Progressive waves

MDCAT Physics · Physics · Waves and oscillations. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A progressive wave carries energy from one place to another, but it does not carry the medium with it. Each particle only vibrates about its rest position. In a transverse wave the vibration is at right angles to the direction of travel (a wave on a string, light). In a longitudinal wave it is along the direction of travel (sound), which gives compressions and rarefactions.

The source fixes the frequency. When a wave enters a new medium its speed and wavelength change together, and the frequency stays the same.

Where two waves meet, their displacements add. This is superposition. Coherent sources give a steady interference pattern of maxima and minima.

Rules to remember

wave speed = frequency × wavelength (v = fλ), and f = 1/T
Phase difference = 2π × distance apart ÷ wavelength (Δφ = 2πx/λ), in radians
Speed of a transverse wave on a string: v = √(T/μ), with tension T in N and mass per unit length μ in kg m⁻¹
Coherent sources have the same frequency and a constant phase difference
Sources in phase: maximum where path difference = nλ, minimum where path difference = (n + ½)λ
Intensity is proportional to amplitude², so I_max/I_min = (A₁ + A₂)²/(A₁ − A₂)²
A compression and the nearest rarefaction are half a wavelength apart

Common mistake

Saying that the frequency changes when a wave passes into another medium. Only the speed and the wavelength change; the frequency is set by the source.

Worked example

A wave of frequency 200 Hz travels at 320 m s⁻¹. What is the phase difference between two points 0.40 m apart along its direction of travel?

Wavelength λ = v ÷ f = 320 ÷ 200 = 1.6 m.
Fraction of a wavelength = 0.40 ÷ 1.6 = 0.25.
Phase difference = 2π × 0.25 = π/2 rad.

Answer: π/2 rad (90°)

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