Transverse Longitudinal Waves

14.1 Introduction

A ripple spreading on a pond, the sound from a sitar, light from a distant star, an earthquake shaking the ground — all are examples of waves. Although they look very different, they share remarkable common mathematical structure. A wave transports energy from one place to another without bulk transport of matter.

If you drop a stone in a still lake, ripples spread outward. Each leaf floating on the lake bobs up and down as the wave passes, but does NOT travel along with the wave. Only the disturbance propagates outward — the water itself stays largely in place. This is the essence of wave motion: it is a way of moving energy without moving the medium en masse.

14.2 Transverse and Longitudinal Waves

14.2.1 Transverse Waves

In a transverse wave, particles of the medium oscillate perpendicular to the direction in which the wave travels. Pluck a stretched string — pull a section sideways and release. A transverse pulse races along the string, while the string itself moves only up and down.

14.2.2 Longitudinal Waves

In a longitudinal wave, particles oscillate along the same direction the wave travels. Push and pull a Slinky lengthwise: alternating dense (compressions) and sparse (rarefactions) regions move outward.

14.3 Displacement Relation in a Progressive Wave

Consider a sinusoidal transverse wave travelling in the +x direction with amplitude a, period T, and wavelength λ. The displacement y of a particle at position x and time t is:

The wave moves with phase velocity:

This fundamental relation \(v=\nu\lambda\) connects the speed of any wave to its frequency and wavelength.

14.3.1 Period, Frequency and Phase

The frequency ν is the number of oscillations per second, related to period by \(\nu=1/T\). The argument \((kx-\omega t+\phi)\) is called the phase of the wave.

14.3.2 Wave Speed v

For a fixed point on the wave (constant phase), \(kx - \omega t = \) constant. Differentiating: \(v=\dfrac{dx}{dt}=\dfrac{\omega}{k}\).