Today I was reading Mr. Musk and his latency shrinking from 550km orbit towards 350km orbit Star Link. My idea for the simmulation was a string that bounces through the jar full of balls. The thing is, we have similar problem among molecules in planet atmoshpere like in static friction. Imagine a box being thrown by hand, then the first pull must be with bigger force. It would be wise to create the first energy bounce by smaller waves to minimize this static friction problem.

As we see left energy spectral area is wider than right ones. It means the bouncer of radiation of Star Link must the first time go through Wan Der Waals bonds and these vapor bonds accepts energy as we know there is not only oxygen, nitrogen, but also hydrogen and steam in the atmosphere.

Now animation of this problem: https://hrubos.tech/simulacie/latency_elastic_wire_bounce.mp4

If we wanna see this slower animation, we still can, and the static first spectrum is present, then as the beam overcomes static friction, the energy is smaller within connection: https://hrubos.tech/simulacie/latency2_elastic_wire_bounce.mp4

using GLMakie
using LinearAlgebra
using Random

# =========================
# PARAMETRE
# =========================

Random.seed!(11)

N = 40
R = 0.04

m_ball = 1.0
I_ball = 0.5 * m_ball * R^2

dt = 0.004
T = 8.0

# pružina drôtu
Nx = 80
kx = 900.0        # tuhosť medzi bodmi
dx = 1.0 / (Nx-1)

wire_mass = 0.3

# kolízie
K = 12000.0
μ = 0.35

# =========================
# GULE (p, v, ω)
# =========================

balls = [
    (
        Point2f(0.2 + 0.6rand(), 0.1 + 0.8rand()),
        Point2f(0, 0),
        0.0
    )
    for _ in 1:N
]

# =========================
# PRUŽNÝ DRÔT (2D VLNA)
# =========================

wire_x = range(0, 1, Nx)
wire_y = [0.5 + 0.1*sin(6*x) for x in wire_x]
wire_v = zeros(Float64, Nx)

function wire_height(xi)
    wire_y[xi]
end

# =========================
# OBSERVABLES
# =========================

ball_obs = Observable(Point2f[])
wire_obs = Observable(Point2f[])

# =========================
# FIGÚRA
# =========================

fig = Figure(size=(1500,700))

ax = Axis(fig[1,1],
    title = "Elastic wire bouncing through granular field",
    aspect = DataAspect()
)

ax2 = Axis(fig[1,2],
    title = "System energy proxy",
    xlabel = "t",
    ylabel = "E"
)

limits!(ax, 0, 1, 0, 1)

E_hist = Float64[]
t_hist = Float64[]

E_obs = Observable(Point2f[])

scatter!(ax, ball_obs, markersize=18)
lines!(ax, wire_obs, linewidth=3)
lines!(ax2, E_obs)

# =========================
# FUNKCIE
# =========================

function update_wire!()
    # spring force (discrete laplacian)
    new_acc = zeros(Float64, Nx)

    for i in 2:Nx-1
        left = wire_y[i-1]
        mid  = wire_y[i]
        right= wire_y[i+1]

        new_acc[i] = kx * ((left + right - 2*mid))
    end

    for i in 2:Nx-1
        wire_v[i] += new_acc[i] * dt
        wire_v[i] *= 0.99
        wire_y[i] += wire_v[i] * dt
    end
end

function update_balls!()
    for i in 1:N
        p, v, w = balls[i]

        v += Point2f(0, -0.0012)

        if p[1] < 0.05 || p[1] > 0.95
            v = Point2f(-0.9v[1], v[2])
        end
        if p[2] < 0.05 || p[2] > 0.95
            v = Point2f(v[1], -0.9v[2])
        end

        balls[i] = (p + v, v, w)
    end
end

function collisions!()
    for i in 1:N, j in i+1:N
        p1, v1, w1 = balls[i]
        p2, v2, w2 = balls[j]

        d = p2 - p1
        dist = norm(d)

        if dist < 2R && dist > 1e-6
            n = d / dist
            t = Point2f(-n[2], n[1])

            rel = v2 - v1

            vn = dot(rel, n)
            vt = dot(rel, t)

            Jn = -1.2 * vn
            Jt = -μ * Jn * sign(vt)

            imp = Jn*n + Jt*t

            v1 -= imp / m_ball
            v2 += imp / m_ball

            w1 -= Jt / I_ball
            w2 += Jt / I_ball

            balls[i] = (p1, v1, w1)
            balls[j] = (p2, v2, w2)
        end
    end
end

function wire_ball_interaction!()
    E = 0.0

    for i in 1:N
        p, v, w = balls[i]

        idx = Int(clamp(round(Int, p[1]*(Nx-1))+1, 2, Nx-1))

        wy = wire_y[idx]
        diff = p[2] - wy

        if abs(diff) < R
            force = K * (R - abs(diff))
            dir = sign(diff + 1e-6)

            v += Point2f(0, force * dir * 0.0004)

            balls[i] = (p, v, w)

            # prenos energie do pružiny
            wire_v[idx] += force * dir * 0.0002

            E += force^2
        end
    end

    push!(E_hist, E)
end

# =========================
# MP4
# =========================

frames = Int(T/dt)

record(fig, "elastic_wire_bounce.mp4", 1:frames; framerate=60) do frame

    t = frame * dt

    update_wire!()
    update_balls!()
    collisions!()
    wire_ball_interaction!()

    ball_obs[] = [p for (p,_,_) in balls]

    wire_obs[] = [
        Point2f(wire_x[i], wire_y[i]) for i in 1:Nx
    ]

    push!(t_hist, t)

    E_obs[] = [
        Point2f(t_hist[i], E_hist[i]) for i in eachindex(E_hist)
    ]

    notify(ball_obs)
    notify(wire_obs)
    notify(E_obs)

    sleep(0.001)
end

The second possibility:

using GLMakie
using LinearAlgebra
using Random

# =========================
# PARAMETRE
# =========================

Random.seed!(11)

N = 400
R = 0.04

m_ball = 100.0
I_ball = 0.5 * m_ball * R^2

dt = 0.004
T = 30.0

# pružina drôtu
Nx = 80
kx = 900.0        # tuhosť medzi bodmi
dx = 1.0 / (Nx-1)

wire_mass = 90.3

# kolízie
K = 12000.0
μ = 0.35

# =========================
# GULE (p, v, ω)
# =========================

balls = [
    (
        Point2f(0.2 + 0.6rand(), 0.1 + 0.8rand()),
        Point2f(0, 0),
        0.0
    )
    for _ in 1:N
]

# =========================
# PRUŽNÝ DRÔT (2D VLNA)
# =========================

wire_x = range(0, 1, Nx)
wire_y = [0.5 + 0.1*sin(6*x) for x in wire_x]
wire_v = zeros(Float64, Nx)

function wire_height(xi)
    wire_y[xi]
end

# =========================
# OBSERVABLES
# =========================

ball_obs = Observable(Point2f[])
wire_obs = Observable(Point2f[])

# =========================
# FIGÚRA
# =========================

fig = Figure(size=(1500,700))

ax = Axis(fig[1,1],
    title = "Elastic wire bouncing through granular field",
    aspect = DataAspect()
)

ax2 = Axis(fig[1,2],
    title = "System energy proxy",
    xlabel = "t",
    ylabel = "E"
)

limits!(ax, 0, 1, 0, 1)

E_hist = Float64[]
t_hist = Float64[]

E_obs = Observable(Point2f[])

scatter!(ax, ball_obs, markersize=18)
lines!(ax, wire_obs, linewidth=3)
lines!(ax2, E_obs)

# =========================
# FUNKCIE
# =========================

function update_wire!()
    # spring force (discrete laplacian)
    new_acc = zeros(Float64, Nx)

    for i in 2:Nx-1
        left = wire_y[i-1]
        mid  = wire_y[i]
        right= wire_y[i+1]

        new_acc[i] = kx * ((left + right - 2*mid))
    end

    for i in 2:Nx-1
        wire_v[i] += new_acc[i] * dt
        wire_v[i] *= 0.82
        wire_y[i] += wire_v[i] * dt
    end
end

function update_balls!()
    for i in 1:N
        p, v, w = balls[i]

        v += Point2f(0, -0.0012)

        if p[1] < 0.05 || p[1] > 0.95
            v = Point2f(-0.9v[1], v[2])
        end
        if p[2] < 0.05 || p[2] > 0.95
            v = Point2f(v[1], -0.9v[2])
        end

        balls[i] = (p + v, v, w)
    end
end

function collisions!()
    for i in 1:N, j in i+1:N
        p1, v1, w1 = balls[i]
        p2, v2, w2 = balls[j]

        d = p2 - p1
        dist = norm(d)

        if dist < 2R && dist > 1e-6
            n = d / dist
            t = Point2f(-n[2], n[1])

            rel = v2 - v1

            vn = dot(rel, n)
            vt = dot(rel, t)

            Jn = -0.4 * vn
            Jt = -μ * Jn * sign(vt)

            imp = Jn*n + Jt*t

            v1 -= imp / m_ball
            v2 += imp / m_ball

            w1 -= Jt / I_ball
            w2 += Jt / I_ball

            balls[i] = (p1, v1, w1)
            balls[j] = (p2, v2, w2)
        end
    end
end

function wire_ball_interaction!()
    E = 0.0

    for i in 1:N
        p, v, w = balls[i]

        #idx = Int(clamp(round(Int, p[1]*(Nx-1))+1, 2, Nx-1))
        idx = clamp(Int(floor(p[1] * (Nx-1)) + 1), 2, Nx-1)
        wy = wire_y[idx]
        diff = p[2] - wy

        if abs(diff) < R
            force = K * (R - abs(diff))
            force = min(force,50.0)
            dir = sign(diff + 1e-6)

            v += Point2f(0, force * dir * 0.0004)

            balls[i] = (p, v, w)

            # prenos energie do pružiny
            wire_v[idx] += force * dir * 0.0002

            #E += force^2 + 1e-6
            E = sum(0.5*m_ball*(v[1]^2 + v[2]^2) for (_, v, _) in balls)
        end
    end

    push!(E_hist, E)
end

# =========================
# MP4
# =========================

frames = Int(T/dt)

record(fig, "elastic_wire_bounce.mp4", 1:frames; framerate=60) do frame

    t = frame * dt

    update_wire!()
    update_balls!()
    collisions!()
    wire_ball_interaction!()

    ball_obs[] = [p for (p,_,_) in balls]

    wire_obs[] = [
        Point2f(wire_x[i], wire_y[i]) for i in 1:Nx
    ]

    push!(t_hist, t)

    E_obs[] = [
        Point2f(t_hist[i], E_hist[i]) for i in eachindex(E_hist)
    ]
    ylims!(ax2, 0, maximum(E_hist) + 1e-6)
    notify(ball_obs)
    notify(wire_obs)
    notify(E_obs)

    sleep(0.001)
end

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