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Lesson 3.2 • Competency 3.2Interactive Simulation
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Wave Motion, Superposition & Stationary Waves

Progressive mechanical waves, wave speed formula v = fλ, phase differences, principle of superposition, standing waves formation with nodes and antinodes, and acoustic beats.

Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.

Learning Objectives

  • Distinguish between progressive energy-transferring waves and standing stationary waves.
  • Apply the principle of superposition to demonstrate standing wave formation and beat frequencies.

Core Theory & Concepts

When two progressive waves of equal frequency and amplitude travel in opposite directions through a medium, their linear superposition produces a stationary (standing) wave. Energy does not travel along the medium; instead, energy oscillates between kinetic and potential forms confined within stationary nodes.

Principle of Superposition

When two or more waves overlap, the resultant displacement at any point is the vector sum of individual displacements: y = y₁ + y₂.

y(x, t) = y₁(x, t) + y₂(x, t)

Standing Wave Equation

Superposing y₁ = A sin(kx - ωt) and y₂ = A sin(kx + ωt) yields y = [2A sin(kx)] cos(ωt). Amplitude is modulated spatially by sin(kx), forming nodes where sin(kx) = 0.

y = 2A sin(kx) cos(ωt)

Common Misconceptions

Misconception

Thinking that particles at nodes are vibrating slowly.

Physical Clarification:In an ideal standing wave, particles at true displacement nodes are permanently at rest with zero displacement and zero velocity at all times.

Syllabus Topic Outline

Competency 3.2
  • 1Transverse versus longitudinal progressive waves
  • 2Wave equation y = A sin(kx - ωt) and phase relationships
  • 3Superposition of two opposing waves: stationary waves
  • 4Nodes, antinodes, and beat frequency f_beat = |f₁ - f₂|

Interactive Wave Superposition & Stationary Waves Laboratory

Explore the Principle of Superposition, standing waves formation with nodes and antinodes, and acoustic beats in real-time.

Wave 1 (Forward)
Wave 2 (Reflected / Second)
Resultant Wave y = y₁ + y₂
Displacement Nodes (Zero Motion)
Antinodes (Max Motion 2A)
Wave 1 Frequency f₁ (Hz)1.0 Hz
Wave 2 Frequency f₂ (Hz)1.0 Hz
Wave 1 Amplitude A₁ (px)35 px
Phase Difference Δϕ (°)0°
Mathematical Physics of Superposition

Principle of Superposition: When two identical waves travel in opposite directions: y₁ = A · sin(kx - ωt) and y₂ = A · sin(kx + ωt) y = y₁ + y₂ = [2A · sin(kx)] · cos(ωt) • Nodes occur where sin(kx) = 0 ⟹ x = n(λ / 2). The medium at nodes is permanently at rest. • Antinodes occur where sin(kx) = ±1 ⟹ x = (2n + 1)(λ / 4). Amplitude reaches 2A. • Distance between two adjacent nodes = λ / 2.

Practice Problems

1

In a standing wave on a string, the distance between two consecutive nodes is 25 cm. What is the wavelength of the waves?