The outline below is also in pdf form here.
Angular velocity and angular lenticity
- Velocity, v = Δs/Δt, lenticity, u = Δt/Δs so u = 1/v and v = 1/u except if u or v are zero
- Zero velocity: no motion but time changes because time is independent
- Zero lenticity: no motion but length changes because length is independent
Circular motion in multidimensional space and time
stance, s; time, t; radius r; period radius q; circumference S = 2πr = wavelength λ; period T = 2πq = wave duration μ; angular velocity, v; angular lenticity, u; arc length, s; arc time, t
- Circle in length space
- length angle θ, arc length s, radius r
- spatial angle: θ ≡ s/r; r = s/θ; s = rθ; 1/θ = r/s; 1/r = θ/s
- angular time rate: ω ≡ θ/t; t = θ/ω; θ = ωt; 1/ω = t/θ; 1/t = ω/θ
- Circle in duration space
- duration angle φ, arc time t, period radius q
- temporal angle, φ ≡ t/q; q = t/φ; t = qφ; 1/φ = q/t; 1/q = φ/t
- angular stance rate: ψ ≡ φ/s; s = φ/ψ; φ = ψs; 1/ψ = s/φ; 1/s = ψ/φ
Angular time rates
- Independent variable is time, dependent variable is length
- Angular velocity: time rate of rotation, ω ≡ θ/t t = θ/ω; θ = ωt
- Wave (phase) velocity: wavelength per unit time, v = s/t = S/T = r/q = ωr
- frequency, f ≡ 1/T = v/S = v/λ = v/s = vh = h/u
- wavelength, λ = v/f = S
- wave (length) number, k = 2π/λ
- Wave velocity normalized
- revolutions: If S = 1, then v = 1/T = f
- space radians: If r = 1, then s = θ = φ and v = θ/t = φ/t = s/t = ω = 1/q = 2π/T = 2πf
- ω = 2πf = 2π/T = θ/t = φ/t; q = T/2π; f = ω/2π
Angular stance rates
- Independent variable is stance, dependent variable is time
- Angular lenticity: space rate of rotation, ψ ≡ φ/s; s = φ/ψ; φ = ψs
- Wave (phase) lenticity: wave duration per unit length, u = t/s = T/S = q/r = ψq
- periodicity, h ≡ 1/S = u/T = u/μ = s/v = uf = f/v
- wave duration, μ = u/h = T
- wave duration number, ℓ = 2π/μ
- Wave lenticity normalized
- revolutions: If T = 1, then u = 1/S = h
- time radians: If q = 1, then t = φ = θ and u = φ/s = θ/s = t/s = ψ = 1/r = 2π/S = 2πh
- ψ = 2πh = 2π/S = θ/s = φ/s; r = S/2π; h = ψ/2π
Wave function for sinusoidal wave
length amplitude, A; duration amplitude, B; length phase, ϕ; duration phase, χ; stance, s; time, t; circumference S = wavelength λ; period T = wave duration μ; (length) wave number, k; duration wave number, ℓ; angular (phase) velocity, v; angular (phase) lenticity, u
length space with time:
x = A cos(ωt + ϕ) a = −ω²x in SHM
y(x = 0, t) = A cos(ωt) = A cos(2πft)
y(x, t) = A cos[ω(t – x/v)] = A cos[2πf (x/v − t)] = sinusoidal wave moving in the +x-direction
y(x, t) = A cos[2π (x/λ − t/T)] = A cos(kx – ωt)
∂²y(x, t)/∂x² = (1/v²) ∂²y(x, t)/∂t² (length) wave equation
duration space with stance:
ξ = B cos(ψs + χ) b = −ψ²ξ in SHM
η(ξ = 0, s) = B cos(ψs) = B cos(2πhs)
η(ξ, s) = B cos[ψ(s – ξ/u)] = B cos[2πh (ξ/u − s)] = sinusoidal wave moving in the + ξ-direction
η(ξ, s) = B cos[2π (ξ/μ − s/S)] = B cos(ℓξ – ψs)
∂²η(ξ, s)/∂ξ² = (1/u²) ∂²η(ξ, s)/∂s² duration wave equation