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## 🚨 Checklist - [x] Read and follow the [Contributing Guidelines](https://github.com/cmichelenstrofer/.github/blob/main/CONTRIBUTING.md). - [x] Provide a more detailed description below, including [referencing or closing the relevant issue(s)](https://docs.github.com/en/issues/tracking-your-work-with-issues/linking-a-pull-request-to-an-issue) 💔 Thanks! ## PR Description - Closes #42. - Adds an equivalence to extend `Periodic` by using a specific, user-supplied, dispersion-relation - This allows converting between temporal and spatial frequencies
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| # equivalences containing angular dimensions | ||
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| # periodic equivalence for both temporal and spatial frequency | ||
| """ | ||
| Periodic() | ||
| Equivalence to convert between temporal or spatial period, frequency, | ||
| [angular frequency](https://en.wikipedia.org/wiki/Angular_frequency), and angular period. | ||
| These quantities are related by ``f = ω/2π = 1/T = 1/(2πT̅)``, where | ||
| - ``f`` is the (temporal) frequency, | ||
| - ``ω`` is the (temporal) angular frequency, | ||
| - ``T`` is the (temporal) period, | ||
| - ``T̄`` is the (temporal) angular period, | ||
| and ``ν = k/2π = 1/λ = 1/(2πλ̄)``, where | ||
| - ``ν`` is the (spatial) frequency (linear wavenumber), | ||
| - ``k`` is the (spatial) angular frequency (angular wavenumber), | ||
| - ``λ`` is the (spatial) period (linear wavelength), and | ||
| - ``λ̄`` is the (spatial) angular period (angular wavelength). | ||
| See also [`DimensionfulAngles.Dispersion`](@ref) | ||
| # Example | ||
| ```jldoctest; filter = r"(\\d*).(\\d{1,10})\\d+" => s"\\1.\\2" | ||
| julia> using Unitful | ||
| julia> using DimensionfulAngles | ||
| julia> uconvert(u"s", 10u"Hz", Periodic()) | ||
| 0.1 s | ||
| julia> uconvert(u"radᵃ/s", 1u"Hz", Periodic()) | ||
| 6.283185307179586 rad s⁻¹ | ||
| ``` | ||
| """ | ||
| struct Periodic <: Equivalence end | ||
| # temporal | ||
| @eqrelation Periodic (AngularVelocity / Frequency = 2π * radᵃ) | ||
| @eqrelation Periodic (Frequency * Time = 1) | ||
| @eqrelation Periodic (AngularVelocity * Time = 2π * radᵃ) | ||
| @eqrelation Periodic (AngularVelocity * AngularPeriod = 1) | ||
| @eqrelation Periodic (Time / AngularPeriod = 2π * radᵃ) | ||
| @eqrelation Periodic (AngularPeriod * Frequency = 1/(2π * radᵃ)) | ||
| # spatial | ||
| @eqrelation Periodic (AngularWavenumber / Wavenumber = 2π * radᵃ) | ||
| @eqrelation Periodic (Wavenumber * Length = 1) | ||
| @eqrelation Periodic (AngularWavenumber * Length = 2π * radᵃ) | ||
| @eqrelation Periodic (AngularWavenumber * AngularWavelength = 1) | ||
| @eqrelation Periodic (Length / AngularWavelength = 2π * radᵃ) | ||
| @eqrelation Periodic (AngularWavelength * Wavenumber = 1/(2π * radᵃ)) | ||
| # default to `uconvert` behavior, temporal | ||
| @eqrelation Periodic (Frequency / Frequency = 1) | ||
| @eqrelation Periodic (Time / Time = 1) | ||
| @eqrelation Periodic (AngularVelocity / AngularVelocity = 1) | ||
| @eqrelation Periodic (AngularPeriod / AngularPeriod = 1) | ||
| # default to `uconvert` behavior, spatial | ||
| @eqrelation Periodic (Wavenumber / Wavenumber = 1) | ||
| @eqrelation Periodic (Length / Length = 1) | ||
| @eqrelation Periodic (AngularWavenumber / AngularWavenumber = 1) | ||
| @eqrelation Periodic (AngularWavelength / AngularWavelength = 1) | ||
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| # periodic equivalence with a specific dispersion relation relating temporal and spatial | ||
| # frequencies | ||
| """ | ||
| Dispersion(; dispersion=nothing, dispersion_inverse=nothing) | ||
| Equivalence to convert between temporal and spatial frequencies using a specific | ||
| [dispersion relation](https://en.wikipedia.org/wiki/Dispersion_relation). | ||
| This extends the Periodic() equivalence to convert between spatial and temporal quantities | ||
| based on the provided dispersion relation. | ||
| See also [`DimensionfulAngles.Periodic`](@ref). | ||
| # Example | ||
| ```jldoctest; filter = r"(\\d*).(\\d{1,10})\\d+" => s"\\1.\\2" | ||
| julia> using DimensionfulAngles, Unitful | ||
| julia> g = Unitful.gn # gravitational acceleration | ||
| 9.80665 m s⁻² | ||
| julia> deepwater = Dispersion( | ||
| dispersion = (k -> √(g*k*θ₀)), dispersion_inverse = (ω -> ω^2/(g*θ₀)) | ||
| ); | ||
| julia> uconvert(u"radᵃ/mm", 1.0u"Hz", deepwater) | ||
| 0.004025678249387654 rad mm⁻¹ | ||
| ``` | ||
| Some dispersion relations do not have an expressible inverse. | ||
| In such cases using `Roots.jl` might be beneficial. | ||
| For example, here is how we could use the | ||
| linear [water wave dispersion](https://en.wikipedia.org/wiki/Dispersion_(water_waves)) | ||
| without deep water approximation: | ||
| ```jldoctest; filter = r"(\\d*).(\\d{1,10})\\d+" => s"\\1.\\2" | ||
| julia> using DimensionfulAngles, Unitful, Roots | ||
| julia> g = Unitful.gn # gravitational acceleration | ||
| 9.80665 m s⁻² | ||
| julia> k0 = (2π)u"radᵃ/m" # initial guess: 1m wavelength | ||
| 6.283185307179586 rad m⁻¹ | ||
| julia> h = 0.5u"m" # water depth | ||
| 0.5 m | ||
| julia> waterwaves = Dispersion( | ||
| dispersion = (k -> √(k*θ₀*g*tanh(k*h/θ₀))), | ||
| dispersion_inverse = (ω -> solve(ZeroProblem(k -> k - ω^2/(g*tanh(k*h/θ₀))/θ₀, k0))) | ||
| ); | ||
| julia> uconvert(u"Hz", 0.004025678249387654u"radᵃ/mm", waterwaves) | ||
| 0.9823052153509486 Hz | ||
| julia> h = (Inf)u"m" # water depth | ||
| Inf m | ||
| julia> waterwaves = Dispersion( | ||
| dispersion = ( k -> √(k*θ₀*g*tanh(k*h/θ₀)) ), | ||
| dispersion_inverse = (ω -> solve(ZeroProblem(k -> k - ω^2/(g*tanh(k*h/θ₀))/θ₀, k0))) | ||
| ); | ||
| julia> uconvert(u"Hz", 0.004025678249387654u"radᵃ/mm", waterwaves) ≈ 1u"Hz" | ||
| true | ||
| ``` | ||
| """ | ||
| struct Dispersion <: Equivalence | ||
| dispersion::Union{Function, Nothing} | ||
| dispersion_inverse::Union{Function, Nothing} | ||
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| function Dispersion(; | ||
| dispersion::Union{Function, Nothing}=nothing, | ||
| dispersion_inverse::Union{Function, Nothing}=nothing | ||
| ) | ||
| return new(dispersion, dispersion_inverse) | ||
| end | ||
| end | ||
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| const _temporal_frequency_dims = [Time, Frequency, AngularPeriod, AngularVelocity] | ||
| const _spatial_frequency_dims = [Length, Wavenumber, AngularWavelength, AngularWavenumber] | ||
| const _temporal_frequency_units = Dict( | ||
| Time => s, Frequency => Hz, AngularPeriod => s/radᵃ, AngularVelocity => radᵃ/s) | ||
| const _spatial_frequency_units = Dict( | ||
| Length => m, Wavenumber => 1/m, AngularWavelength => m/radᵃ, AngularWavenumber => radᵃ/m | ||
| ) | ||
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| # use all the equivalences in Periodic | ||
| for T1 ∈ _temporal_frequency_dims, T2 ∈ _temporal_frequency_dims | ||
| @eval begin | ||
| function UnitfulEquivalences.edconvert(d::dimtype($T1), x::$T2, ::Dispersion) | ||
| return edconvert(d, x, Periodic()) | ||
| end | ||
| end | ||
| end | ||
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| for T1 ∈ _spatial_frequency_dims, T2 ∈ _spatial_frequency_dims | ||
| @eval begin | ||
| function UnitfulEquivalences.edconvert(d::dimtype($T1), x::$T2, ::Dispersion) | ||
| return edconvert(d, x, Periodic()) | ||
| end | ||
| end | ||
| end | ||
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| # add new equivalences between temporal <-> spatial frequencies | ||
| for D_in ∈ _spatial_frequency_dims, D_out ∈ _temporal_frequency_dims | ||
| @eval begin | ||
| function UnitfulEquivalences.edconvert( | ||
| ::dimtype($D_out), x::$D_in, equivalence::Dispersion | ||
| ) | ||
| isnothing(equivalence.dispersion) && throw(ArgumentError( | ||
| "`dispersion` function not defined")) | ||
| k = uconvert(radᵃ/m, x, Periodic()) | ||
| ω = equivalence.dispersion(k) | ||
| u = _temporal_frequency_units[$D_out] | ||
| return uconvert(u, ω, Periodic()) | ||
| end | ||
| end | ||
| end | ||
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| for D_in ∈ _temporal_frequency_dims, D_out ∈ _spatial_frequency_dims | ||
| @eval begin | ||
| function UnitfulEquivalences.edconvert( | ||
| ::dimtype($D_out), x::$D_in, equivalence::Dispersion | ||
| ) | ||
| isnothing(equivalence.dispersion_inverse) && throw(ArgumentError( | ||
| "`dispersion_inverse` function not defined")) | ||
| ω = uconvert(radᵃ/s, x, Periodic()) | ||
| k = equivalence.dispersion_inverse(ω) | ||
| u = _spatial_frequency_units[$D_out] | ||
| return uconvert(u, k, Periodic()) | ||
| end | ||
| end | ||
| end |
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