Displacement Relation in a Progressive Wave 🌊

Picture a travelling wave as a moving “wiggle” that carries energy without dragging the material along. For a smooth (sinusoidal) wave moving in the +x-direction you use

\(y(x,t)=a\sin\bigl(kx-\omega t+\varphi\bigr)\)

:contentReference[oaicite:0]{index=0}
  • \(y(x,t)\): displacement of the particle at position \(x\) and time \(t\).
  • \(a\): amplitude – the largest distance a particle swings from the rest line. 🎉
  • \(\varphi\): initial phase angle – sets where the wave “starts” at \(x=0,\;t=0\).
  • \(k\): angular wave number or propagation constant. 📏 It tells how many wave “bumps” fit in a metre: \(k=\dfrac{2\pi}{\lambda}\).
  • \(\omega\): angular frequency – how fast each particle cycles. ⏱️ It links to the time period: \(\omega=\dfrac{2\pi}{T}\).
:contentReference[oaicite:1]{index=1}

Key Distances & Times

  • Wavelength \( \lambda\): the shortest distance between identical points (crest-to-crest or trough-to-trough). You get \( \lambda = \dfrac{2\pi}{k}\). 🛤️
  • Period \(T\): time for one full oscillation → \(T=\dfrac{2\pi}{\omega}\).
  • Frequency \( \nu \): oscillations each second → \( \nu =\dfrac{1}{T} = \dfrac{\omega}{2\pi}\). 🎶
:contentReference[oaicite:2]{index=2}

Wave Speed 🚀

Follow any crest (or any point of fixed phase). Because the crest keeps \(kx-\omega t = \text{constant}\), it slides with speed

\(v = \dfrac{\omega}{k} = \lambda \nu\)

:contentReference[oaicite:3]{index=3}

This relation works for every progressive wave – transverse or longitudinal. 👏

Transverse vs. Longitudinal

  • Transverse: particles swing across the direction of travel. Use \(y(x,t)\) for the displacement.
  • Longitudinal: particles shuffle along the direction of travel. Use \(s(x,t)=a\sin(kx-\omega t+\varphi)\).
:contentReference[oaicite:4]{index=4}

Worked Example 📝

Given the wave \(y(x,t)=0.005\sin(80x-3t)\):

  • Amplitude: \(a = 0.005\;\text{m} = 5\;\text{mm}\).
  • \(k = 80\;\text{m}^{-1}\) ⇒ \(\lambda = \dfrac{2\pi}{k} \approx 7.85\;\text{cm}\).
  • \(\omega = 3\;\text{s}^{-1}\) ⇒ \(T=\dfrac{2\pi}{\omega}\approx 2.09\;\text{s}\) and \( \nu \approx 0.48\;\text{Hz}\).
  • At \(x = 0.30\;\text{m},\;t = 20\;\text{s}\): \(y \approx 5\;\text{mm}\,\sin(1.699)\).
:contentReference[oaicite:5]{index=5}

Important Concepts for NEET 🎯

  1. The displacement equation \(y(x,t)=a\sin(kx-\omega t+\varphi)\) and the meaning of every symbol.
  2. Core links: \(k=2\pi/\lambda\) and \(\omega = 2\pi/T\).
  3. Universal speed rule \(v=\omega/k=\lambda \nu\).
  4. Clear difference between transverse (\(y\)) and longitudinal (\(s\)) descriptions.
  5. Amplitude and phase – how they set the wave’s size and starting point.
:contentReference[oaicite:6]{index=6}