Find Wave Speed from the Medium, Then Use f and λ
A wave's speed is set by the medium, its frequency by the source, and its wavelength follows from v = f λ. Most speed questions are about which of these is unchanged when something else changes.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Speed on strings and ropes
For y = 0.02 sin[2π(t/0.04 - x/0.50)], the period is 0.04 s and the wavelength 0.50 m, so v = 12.5 m/s and, with μ = 0.04 kg/m, the tension is T = μ v² = 6.25 N. In a uniform rope hanging from a ceiling, the tension at a distance x from the free lower end is μ g x, so v = sqrt(g x): at the top of a 2.5 m rope v = sqrt(10 x 2.5) = 5 m/s and at 0.5 m from the lower end v = sqrt(10 x 0.5) = 2.24 m/s, so the wave slows toward the bottom.
Sound in different media; P and S waves
Sound travels faster in moist air than in dry air, because moist air is less dense. When sound passes from water (1450 m/s) into air (330 m/s) the frequency is unchanged, since it is set by the source, while the speed falls, so the wavelength falls too. For earthquake waves, the difference in arrival time is Δt = d (1/v_S - 1/v_P), so d = Δt v_S v_P / (v_P - v_S): with Δt = 4 min = 240 s, v_S = 4.5 km/s and v_P = 8.0 km/s, d is about 2500 km.
Echoes and intensity
The intensity of a wave is proportional to the square of the amplitude and the square of the frequency, I proportional to A² f². If the amplitude doubles the intensity is multiplied by 4, and if the frequency becomes one fourth the intensity is multiplied by 1/16, so the net intensity is 1/4 of the original.
OlympIQ exam lens
- ✓String wave speed v = sqrt(T/μ); for a hanging rope v = sqrt(g x), x from the free end
- ✓Frequency is fixed by the source; speed and wavelength change from one medium to another, with v = f λ
- ✓Sound is faster in moist air than in dry air
- ✓P-S arrival gap: d = Δt v_S v_P / (v_P - v_S)
- ✓Echo distance = v t / 2; intensity is proportional to A² f²
Do not say the frequency changes when a wave enters another medium. The frequency stays the same and the speed changes, so the wavelength changes in proportion to the speed.
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