Wave Model

A wave is a periodic disturbance that carries energy and information from one place to another without any net transfer of matter. Waves are classified by comparing the direction of oscillation with the direction of energy propagation.

In a transverse wave the oscillations are perpendicular to the direction the energy travels: waves on a stretched string, ripples on a water surface and all electromagnetic waves behave this way, showing alternating crests and troughs.

In a longitudinal wave the oscillations are parallel to the energy direction, so the medium is alternately squeezed into compressions and stretched into rarefactions; sound in air is the standard example.

A slinky spring demonstrates both types: flicking one end side to side sends a transverse pulse along it, while pushing and pulling along its length sends a longitudinal pulse.

You should be able to:

• define both wave types
• classify given examples
• sketch and label each profile (crest, trough, compression, rarefaction)
• explain that in both cases it is energy, not matter, that moves along the wave

Four quantities describe any periodic wave:

• the wavelength $\lambda$ is the shortest distance between two points oscillating in phase
• the frequency $f$ is the number of cycles passing a point each second
• the period $T = \frac{1}{f}$ is the time for one complete cycle
• the amplitude is the maximum displacement from equilibrium

Because exactly one wavelength passes a fixed point during one period, the wave speed is:

$$v = f\lambda = \frac{\lambda}{T}$$

These quantities are read from two different graphs. A displacement-distance graph is a snapshot of the whole wave at one instant, so its repeat length is $\lambda$; a displacement-time graph follows a single point, so its repeat interval is $T$.

As a concrete example, a water wave with $f = 2.0\ \text{Hz}$ and $\lambda = 0.75\ \text{m}$ travels at $v = 2.0 \times 0.75 = 1.5\ \text{m s}^{-1}$.

You must be able to:

• rearrange $v = f\lambda$ fluently
• convert unit prefixes such as $\text{kHz}$ and $\text{ms}$ before substituting
• extract $\lambda$, $T$ and amplitude from the correct type of graph

Sound is a longitudinal mechanical wave. A vibrating source, such as a loudspeaker cone or a tuning fork, pushes repeatedly on the air particles next to it, and energy is then passed forward by particle collisions. This creates travelling regions of high density and pressure (compressions) separated by regions of low density and pressure (rarefactions).

Because the mechanism depends entirely on particles colliding, propagation is impossible in a vacuum: a ringing bell inside an evacuated jar falls silent even though it is still visibly vibrating.

The speed of sound depends on the medium:

• roughly $340\ \text{m s}^{-1}$ in air at room temperature
• faster in liquids
• fastest in solids, where the particles are most strongly coupled

The frequency of a sound determines its pitch and the amplitude its loudness; a typical human ear responds between about $20\ \text{Hz}$ and $20\ \text{kHz}$. The few-second gap between seeing lightning and hearing thunder is an everyday consequence of this finite speed.

You should be able to:

• explain the propagation mechanism in terms of compressions and rarefactions
• justify why sound cannot cross a vacuum
• apply $v = f\lambda$ to sound problems

Electromagnetic waves are transverse waves in which oscillating electric and magnetic fields, perpendicular both to each other and to the direction of travel, carry energy without needing any medium.

The mechanism is a self-sustaining loop: a time-varying electric field generates a perpendicular magnetic field, which in turn regenerates the electric field, so the wave drives itself through empty space at the universal speed $c = 3.00 \times 10^8\ \text{m s}^{-1}$.

The electromagnetic spectrum is the family of these waves ordered by wavelength:

• radio waves
• microwaves
• infrared
• visible light (from about $400\ \text{nm}$ at the violet end to $700\ \text{nm}$ at the red end)
• ultraviolet
• X-rays
• gamma rays

All regions travel at the same speed $c$ in a vacuum, so a higher frequency always means a shorter wavelength through:

$$c = f\lambda$$

When an electromagnetic wave enters a medium such as glass, its speed and wavelength decrease while its frequency stays fixed by the source.

You should be able to:

• list the spectrum regions in order
• recall the visible wavelength range
• calculate any one of frequency, wavelength or speed from the other two