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Waves and Wave Motion

Part of the JAMB UTME study roadmap. Physics topic phy-6 of Physics.

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Waves and Wave Motion

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Waves are disturbances that carry energy from one point to another without any net transfer of matter. The master relation is v = fλ, where v is wave speed (m/s), f is frequency (Hz), and λ is wavelength (m); frequency and period obey T = 1/f.

QuantitySymbolUnit
Wavelengthλm
FrequencyfHz
PeriodTs
AmplitudeAm
Wave speedvm/s
  • Transverse waves vibrate perpendicular to propagation (light, waves on a stretched string).
  • Longitudinal waves vibrate parallel to propagation (sound, compression on a slinky).
  • In a standing wave, distance between adjacent nodes equals λ/2 — a frequent JAMB trap.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Progressive waves and the wave equation

A travelling wave moving in the +x direction can be written as y = A sin(2π/λ)(vt − x), where y is the displacement at position x and time t, A is amplitude, and the angular wave number k = 2π/λ and angular frequency ω = 2πf describe how rapidly the wave varies in space and time. Wave speed in a non-dispersive medium depends only on the medium itself, not on frequency or amplitude.

Transverse vs longitudinal

In transverse waves the particle oscillation is perpendicular to energy propagation, producing crests and troughs. In longitudinal waves particles oscillate along the propagation direction, producing compressions and rarefactions. Light and ripples are transverse; sound in air is longitudinal.

Standing (stationary) waves

Formed when two identical progressive waves travel in opposite directions and superpose. Nodes are positions of permanent zero displacement; antinodes are positions of maximum displacement. The node-to-node spacing is λ/2, and the string length must equal an integer number of half-wavelengths: L = n(λ/2), where n = 1 gives the fundamental.

Beats and Doppler effect

Two close frequencies f₁ and f₂ superpose to produce beats at frequency f_b = |f₁ − f₂|, heard as rhythmic loudness variation. The Doppler effect is f′ = f(v ± v_o)/(v ∓ v_s), with upper signs when observer moves toward the source and lower signs when source moves away — sign handling is the most common JAMB trap.

JAMB favourite: identify whether a given wave is transverse or longitudinal, then compute speed, frequency or wavelength from v = fλ.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge cases and deeper mechanisms

On a stretched string the wave speed obeys v = √(T/μ), where T is tension (N) and μ is mass per unit length (kg/m). Raising T quadruples v², so doubling tension increases speed by only √2 — a common multiple-choice distractor. Sound in air travels faster in warm air, roughly v ≈ 330 + 0.6T °C m/s, because hotter air has a higher bulk modulus-to-density ratio.

Common mistakes to avoid

  • Confusing frequency with period: they are reciprocals, not synonyms.
  • Writing v = fλ as v = f/λ — multiplication, not division.
  • Claiming waves transport matter; only energy and momentum propagate.
  • Using λ instead of λ/2 when counting nodes on a standing-wave diagram.

Worked micro-example

A 0.80 m guitar string fixed at both ends has μ = 0.005 kg/m under tension 80 N. Speed: v = √(80/0.005) = √16000 ≈ 126.5 m/s. Fundamental wavelength: λ₁ = 2L = 1.60 m, so f₁ = v/λ₁ ≈ 79.1 Hz. The second harmonic has f₂ ≈ 158 Hz, confirming f_n = nf₁.

Practice prompts

  1. Two tuning forks at 256 Hz and 260 Hz sound together. How many beats per second are heard, and what must be done to eliminate them?
  2. An ambulance approaches a stationary observer at 30 m/s emitting 800 Hz. Using v_sound = 330 m/s, compute the observed frequency.

Connections

Waves link directly to sound (acoustics), light optics, and simple harmonic motion — the displacement of any wave particle obeys SHM with the same period T as the wave.

Exam strategy: spend under 90 seconds per direct question; the v = fλ trio and Doppler signs are worth roughly 2 of the 40 Physics items in JAMB UTME.

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