boats, sound waves around corners, light diffraction by small particles.

Applications of Reflection, Refraction, Diffraction and Interference in Daily Life

Reflection

1. Using ultrasound reflection to measure distances (e.g., seabed depth).
2. Sonar detects underwater objects by echo timing.
3. Mirror design relies on light reflection.
4. Detecting cracks in metals.
5. Determining frequency of alternating currents.

Refraction

1. Lenses form images by refraction.
2. Prisms create color spectra.
3. Mirages and optical illusions involve refraction.

Interference

1. Holography uses interference patterns to create 3D images.
2. Noise-cancelling earphones use destructive interference.
3. Concert halls are designed to minimize destructive interference.

Diffraction

1. Diffraction gratings analyze atomic and molecular structures.
2. X-ray diffraction determines crystal structures.
3. Holography uses diffraction of laser beams to produce 3D images.

The Behaviour of Waves

Waves exhibit reflection, refraction, diffraction, and interference, which define their behavior.

Propagation of Waves

The Propagation of Mechanical Waves

Mechanical waves propagate in three ways:

• Transverse waves: vibrations perpendicular to wave direction (e.g., shaking a slinky up and down).

Transverse wave

• Longitudinal waves: vibrations parallel to wave direction (e.g., pushing a slinky lengthwise). These have compressions and rarefactions. Sound waves are longitudinal.

Longitudinal wave

• Surface waves: travel along boundaries between materials, combining transverse and longitudinal motions (e.g., water surface waves).

Surface wave

Earthquakes produce a mixture of these waves: primary waves (longitudinal), secondary waves (transverse), and surface waves.

The Propagation of Electromagnetic Waves

Electromagnetic waves are transverse and travel at the speed of light (3×10⁸ m/s). They do not require a medium and are arranged in the electromagnetic spectrum by frequency or wavelength.

The Relationship between Frequency, Speed and Wavelength of a Wave

Frequency (f) is inversely proportional to wavelength (λ) when speed (v) is constant: f = v/λ.

Speed is the product of frequency and wavelength: v = f × λ.

Period (T) is the reciprocal of frequency.

Mathematically: v = f × λ

• v = speed of wave
• λ = wavelength
• f = frequency

The Refractive Index of a Medium

Refractive index (n) is the ratio of wave velocity in air (V_a) to velocity in medium (V_m): n = V_a/V_m.

It can also be expressed as the ratio of sine of incidence angle to sine of refraction angle.

Refractive index measures how much light slows in a material compared to vacuum.

Higher refractive index means slower light speed in the material.

Example 1

Light of frequency 4.6 × 10¹⁴ Hz travels at 1.24 × 10⁸ m/s in diamond. Calculate the refractive index.

Solution

V_diamond = 1.24 × 10⁸ m/s

c = 3.0 × 10⁸ m/s

Refractive index of diamond = 2.42

Sound Waves

Source of Sound Wave

Sound is oscillation in pressure and particle displacement propagated in a medium with internal forces.

Sources include:

• Vibrating solids
• Rapid expansion or compression (explosions)
• Vortex shedding from airflow around obstacles (e.g., whistles, flutes, singing power lines)

The Concept of Audibility

Audibility range is the frequency range audible to humans or animals.

Human range: approximately 20 Hz to 20,000 Hz, varying with age and individual.

Some animals (dolphins, bats) hear up to 100 kHz.

The Perception of Hearing

Hearing thresholds are measured by audiograms, showing minimum sound levels at various frequencies.

Tests involve presenting tones and recording the lowest audible intensity.

The Human Ear

The ear detects sound and aids balance. It consists of three parts:

The Outer Ear

Includes the pinna (auricle) made of cartilage, ear canal with hairs and wax, ending at the eardrum (tympanic membrane).

The Middle Ear (Tympanum)

The eardrum is a thin, slightly conical membrane separating outer and middle ear.

The Inner Ear (Labyrinth)

The inner ear converts sound into electrical impulses sent to the brain and controls balance.

The cochlea contains hair cells that transduce sound vibrations into nerve signals.

Hair cells are arranged in inner and outer rows, with stereocilia projecting into endolymph fluid.

The organ of Corti sits on the basilar membrane and contains supporting cells and hair cells.

Nerve fibers carry signals to the brainstem and auditory cortex in the temporal lobe.

The Concept of Echo and Reverberation

Echo

An echo is a reflected sound wave arriving after a delay.

Occurs in large empty spaces, caves, or mountains.

Application of Echo:

Measuring distance

Using speed of sound and echo time, distance can be calculated.

Sonar devices emit sound and measure echo return time to find underwater objects.

Example: If speed of sound in water is 1500 m/s and echo returns in 0.02 s, distance = 1500 × 0.02 = 30 m (round trip), so object is 15 m away.

Sonar uses sound waves; radar uses electromagnetic waves.

Velocity

The Doppler Effect causes frequency changes when waves reflect off moving objects.

Frequency increases if object approaches, decreases if it recedes.

Used to measure object velocity with sonar or radar.

Bats can find moths

Bats emit sounds and listen for echoes to locate prey.

Reverberation

Reverberation is the collection of reflected sounds in an enclosed space, aiding sound intensity but causing muddiness if excessive.

Reverberation delay is less than 0.1 second, so reflected sound blends with original.

Question Time 1

Echo differs from reverberation by distance and delay time; echo involves longer distances and delays, reverberation involves shorter ones and multiple reflections.

The Speed of Sound in Air

Sound travels in gases, liquids, and solids but not in vacuum.

Speed in air is about 340 m/s and increases with temperature.

Sound travels fastest in solids, slower in liquids, and slowest in gases.

Musical Sound

The Concept of a Musical Sound

Music is organized sound with patterns, using specific frequencies called musical scales.

Noise is random sound without structure.

Factors Affecting Loudness, Pitch and Quality of Musical Sound

Musical sounds have properties:

• Loudness: Perceived intensity, related to amplitude and number of auditory nerves activated.
• Pitch: Perceived frequency of sound; subjective sensation mapped from frequency.
• Timbre: Tone quality distinguishing sounds with same pitch and loudness.

Different Musical Instruments

Musical instruments are categorized as:

1. Wind Instruments: Sound produced by blowing air; pitch depends on air column length. Examples: flute, clarinet, saxophone.
2. Percussion Instruments: Sound produced by striking. Tuned percussion produce definite pitches (xylophone), indefinite pitch percussion produce noise (cymbals).
3. String Instruments: Sound produced by vibrating strings; pitch depends on string length, thickness, and tension. Examples: guitar, violin, harp.

The Terms Stationary Wave, Nodes and Antinodes

Stationary wave

A stationary wave has points with constant amplitude due to interference of waves traveling in opposite directions.

Standing waves do not propagate energy but have nodes (minimum amplitude) and antinodes (maximum amplitude).

Nodes

Points of minimum amplitude.

Antinodes

Points of maximum amplitude.

The Frequency of a Musical Note

Frequency depends on vibrating string length, tension, and linear mass density (mass per unit length).

The Difference between the Fundamental Note and Overtones

Fundamental Note

The lowest resonant frequency of a vibrating object; harmonics are integer multiples of this frequency.

Overtone

Any frequency higher than the fundamental. Harmonics are overtones that are integer multiples of the fundamental.

The first overtone is the second harmonic. The terms overtone, harmonic, and partial have specific meanings and should not be confused.

The Concept of Resonance as Applied to Sound

Resonance occurs when a system oscillates with greater amplitude at specific frequencies called resonant frequencies.

Small periodic forces can produce large oscillations due to energy storage and transfer between kinetic and potential forms.

Damping reduces amplitude over time.

Resonance in Closed Ended Pipes

Closed pipes have one end closed and one end open (e.g., some organ pipes, flutes).

• Closed end is a node (no air movement).
• Open end is an antinode (maximum air movement).
• Fundamental frequency corresponds to a quarter wavelength fitting in the pipe: L = ¼ λ.

Using v = fλ, frequency formulas can be derived.

Higher harmonics in closed pipes are odd multiples (3rd, 5th, etc.).

3rd harmonic: L = ¾ λ, frequency formula shown below.

5th harmonic: L = 5/4 λ, frequency formula f = 5v/4L.

Open End Pipes

Open pipes have both ends open (e.g., trumpet).

Fundamental

• Antinodes at both ends.
• Smallest wave fits half a wavelength: L = ½ λ.

f = v/2L

2nd Harmonic

• Open pipes have even and odd harmonics.
• 2nd harmonic fits one full wavelength: L = λ.
• Frequency formula: f = 2v/2L.

3rd harmonic formula: L = 3/2 λ, f = 3v/2L.

A Simple Musical Instrument

Musical instruments produce sound based on pitch, frequency, intensity, loudness, and quality.

Activity 2

Construct a simple musical instrument.

Electromagnetic Spectrum

The Concept of Electromagnetic Spectrum

The electromagnetic spectrum includes all frequencies of electromagnetic radiation, from radio waves to gamma rays.

It covers wavelengths from thousands of kilometers to fractions of atomic size.

Properties of electromagnetic spectrum

1. Continuous spectrum with no gaps; radiation types gradually change.
2. Some wavelength ranges overlap due to source-based naming (e.g., x-rays and gamma rays).

The Main Bands of the Electromagnetic Spectrum

Radio waves

Longest wavelengths, frequencies from 3 kHz to 300 GHz.

Produced naturally by lightning and astronomical objects; artificially by transmitters.

• Sources: alternating currents in antennas, oscillators, celestial bodies.

Microwaves

Wavelengths between 10^-4 m and 0.1 m.

• Produced by oscillating charges in antennas and magnetrons.

Infrared waves

Frequencies between 10¹¹ and 10¹⁴ Hz; lie between microwaves and visible light; have heating effects.

Sources: thermal vibrations of atoms and molecules; all hot bodies emit infrared.

Visible light

Narrow frequency range detectable by human eyes.

Detection of Infra-red, Visible and Ultra-violet Rays

Infrared waves

Invisible to humans but sensed as heat; detected by black bulb thermometers, photographic films, thermistors.

Visible light

Seen by humans due to emission or reflection.

Ultraviolet light

Detected by photographic films and fluorescent materials.

Application of Electromagnetic Waves in Daily Life

The Application of Microwaves, Radio Waves, Gamma Rays and X-rays

Radio waves

1. Used in communication, broadcasting, radar, satellites, and networks.
2. Astronomy uses radio telescopes to study celestial bodies.

Microwaves

1. Cooking
2. Radar systems
3. Long-distance communication

Gamma rays

1. Medical applications similar to x-rays
2. Agriculture (e.g., pest control)

X-rays

1. X-ray photography
2. Cancer diagnosis and treatment

The Importance of Electromagnetic Waves in Agriculture and Climate

Electromagnetic radiation supports photosynthesis, oxygen production, and pest control.

Gamma rays sterilize pests and produce disease-resistant plants.

Ultraviolet light purifies water and air.