Boundary & Energy LabWaves · H2 9749
Three wave facts from the notes, made undeniable with sliders.
Station 1 — Medium Boundary
λ changes, f cannot| before (medium 1) | after (medium 2) | |
|---|---|---|
| I | I₁ | 0.25 I₁ (I ∝ A² : (0.50)²) |
| f | 2.0 Hz | 2.0 Hz (unchanged — always) |
| λ = v/f | 2.00 m | 1.00 m (0.50 λ₁) |
Never drift apart — f is fixed by the source.
Station 2 — The Intensity Chain
I ∝ f²A²ⓘ why the squares
Every particle of the medium does SHM with energy E = ½mω²y₀². ω = 2πf and y₀ = A, so the energy each particle carries — and the rate the wave delivers it through unit area — scales as f²A². Meter computes the real thing: I = ½ρvω²A² with ρ = 1000 kg/m³, v = 1.5 m/s fixed.
Snapshot, then drag — I₂/I₁ computes live.
Station 3 — Spreading vs Damping
two opposite storiesⓘ exam phrasing
The source's power is spread over wavefronts of increasing area 4πr², so intensity falls as 1/r² and, since I ∝ A², amplitude falls as 1/r. No energy is lost — it is only spread more thinly; total power through every wavefront stays constant (A here = 40 mm at r₀ = 1 m).
ⓘ exam phrasing
The particle does work against resistive forces, so oscillation energy is progressively lost as thermal energy; amplitude decays exponentially with time. Energy lost between two peaks = ½mω²(y₀₁² − y₀₂²).
What this confirmsWave Motion — Everything to Memorise
- ▸“Boundary crossing — f is sacred” — the two cycle counters either side of the boundary tick in sync forever, at any v₂/v₁: frequency never changes at a boundary.
- ▸“Wave speed is a property of the medium” — you set v₂ with the medium slider, the source cannot; λ₂ = v₂/f stretches or compresses instantly to match.
- ▸“The intensity chain from SHM” — E = ½mω²y₀² per particle forces I ∝ f²A²: the A ×2 and f ×2 buttons each jump the meter ×4, both together ×16.
- ▸“Proportionality → ratio, always” — the snapshot panel live-computes I₂/I₁ = (A₂/A₁)²(f₂/f₁)² and the meter ratio always equals the product; Station 1's exam row (v and A halved → 0.25 I, f, 0.50 λ) is the same move.
- ▸“Shrinking amplitude — two opposite stories” — spreading: A ∝ 1/r with total power P = I·4πr² pinned constant (energy NOT lost); damping: exponential y–t decay with energy converted to thermal (energy IS lost).
- ▸“Energy lost between two amplitudes” — pick any two peaks on the damping trace and ΔE = ½mω²(y₀₁² − y₀₂²) is evaluated live in real units.