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Waves

Syllabus reference

Unit 2, Topic 2 — 20 hours (including practicals)

Mechanical waves

A wave is a disturbance that transfers energy from one place to another without transferring matter. Mechanical waves require a medium to travel through.

Transverse waves — the oscillation is perpendicular to the direction of energy transfer (e.g. waves on a string, light).

Longitudinal waves — the oscillation is parallel to the direction of energy transfer (e.g. sound, compression waves on a spring).

Wave characteristics

Key formulas

[ v = f\lambda ] [ T = \frac{1}{f} ]

where:

  • \(v\) = wave speed (m s⁻¹)
  • \(f\) = frequency (Hz)
  • \(\lambda\) = wavelength (m)
  • \(T\) = period (s)

Amplitude — the maximum displacement from the equilibrium position.

Wavelength (\(\lambda\)) — the distance between two consecutive points in phase (e.g. crest to crest).

Frequency (\(f\)) — the number of complete oscillations per second.

Period (\(T\)) — the time for one complete oscillation.

Interpreting wave graphs

Displacement–distance graph — shows the wave at a snapshot in time. Read off wavelength and amplitude.

Displacement–time graph — shows the motion of a single point over time. Read off period and amplitude.

Waves at boundaries

When a wave reaches a boundary between two media:

  • Fixed end — the pulse is reflected and inverted
  • Free end — the pulse is reflected without inversion
  • At a boundary between two media, the wave is partially reflected and partially transmitted. The speed changes but the frequency stays the same, so the wavelength changes.

Superposition

When two waves meet, the resultant displacement is the sum of the individual displacements (principle of superposition).

Constructive interference — waves in phase combine to produce a larger amplitude.

Destructive interference — waves out of phase combine to produce a smaller (or zero) amplitude.

Mandatory practical: Waves on springs

Investigate the behaviour of longitudinal and transverse waves on springs — reflection from fixed and free ends, and transmission/reflection at a medium boundary.

Sound

Sound is a longitudinal mechanical wave — it requires a medium and cannot travel through a vacuum.

The speed of sound depends on the medium: fastest in solids, slower in liquids, slowest in gases. In air at room temperature, \(v \approx 340\) m s⁻¹.

Intensity and the inverse-square law

The intensity of a wave is the power per unit area. For a point source radiating equally in all directions:

\[ I \propto \frac{1}{d^2} \]

As you double the distance from the source, the intensity drops to one quarter.

Light

Light is an electromagnetic wave — it does not require a medium and travels at \(c = 3 \times 10^8\) m s⁻¹ in a vacuum.

The electromagnetic spectrum (in order of increasing frequency): radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays.

Reflection

The angle of incidence equals the angle of reflection. Both are measured from the normal to the surface.

Refraction and Snell's law

When light passes from one medium to another, it changes speed and may change direction. Snell's law:

Key formula

\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]

where \(n\) = refractive index, \(\theta\) = angle to the normal

Total internal reflection

Occurs when light travels from a denser to a less dense medium at an angle greater than the critical angle \(\theta_c\):

\[ \sin\theta_c = \frac{n_2}{n_1} \]

Applications include fibre optics and endoscopes.

Polarisation

Polarisation is the restriction of wave oscillations to a single plane. Only transverse waves can be polarised — this provides evidence that light is a transverse wave.

Lenses and mirrors

Use ray diagrams to locate images formed by:

  • Plane mirrors
  • Concave and convex mirrors
  • Convex and concave lenses

Key features: focal point, centre of curvature, principal axis. Images can be real or virtual, upright or inverted, magnified or diminished.

Diffraction and interference

Diffraction is the spreading of waves as they pass through an opening or around an obstacle. It is most significant when the opening is comparable in size to the wavelength.

When light passes through a narrow slit, a diffraction pattern of bright and dark fringes is produced due to constructive and destructive interference.

Simulations and videos

PhET Simulations:

Crash Course Physics:

External resources: