Wave Motion

SYLLABUS

Progressive waves, Mathematical description of a wave, Stationary waves

What is This Chapter About?

Drop a stone into a still pond and watch what happens. Circular ripples spread outward from the point of impact — not because water molecules travel outward, but because a disturbance travels outward, carrying energy with it while the water itself moves up and down. This is the essence of a wave: a disturbance that propagates through a medium (or through space), transferring energy from one place to another without any net transfer of matter.

Waves are everywhere. Sound is a wave. Light is a wave. The vibrations of a guitar string are waves. Even the quantum mechanical description of particles involves waves. Chapter 6 of NEB Grade 12 Physics lays the mathematical and conceptual foundation for understanding all wave phenomena, carrying 2 teaching hours as a compact but essential introduction to the broader wave and optics section of the course.

Key Concepts You Will Learn

Progressive Waves

A progressive wave (also called a travelling wave) is a wave that continuously moves through a medium, carrying energy in the direction of propagation. Unlike a stationary object, the wave pattern itself moves — though the medium particles only oscillate about their equilibrium positions.

Every progressive wave is characterised by several key quantities. The amplitude (A) is the maximum displacement of a particle from its equilibrium position. The wavelength ([latex]\lambda[/latex]) is the distance between two consecutive points in phase — for example, two consecutive crests. The time period (T) is the time for one complete oscillation. The frequency (f) is the number of complete oscillations per second: [latex]f = \frac{1}{T}[/latex]. The wave speed (v) is the speed at which the wave pattern travels: [latex]v = f\lambda[/latex].

Mathematical Description of a Progressive Wave

The displacement of any particle in a progressive wave at position x and time t is described by the wave equation:

[latex]y = A\sin(\omega t – kx)[/latex]

where [latex]\omega = 2\pi f[/latex] is the angular frequency and [latex]k = \frac{2\pi}{\lambda}[/latex] is the wave number (also called propagation constant). This single equation contains all the information about the wave — its amplitude, frequency, wavelength, and direction of travel.

For a wave travelling in the negative x-direction, the equation becomes [latex]y = A\sin(\omega t + kx)[/latex]. Understanding how to read physical information from these equations — extracting amplitude, frequency, wavelength, and speed — is a key skill tested in NEB examinations.

Stationary Waves

When two progressive waves of the same amplitude and frequency travel in opposite directions through the same medium, they superpose to produce a stationary wave (also called a standing wave). Unlike a progressive wave, a stationary wave does not travel — its pattern stays fixed in space.

In a stationary wave, certain points called nodes remain permanently at rest (zero displacement), while points called antinodes oscillate with maximum amplitude. The distance between two consecutive nodes (or antinodes) is [latex]\frac{\lambda}{2}[/latex].

The mathematical form of a stationary wave is:

[latex]y = 2A\cos(kx)\sin(\omega t)[/latex]

This equation factors into a space-dependent part [latex]2A\cos(kx)[/latex] (which gives the fixed pattern of nodes and antinodes) and a time-dependent part [latex]\sin(\omega t)[/latex] (which gives the oscillation). This separation is what makes standing waves “stationary” — the spatial pattern does not move.

The condition for forming stationary waves is that the two superposing waves must have the same amplitude, frequency, and wavelength but travel in opposite directions. In practice, this occurs when a wave reflects from a boundary and interferes with the incoming wave.

Why This Chapter Matters for NEB Exams

Wave Motion provides the mathematical language for Chapters 7, 8, and 9 that follow. NEB examinations regularly test the wave equation — asking students to identify amplitude, frequency, wavelength, and speed from a given equation, or to write the equation for a wave with specified properties. The distinction between progressive and stationary waves, and the derivation of the stationary wave equation from two progressive waves, are also standard examination questions.

Important Topics to Focus On

• Characteristics of progressive waves: amplitude, wavelength, frequency, period, wave speed
• The wave relation: [latex]v = f\lambda[/latex]
• Mathematical equation of a progressive wave: [latex]y = A\sin(\omega t – kx)[/latex]
• Definitions of angular frequency [latex]\omega[/latex] and wave number k
• Extracting physical quantities from a given wave equation
• Conditions for formation of stationary waves
• Mathematical derivation of stationary wave equation from two progressive waves
• Properties of nodes and antinodes
• Key differences between progressive and stationary waves
• Numerical problems on wave speed, frequency, wavelength, and wave equations

Connection to Other Chapters

Wave Motion connects directly forward to Chapter 7 (Mechanical Waves), where the speed of sound in different media is derived, and Chapter 8 (Waves in Pipes and Strings), where stationary waves explain musical harmonics. Chapter 9 (Acoustic Phenomena) applies wave concepts to sound. The wave nature of light introduced here extends to Chapters 10 through 13 (Nature of Light, Interference, Diffraction, Polarization). The mathematical wave equation also connects back to Chapter 2 (Periodic Motion) — each particle in a wave executes simple harmonic motion, making SHM and wave motion deeply interrelated.