5.1 Fundamentals of Waves
Definition and Types of Waves
A wave is a disturbance that propagates from one point to another, transferring energy without transferring matter. Waves are broadly classified into two main types:
- Transverse Waves: Waves in which the direction of vibration of particles is perpendicular to the direction of propagation of the wave (e.g., water waves, light waves, electromagnetic waves). Features include crests and troughs.
- Longitudinal Waves: Waves in which the direction of vibration of particles is parallel to the direction of propagation of the wave (e.g., sound waves). Features include compressions and rarefactions.
Wave Terminology and Parameters
- Displacement ($y$): The distance of a vibrating particle from its equilibrium position.
- Amplitude ($A$): The maximum displacement of a particle from its equilibrium position.
- Period ($T$): The time taken to complete one full oscillation or wave cycle, measured in seconds ($\text{s}$).
- Frequency ($f$): The number of complete oscillations or wave cycles completed per second, measured in Hertz ($\text{Hz}$). $f = \frac{1}{T}$.
- Wavelength ($\lambda$): The distance between two successive points in phase (e.g., crest to crest or compression to compression), measured in metres ($\text{m}$).
- Wave Speed ($v$): The distance travelled by a wave profile per unit time.
The universal wave equation relating speed, frequency, and wavelength is:
$$v = f \lambda$$
5.2 Damping and Resonance
Damping
Damping is the reduction in amplitude of an oscillating system due to the loss of energy to the surroundings (usually converted into heat due to friction or air resistance).
- External Damping: Caused by external resistive forces like air resistance or friction.
- Internal Damping: Caused by the stretching and compression of the material of the oscillating system itself.
- In a damped system, the frequency remains constant while the amplitude decreases progressively over time.
Resonance
Resonance occurs when a system is driven by an external periodic force at a frequency equal to its natural frequency ($f_0$).
- At resonance, the oscillating system absorbs maximum energy from the driver.
- The system oscillates with its maximum amplitude.
5.3 Reflection of Waves
Law of Reflection
When a wave strikes a barrier, it bounces back into the same medium. The reflection of waves obeys the Law of Reflection:
- The angle of incidence ($i$) is equal to the angle of reflection ($r$).
- The incident wave, reflected wave, and normal all lie on the same plane.
During reflection, characteristics such as wavelength, frequency, and speed remain unchanged, but the direction of propagation changes.
5.4 Refraction of Waves
Refraction and Wavefronts
Refraction of waves is the change in direction of propagation of a wave when it passes from one medium to another due to a change in speed caused by a change in medium density or depth (in water waves).
- Deep water to shallow water: Speed ($v$) decreases, wavelength ($\lambda$) decreases, frequency ($f$) remains constant, and wave bends towards the normal.
- Shallow water to deep water: Speed ($v$) increases, wavelength ($\lambda$) increases, frequency ($f$) remains constant, and wave bends away from the normal.
$${\lambda_1}{\lambda_2} = \frac{v_1}{v_2}$$
5.5 Diffraction of Waves
Diffraction Phenomenon
Diffraction of waves is the spreading out of waves when they pass through an aperture (gap) or round an obstacle.
- Diffraction changes the direction of propagation and spreading pattern, but does not change the frequency, wavelength, or speed of the wave.
- Key Factors affecting Diffraction:
- Smaller aperture size relative to wavelength ($\lambda$) results in more pronounced diffraction (wider spreading).
- Obstacles or gaps much larger than the wavelength result in little to no noticeable diffraction (producing sharp shadows).
5.6 Interference of Waves
Principle of Superposition
The Principle of Superposition states that when two or more waves overlap at a point, the resultant displacement is the vector sum of the individual displacements of the overlapping waves.
Types of Interference
- Constructive Interference: Occurs when two crests or two troughs meet. Displacements reinforce each other, producing a wave of maximum amplitude (antinode).
- Destructive Interference: Occurs when a crest meets a trough. Displacements cancel each other out, producing a wave of zero amplitude (node).
Young's Double-Slit / Interference Formula
For coherent water, sound, or light waves, the wavelength $\lambda$ can be determined using:
$$\lambda = \frac{ax}{D}$$
- $\lambda$: Wavelength of the waves ($\text{m}$)
- $a$: Distance between the two coherent sources or slits ($\text{m}$)
- $x$: Distance between consecutive bright/antinodal lines or dark/nodal lines ($\text{m}$)
- $D$: Perpendicular distance from the sources to the observation screen ($\text{m}$)
5.7 Electromagnetic Waves
The Electromagnetic Spectrum
Electromagnetic waves consist of oscillating electric and magnetic fields perpendicular to each other and to the direction of wave travel. They are transverse waves that can travel through a vacuum at the speed of light ($c = 3.0 \times 10^8 \text{ m s}^{-1}$).
Arranged in order of increasing frequency and decreasing wavelength:
- Radio Waves: Lowest frequency, longest wavelength. Used for broadcasting and telecommunications.
- Microwaves: Used in satellite communications and cooking.
- Infrared Rays: Thermal radiation used in remote controls and night vision.
- Visible Light: Detected by the human eye, enabling sight.
- Ultraviolet Rays: Causes fluorescence and sunburn, used in sterilization.
- X-rays: High penetration power, used in medical imaging and security screening.
- Gamma Rays: Highest frequency, shortest wavelength, highest energy. Produced by radioactive decay, used in cancer treatment.