Waves investigates how energy propagates through space and matter without permanent displacement of the medium. Students explore wave characteristics including wavelength, frequency, speed, and amplitude while distinguishing between transverse and longitudinal waves. This chapter covers wave phenomena such as superposition, interference, standing waves, and resonance, connecting theoretical principles to practical applications in music, communication technology, and medical imaging systems.
Chapter 15: Waves
1. Wave Motion
A wave is a disturbance that travels through a medium or space, transferring energy without transferring matter. Wave motion is one of the most important phenomena in physics, describing everything from sound and light to quantum particles.
Types of waves based on duration:
- Pulse: Single disturbance traveling through medium
- Wave train: Continuous succession of pulses
- Periodic wave: Wave with pattern repeating at regular intervals
The wave function y(x,t) describes the displacement of the medium at position x and time t. Key terms to understand wave motion include:
- Wavelength (λ): Distance between two consecutive identical points
- Period (T): Time taken for one complete oscillation
- Frequency (f): Number of oscillations per second (f = 1/T)
- Wave speed (v): Speed at which wave disturbance travels (v = λf)
- Amplitude (A): Maximum displacement from equilibrium position
2. Types of Waves
Waves can be classified in several ways:
Based on medium requirement:
- Mechanical waves: Require material medium (e.g., sound, water waves)
- Electromagnetic waves: Don’t require medium (e.g., light, radio waves)
- Matter waves: Associated with moving particles (de Broglie waves)
Based on direction of particle motion relative to wave propagation:
- Transverse waves: Particles move perpendicular to wave direction (e.g., light, vibrating string)
- Longitudinal waves: Particles move parallel to wave direction (e.g., sound, compression springs)
- Surface waves: Combination of transverse and longitudinal motion (e.g., water waves)
Based on dimensions:
- One-dimensional: Waves on strings
- Two-dimensional: Ripples on water surface
- Three-dimensional: Sound waves in air
3. Mathematical Representation of Waves
For a sinusoidal wave traveling in the positive x-direction:
- y(x,t) = A sin(kx – ωt + φ)
- Where k = 2π/λ is wave number and ω = 2π/T is angular frequency
For a wave traveling in the negative x-direction:
- y(x,t) = A sin(kx + ωt + φ)
The phase velocity of the wave is v = ω/k = λf.
The phase of the wave (kx – ωt + φ) determines the state of oscillation at position x and time t. Points with the same phase are in the same state of oscillation.
4. Wave Speed
The speed of a wave depends on the properties of the medium. For different types of waves:
Waves on a stretched string:
- v = √(T/μ)
- Where T is tension and μ is linear mass density
Sound waves in gases:
- v = √(γP/ρ) = √(γRT/M)
- Where γ is ratio of specific heats, P is pressure, ρ is density
- For air at 20°C, v ≈ 343 m/s
Sound waves in solids:
- v = √(Y/ρ)
- Where Y is Young’s modulus and ρ is density
Electromagnetic waves in vacuum:
- v = c = 3 × 10⁸ m/s (speed of light)
Water waves:
- For deep water: v = √(gλ/2π)
- For shallow water: v = √(gd) where d is depth
5. Principle of Superposition
The principle of superposition states that when two or more waves overlap, the resultant displacement at any point is the vector sum of the displacements due to each individual wave.
Mathematically, if y₁(x,t) and y₂(x,t) are two waves, then:
- y(x,t) = y₁(x,t) + y₂(x,t)
This principle applies to all linear wave systems and is the basis for understanding interference, standing waves, and beats.
Constructive interference occurs when waves are in phase, resulting in enhanced amplitude. Destructive interference occurs when waves are out of phase, resulting in reduced amplitude.
6. Reflection and Refraction
When a wave encounters a boundary between two media, part of it is reflected and part is transmitted (refracted).
Reflection follows these laws:
- Angle of incidence equals angle of reflection
- For a fixed end reflection (higher density medium), there is a phase change of π
- For a free end reflection (lower density medium), there is no phase change
Refraction follows Snell’s Law:
- n₁sin(θ₁) = n₂sin(θ₂)
- Where n₁ and n₂ are refractive indices of the media
The change in wavelength during refraction:
- λ₂/λ₁ = v₂/v₁ = n₁/n₂
- Frequency remains unchanged during refraction
These phenomena are observed in all types of waves, from mechanical to electromagnetic.
7. Standing Waves
Standing waves form when two identical waves traveling in opposite directions superimpose. This commonly occurs in fixed-length systems like strings fixed at both ends, air columns in musical instruments, and electromagnetic waves in cavities.
The standing wave equation:
- y(x,t) = 2A sin(kx) cos(ωt)
- This represents a wave where each point oscillates with an amplitude that depends on position
Key features of standing waves:
- Nodes: Points of zero amplitude (sin(kx) = 0
Complete Chapter-wise Hsslive Plus One Physics Notes
Our Hsslive Plus One physics notes cover all chapters with key focus areas to help you organize your study effectively:
- 1. Physical World Notes
- 2. Units and Measurements Notes
- 3. Motion in a Straight Line Notes
- 4. Motion in a Plane Notes
- 5. Laws of Motion Notes
- 6. Work, Energy and Power Notes
- 7. System of Particles and Rotational Motion Notes
- 8. Gravitation Notes
- 9. Mechanical Properties of Solids Notes
- 10. Mechanical Properties of Fluids Notes
- 11. Thermal Properties of Matter Notes
- 12. Thermodynamics Notes
- 13. Kinetic Theory Notes
- 14. Oscillations Notes
- 15. Waves Notes