Waves • Topic 1 of 3

Nature & Types of Waves

A wave is a disturbance that travels through a medium, carrying energy and momentum from one place to another without any bulk movement of the medium itself. When you drop a stone in a pond, the ripples spread outward, but a floating leaf only bobs up and down — it does not travel with the ripple. That single observation captures the heart of wave motion: the pattern moves, the particles do not.

Mechanical waves need a material medium (solid, liquid or gas) to travel through, because they rely on the elasticity and inertia of the medium. Sound, water ripples and waves on a string are all mechanical. They cannot travel through a vacuum — which is why no one can hear an explosion in outer space. (Electromagnetic waves like light need no medium, but in Class 11 we focus on mechanical waves.)

Mechanical waves come in two kinds, classified by the direction in which particles vibrate relative to the direction the wave travels:

  • Transverse waves: particles of the medium vibrate perpendicular to the direction of wave propagation. A wave on a stretched string is transverse — you shake the end up and down, but the pulse runs along the string. Transverse waves show crests (high points) and troughs (low points). They can travel through solids and on liquid surfaces, but not through the body of a gas or liquid.
  • Longitudinal waves: particles vibrate parallel to the direction of propagation. Sound in air is longitudinal — the air is alternately squeezed and stretched, forming regions of compression (high pressure) and rarefaction (low pressure). These can travel through solids, liquids and gases.

Wave parameters. Any periodic wave is described by a few key quantities:

  • Wavelength $\lambda$: the distance between two consecutive identical points (crest to crest, or compression to compression), measured in metres.
  • Frequency $f$: the number of complete oscillations per second, in hertz (Hz). The time period $T=\frac{1}{f}$ is the time for one full oscillation.
  • Amplitude $A$: the maximum displacement of a particle from its rest position.
  • Wave speed $v$: how fast the disturbance travels. The fundamental relation is $v=f\lambda$.

A travelling (progressive) wave moving in the $+x$ direction is written as $y=A\sin(\omega t-kx)$, where $\omega=2\pi f$ is the angular frequency and $k=\frac{2\pi}{\lambda}$ is the angular wave number. From these, $v=\frac{\omega}{k}=f\lambda$.

The speed of a wave depends only on the medium, not on the frequency. On a stretched string of tension $T$ and linear mass density $\mu$ (mass per unit length), the speed is $v=\sqrt{\frac{T}{\mu}}$. For sound, the speed depends on the elasticity and density of the medium: it is fastest in solids (about 5000 m/s in steel), slower in liquids (about 1500 m/s in water) and slowest in gases (about 340 m/s in air at room temperature).

Transverse wave with wavelength and amplitude labelledAλcresttroughwave travels →
Solved Examples
1
Worked Example
A wave has a frequency of 500 Hz and a wavelength of 0.68 m. Calculate its speed.
Solution
  1. Step 1: Use $v=f\lambda$.
  2. Step 2: Substitute: $v=500\times0.68$.
  3. Step 3: Compute: $v=340\ \text{m/s}$.

Answer: $v=340\ \text{m/s}$ (the speed of sound in air).

2
Worked Example
A radio station broadcasts at a frequency of 100 MHz. Find the wavelength of the waves. (Speed of EM waves $=3\times10^{8}\ \text{m/s}$.)
Solution
  1. Step 1: Use $\lambda=\frac{v}{f}$.
  2. Step 2: Substitute: $\lambda=\frac{3\times10^{8}}{100\times10^{6}}=\frac{3\times10^{8}}{1\times10^{8}}$.
  3. Step 3: Compute: $\lambda=3\ \text{m}$.

Answer: $\lambda=3\ \text{m}$.

3
Worked Example
A string of mass 5 g and length 1 m is stretched with a tension of 80 N. Find the speed of a transverse wave on it.
Solution
  1. Step 1: Linear mass density $\mu=\frac{m}{L}=\frac{5\times10^{-3}}{1}=5\times10^{-3}\ \text{kg/m}$.
  2. Step 2: Use $v=\sqrt{\frac{T}{\mu}}=\sqrt{\frac{80}{5\times10^{-3}}}=\sqrt{1.6\times10^{4}}$.
  3. Step 3: Compute: $v=126\ \text{m/s}$.

Answer: $v\approx126\ \text{m/s}$.

4
Worked Example
A progressive wave is given by $y=0.02\sin(300t-2x)$ (SI units). Find its amplitude, angular frequency, wave number, wavelength and speed.
Solution
  1. Step 1: Compare with $y=A\sin(\omega t-kx)$: amplitude $A=0.02\ \text{m}$, $\omega=300\ \text{rad/s}$, $k=2\ \text{rad/m}$.
  2. Step 2: Wavelength $\lambda=\frac{2\pi}{k}=\frac{2\pi}{2}=\pi\approx3.14\ \text{m}$.
  3. Step 3: Speed $v=\frac{\omega}{k}=\frac{300}{2}=150\ \text{m/s}$.

Answer: $A=0.02\ \text{m}$, $\omega=300\ \text{rad/s}$, $k=2\ \text{rad/m}$, $\lambda\approx3.14\ \text{m}$, $v=150\ \text{m/s}$.

5
Worked Example
The time period of a wave is 0.004 s and its speed is 350 m/s. Find its frequency and wavelength.
Solution
  1. Step 1: Frequency $f=\frac{1}{T}=\frac{1}{0.004}=250\ \text{Hz}$.
  2. Step 2: Use $\lambda=\frac{v}{f}=\frac{350}{250}$.
  3. Step 3: Compute: $\lambda=1.4\ \text{m}$.

Answer: $f=250\ \text{Hz}$, $\lambda=1.4\ \text{m}$.

6
Worked Example
The tension in a string is increased four times without changing anything else. How does the speed of a transverse wave on it change?
Solution
  1. Step 1: Since $v=\sqrt{\frac{T}{\mu}}$, we have $v\propto\sqrt{T}$.
  2. Step 2: Replacing $T$ by $4T$: $v'\propto\sqrt{4T}=2\sqrt{T}$.
  3. Step 3: Therefore $v'=2v$.

Answer: The wave speed doubles.

Key Points

  • A wave transports energy and momentum through a medium without bulk transport of the medium; mechanical waves need a material medium.
  • Transverse waves: particles vibrate perpendicular to propagation (crests & troughs); longitudinal waves: particles vibrate parallel (compressions & rarefactions).
  • Key relation: $v=f\lambda$, with $\omega=2\pi f$ and $k=\frac{2\pi}{\lambda}$; a progressive wave is $y=A\sin(\omega t-kx)$.
  • Speed of a transverse wave on a string: $v=\sqrt{\frac{T}{\mu}}$, where $\mu$ is the linear mass density.
  • Wave speed depends on the medium, not on the frequency; sound is fastest in solids, slowest in gases.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.In a longitudinal wave, the particles of the medium vibrate:
Explanation: Longitudinal waves (like sound) have particle vibrations parallel to the wave's travel, forming compressions and rarefactions.
Q2.The relation between wave speed, frequency and wavelength is:
Explanation: A wave travels one wavelength in one time period, so $v=\frac{\lambda}{T}=f\lambda$.
Q3.The speed of a transverse wave on a stretched string is given by:
Explanation: Wave speed on a string is $v=\sqrt{\frac{T}{\mu}}$, where $T$ is tension and $\mu$ is mass per unit length.
Q4.Sound waves cannot travel through:
Explanation: Sound is a mechanical wave and needs a material medium; it cannot travel through a vacuum.
Practice more MCQs → 📝 Assignment · 30 marks