Waves and Optics: Refraction, Decibels, and Pendulum Periods
Snell's Law, decibel sound intensity, wave period, and pendulum period — four calculations spanning optics and wave mechanics.
Light and sound both travel as waves, and these four calculations capture some of their most commonly tested behaviors.
Snell's Law: how light bends between materials
Light passing from a material with refractive index 1.0 at a 30° angle of incidence into a material with index 1.5 bends to a smaller angle on the other side, following n₁sin(θ₁) = n₂sin(θ₂) — the formula behind everything from eyeglass lens design to why a straw looks bent in a glass of water.
Decibels: a logarithmic scale for a huge intensity range
Sound level in decibels is 10 × log₁₀(intensity / reference threshold). A sound at 10⁻⁶ W/m², compared to the standard 10⁻¹² W/m² threshold of hearing, measures 60 dB — the logarithmic scale is necessary because audible sound intensities span many orders of magnitude.
Wave period: the reciprocal of frequency
Period is simply 1/frequency — a 500 Hz wave has a period of 0.002 seconds. This reciprocal relationship holds for any periodic wave, whether sound, light, or mechanical oscillation.
Pendulum period: length and gravity, nothing else
A simple pendulum's period depends only on its length and local gravity: T = 2π√(L/g). Mass doesn't affect the period at all — a fact that surprises many students first encountering the formula, since intuition often suggests a heavier pendulum should swing differently.
Wave behavior, calculated directly
Refraction, sound intensity, and periodic motion all follow predictable formulas once you have the right inputs. Try the Snell's Law calculator, decibel calculator, wave period calculator, and pendulum period calculator.