Waves

Revision notes

A sinusoidal wave can be described by amplitude, frequency, wavelength and phase. A path difference of one wavelength corresponds to a phase difference of 2π rad (360°). Coherent sources maintain a constant phase relationship, allowing stable interference. Constructive interference occurs when waves arrive in phase; destructive interference occurs when they arrive in antiphase for equal amplitudes.

Stationary waves form from superposition of waves of the same frequency travelling in opposite directions, producing nodes and antinodes. Diffraction becomes important when apertures are comparable with wavelength. For a diffraction grating, maxima satisfy d sin θ = nλ, where d is the grating spacing and n is the order. Transverse waves can be polarised; longitudinal waves cannot.

Describing waves

A progressive wave transfers energy and phase through space. Amplitude is maximum displacement from equilibrium, frequency is oscillations per second, wavelength is the distance over which phase changes by one complete cycle, and wave speed satisfies v = fλ.

Phase difference can be expressed in radians or degrees and related to path difference. A path difference of one wavelength corresponds to 2π radians; half a wavelength corresponds to π radians. This language is essential for interference and stationary-wave questions.

Superposition and interference

The principle of superposition states that when waves overlap, the resultant displacement is the vector sum of the individual displacements. Coherent sources maintain a constant phase relationship. Constructive interference occurs when waves arrive in phase; destructive interference occurs when they arrive in antiphase.

For two-source interference, maxima occur when path difference is an integer multiple of λ, while minima occur at half-integer multiples for sources initially in phase.

Diffraction

Diffraction is spreading caused by wavefronts passing through a gap or around an obstacle. It becomes more pronounced when the gap dimension is comparable with the wavelength. This explains why long-wavelength waves bend around obstacles more readily than short-wavelength waves.

Stationary waves

Stationary waves form from two waves of the same frequency travelling in opposite directions and superposing. Nodes have zero amplitude; antinodes have maximum amplitude. Adjacent nodes are separated by \(\dfrac{\lambda }{2}\) and a node-to-neighbouring-antinode separation is \(\dfrac{\lambda }{4}\).

There is no net energy transfer along an ideal stationary-wave pattern. For strings and air columns, boundary conditions determine which wavelengths and harmonics are allowed.

Refraction and refractive index

Refraction occurs because wave speed changes. Frequency remains fixed by the source, so wavelength changes with speed. The refractive index of a material is n = \(\dfrac{c}{v}\). At a boundary, n₁ sin θ₁ = n₂ sin θ₂ (Snell's law), and the critical angle c satisfies sin c = \(\dfrac{n_{2}}{n_{1}}\). Total internal reflection occurs only when light travels from higher refractive index towards lower refractive index and the incidence angle exceeds the critical angle.

Polarisation

Polarisation is evidence that electromagnetic waves are transverse. A polariser selects an oscillation direction. For an ideal polariser and analyser at angle θ, the transmitted intensity is I = I₀cos²θ (Malus' law).

Exam technique

Translate between phase difference and path difference before deciding whether interference is constructive or destructive. For stationary waves, distinguish displacement amplitude from energy transfer. In refraction questions, state explicitly that frequency remains constant.

Refraction and optical fibres

An optical fibre has a high-refractive-index core surrounded by lower-index cladding. Light meeting the core–cladding boundary at more than the critical angle is totally internally reflected along the fibre. The cladding protects the core and prevents signal crossover between fibres; material and modal dispersion limit how quickly pulses can be sent.

Stationary waves on strings and intensity

For a string fixed at both ends, the fundamental has a node at each end and λ = 2L. Its frequency is f = \(\dfrac{1}{2L}\)\(\sqrt{\dfrac{T}{\mu }}\), where T is the tension and μ the mass per unit length. Higher harmonics have frequencies that are whole-number multiples of the fundamental.

The intensity of a wave is the power per unit area, and it is proportional to amplitude squared: doubling the amplitude quadruples the intensity.

Young's double-slit experiment

Young's double-slit experiment shows interference of light. For slit separation s and screen distance D, the fringe spacing is w = \(\dfrac{\lambda D}{s}\). Using a laser as a coherent, monochromatic source gives clear fringes; never look directly into the beam.