Waves: Properties and Equations

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your WAEC Physics paper.

A wave is a periodic disturbance that transfers energy, not matter, from one point to another. Particles of the medium only oscillate about their equilibrium positions while the wave pattern travels onward. The single most-tested relation in this topic is v = fλ, where v is wave speed in m/s, f is frequency in Hz, and λ is wavelength in metres.

Transverse waves vibrate perpendicular to propagation (light, waves on a stretched string); they can be polarised.
Longitudinal waves vibrate parallel to propagation (sound, compression on a spring); they cannot be polarised.
Frequency f is fixed by the source, so wavelength shifts when v changes in a new medium (refraction).

WAEC Paper 2 and Paper 3 questions on this topic typically ask for the definition of v = fλ, classification of waves, or a 2–3 mark sketch of a transverse waveform with λ and A labelled.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months before WASSCE.

Classification of Waves

Waves fall into two pairs of categories. Mechanical waves (sound, water waves, waves on a string) need a material medium; electromagnetic waves (light, radio, X-rays) do not. By motion they are transverse (vibration perpendicular to travel) or longitudinal (vibration parallel to travel). The cleanest discriminator in WAEC questions is polarisation: only transverse waves can be polarised.

The Fundamental Wave Equation

The relation v = fλ is derived from the definitions of speed, frequency and wavelength. In one period T the wave advances by exactly one wavelength, so v = λ/T. Since T = 1/f, substituting gives v = fλ. This single line lets you solve most WAEC numerical items on echo problems, ripple tanks, and radio waves.

Speed in Specific Media

Speed depends on the medium, not the source. For a stretched string the wave speed is v = √(T/μ), where T is tension in N and μ is mass per unit length in kg/m. For sound in a gas, v = √(γP/ρ), where γ = Cp/Cv, P is pressure, and ρ is density. Numerically, sound travels slowest in gases (~330–360 m/s in air), faster in liquids (~1500 m/s in water), and fastest in solids (~5000 m/s in steel).

Quantity	Symbol	Unit	Depends on
Wavelength	λ	m	Medium (changes when v changes)
Frequency	f	Hz	Source only
Period	T	s	Source only (T = 1/f)
Amplitude	A	m	Energy of source
Wave speed	v	m/s	Medium properties

Standing Waves

When two identical progressive waves travel in opposite directions they superpose to form a stationary (standing) wave. Points of zero amplitude are nodes; points of maximum amplitude are antinodes. For a string fixed at both ends, the allowed frequencies are fₙ = nv/(2L), n = 1, 2, 3, …. This is the basis of the sonometer experiment in WAEC Paper 3.

v = fλ is derived, not assumed — write the steps on Paper 2 theory.
When sketching, label λ from crest to crest and A from equilibrium to crest.
For standing waves, the string length L must contain an integer number of half-wavelengths.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Wave Equation in Time and Space

A one-dimensional travelling wave can be written as y(x,t) = A sin[2π(ft − x/λ)]. The argument inside the sine is the phase. A positive sign with the minus direction (−x/λ) means the wave travels in the +x direction; flipping the sign reverses propagation. The factor 2π converts cycles into radians; many WAEC candidates drop it and lose a mark on derivation items. Differentiating twice with respect to t and twice with respect to x gives the wave equation ∂²y/∂x² = (1/v²) ∂²y/∂t², which holds for any non-dispersive medium.

Resonance and Natural Frequencies

Resonance occurs when a system is driven at one of its natural frequencies fₙ, producing large-amplitude oscillations. On a sonometer wire, the rider at the middle falls off when the driving fork matches fₙ = nv/(2L). The corresponding wavelength condition is L = nλ/2, so n half-wavelengths fit on the string. This is the most-tested practical setup in WAEC Physics Paper 3.

Common Mistakes and Exam Traps

Mistake	Correct idea
Writing v = f/λ	v = fλ — speed grows with both f and λ
Saying waves carry matter	Only energy is transferred; medium particles oscillate
Sound faster in air than steel	v increases from gas → liquid → solid for sound
Calling nodes the loud points	Nodes are zero amplitude; antinodes are maximum

Practice Prompts

A ripple tank source vibrates at 12 Hz and the waves travel 0.36 m in one second. Calculate the wavelength.
A sonometer wire of length 0.60 m and mass 3.0 × 10⁻⁴ kg is stretched by a 60 N tension. Find the frequency of the second harmonic.

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Waves: Properties and Equations