Mechanical Waves

What is a Mechanical Wave?

A mechanical wave is a disturbance in matter that carries energy from one place to another. The matter through which a mechanical wave travels is called the medium (such as air, water, or a solid string).

Crucially, while the energy travels across the medium, the particles of the medium do not permanently move from their positions; they simply oscillate back and forth around an equilibrium point.

Learning Goals: By the end of this guide, you should be able to:

1. Distinguish between transverse and longitudinal waves.
2. Define key wave parameters: amplitude, wavelength, frequency, and period.
3. Apply the wave equation $v = f\lambda$ to solve problems.

Types of Mechanical Waves

Mechanical waves are classified based on how the particles of the medium vibrate relative to the direction of wave propagation.

Transverse Waves

In a transverse wave, the particles of the medium vibrate perpendicular to the direction that the wave travels.

• Examples: Waves on a string, water surface waves.
• Key Features: They have crests (highest points) and troughs (lowest points).

Longitudinal Waves

In a longitudinal wave, the particles of the medium vibrate parallel to the direction that the wave travels.

• Examples: Sound waves, seismic P-waves.
• Key Features: They consist of compressions (areas of high pressure/density where particles are close) and rarefactions (areas of low pressure/density where particles are spread out).

Wave Properties

To mathematically describe a wave, we use several key parameters:

• Amplitude ($A$): The maximum displacement of a particle from its rest position. It relates to the energy of the wave (e.g., loudness of sound).
• Wavelength ($\lambda$): The distance between two consecutive corresponding points on a wave (e.g., crest to crest, or compression to compression). Measured in metres (m).
• Frequency ($f$): The number of complete wave cycles that pass a given point in one second. Measured in Hertz (Hz).
• Period ($T$): The time it takes for one complete wave cycle to pass. Measured in seconds (s). $T = \frac{1}{f}$.

The Wave Equation

The speed at which a wave travels through a medium depends entirely on the properties of that medium. However, for any periodic wave, the wave speed ($v$), frequency ($f$), and wavelength ($\lambda$) are always related by the universal wave equation:

Example Problem

Question: A sound wave in air has a frequency of $256\ \text{Hz}$ and a wavelength of $1.34\ \text{m}$. What is the speed of sound in air?

Step 1: Identify knowns. $f = 256\ \text{Hz}$, $\lambda = 1.34\ \text{m}$.

Step 2: Apply the wave equation.

The speed of sound is approximately $343\ \text{m/s}$.

Common Mistakes

1. Confusing particle speed with wave speed: The wave speed ($v$) is how fast the energy disturbance moves forward. The particles themselves are oscillating up and down (or back and forth) with varying speeds, momentarily stopping at the amplitude peaks.
2. Thinking the medium travels: A wave on the ocean moves toward the shore, but the water itself is just bobbing in circular paths. Waves transfer energy, not matter.
3. Wavelength vs. Period graphs: In a Displacement-Position ($y-x$) graph, the distance between peaks is the wavelength. In a Displacement-Time ($y-t$) graph of a single particle, the distance between peaks is the period. Do not mix these up!

Related Topics

• Simple Harmonic Motion — The mathematics of the individual oscillating particles in a wave medium.