Veranschaulichung der Entstehung, Ausbreitung und Überlagerung zweier Kreiswellen (eine hyperbolische Interferenz)
Author: Chijner Roman,Heidelberg
Date: 22. September 2011

► 2D oder 3D Ansicht

P_1 move me

P_2 move me

P_3 move me

P_4 move me

S1 move me

S2 move me

Interferenztyp

R_{X}

R_{Y}

R_{Z}

1

$\scalebox{1.4} \text{\copyright Roman Chijner, Heidelberg 2013}$
$\scalebox{1.3} \text{     Gestaltungshilfe:Werner Knobloch}$

O

X

Z

Y

X

Y

Z

X

Y

Z

►Achsen

►Animation "Zeit"

Überlagerung von Wellen, die von zwei gleichzeitig schwingenden Wellenzentren ausgehen 

Überlagerung des reflektierten mit dem einfallenden Wellenzug

► Punktquellen  S_1, S_2

► Beabachtungspunkte P_1, P_2, P_3, P_4

$\scalebox{1.5}{\thicksim} Wellenbäuche

$\scalebox{1.5}{\cdot }\; \;  Wellenknoten$

x

II←

II←

►

►

►Gitternetz 2

►Gitternetz 1

► "move me"

P_1

P_2

P_3

P_4

Ansicht nur für δ = 0: ortsfeste und zeitunab-
hängige Wellenknoten und Wellenbäuche 

S_1

S_2

Sichtbarkeit

Aufbau

Abbau

N (≤ 90) =:

x_{max}=:

x_{min}=:

y_{max}=:

y_{min}=:

.                                                                                                      .





.

+10

-10

I (für N≤30) oder II

imSkript Oberfläche

dN =:

                                                                                                              .







.

s_z =:

δ =:

T =:

λ =:

Δφ =:

t =:

►

zeit- und ortsabhängige Wellenknoten

► Wellenparameter:

► 3D-Flächen:

I. In einem Durchgang (für N≤30)

► Grenzen:

► Winkellage:

 Interferenzmuster gedämpfter Kreiswellen 

Kacheln / Teilschritt:

N>30 nur mit Methode II

γ = 0.2617993877991494 rad (slider from 0*° to 6.283185307179586, step 0.017453292519943295)
α = 5.916666164260777 rad (slider from 0*° to 6.283185307179586, step 0.017453292519943295)
β = 3.3161255787892263 rad (slider from 0*° to 6.283185307179586, step 0.017453292519943295)
d = true
γ0 = If(¬d, γ, 0*°)
α0 = If(¬d, α, 0*°)
β0 = If(¬d, β, 90*°)
a13: ((cos(α0) * sin(β0)) * cos(γ0)) + (sin(α0) * sin(γ0))
P13: (((sin(α0) * sin(β0)) * cos(γ0)) - (cos(α0) * sin(γ0)), (cos(β0) * cos(γ0))) + ((-a13)) / sqrt(8)
a11: (cos(α0) * cos(β0))
P11: ((sin(α0) * cos(β0)), (-sin(β0))) + ((-a11)) / sqrt(8)
a12: ((cos(α0) * sin(β0)) * sin(γ0)) - (sin(α0) * cos(γ0))
P12: (((sin(α0) * sin(β0)) * sin(γ0)) + (cos(α0) * cos(γ0)), (cos(β0) * sin(γ0))) + ((-a12)) / sqrt(8)
P13r' = (0.35355339059327373, 0.35355339059327373)
P12r' = (1, 0)
BP1 = (14.985290901853727, -0.02673070970372511)
BP2 = (4.855012632291535, -0.02673070970372511)
BP3 = (-15.085640382530883, 0.1865383064975845)
BP4 = (-5.061996621069346, 0.0799037983969297)
Ps1 = (-30, 0)
Ps2 = (30, 0)
P_{BP}: {BP1, BP2, BP3, BP4, Ps1, Ps2}
Pl1_{BP} = Sequence((P12r' * x(Element[P_{BP}, i])) + (P13r' * y(Element[P_{BP}, i])), i, 1, 6)
Plxy0 = Sequence((((-y(Element[Pl1_{BP}, i])) * sqrt(8)), x(Element[Pl1_{BP}, i]) - y(Element[Pl1_{BP}, i])), i, 1, 6)
N = 10 (slider from 1 to 90, step 1)
N_y: N
x_{max} = 30 (slider from 0 to 50, step 0.1)
x_{min} = -30 (slider from -30 to 5, step 0.1)
dx: (x_{max} - x_{min}) / N
y_{min} = -30 (slider from -30 to 5, step 0.1)
y_{max} = 30 (slider from -20 to 30, step 0.1)
dy: (y_{max} - y_{min}) / N_y
l1a = Sequence(Sequence[(x_{min} + (i * dx), y_{min} + (j * dy)), i, 0, N], j, 0, N_y)
uv = Join(l1a)
Y1: y(uv)
X1: x(uv)
u_1: x(Element[Plxy0, 5])
v_1: y(Element[Plxy0, 5])
δ = 0.04 (slider from 0 to 1, step 0.01)
c = false
cc: ¬c
Δφ = 0 (slider from 0 to 6.283185307179586, step 0.1)
s_z = 3 (slider from -10 to 35, step 0.1)
t = 0 (slider from 0 to (20 * T), step 0.1)
T = 10 (slider from 0 to 20, step 0.1)
ω: (2 * 3.141592653589793 / T)
λ = 20 (slider from 0 to 100, step 0.1)
c_0: λ / T
k: (2 * 3.141592653589793 / λ)
L = Distance(Element[Plxy0, 5], Element[Plxy0, 6])
t_{12}: (L / c_0 * cc)
u_2: x(Element[Plxy0, 6])
v_2: y(Element[Plxy0, 6])
r_{S2}: sqrt((X1 - u_2)^(2) + (Y1 - v_2)^(2))
A_2 = Sequence(If[t  <  t_{12}, 0, If[Element[r_{S2}, i] ≤ (c_0 * (t - t_{12})), s_z, 0]], i, 1, ((N + 1) * (N_y + 1)))
r_{S1}: sqrt((X1 - u_1)^(2) + (Y1 - v_1)^(2))
A_1 = Sequence(If[Element[r_{S1}, i]  <  (c_0 * t), s_z, 0], i, 1, ((N + 1) * (N_y + 1)))
s_x = 1 (slider from -30 to 55, step 0.1)
s_y = 1 (slider from -50 to 50, step 0.1)
Z1: ((A_1 * exp(((-δ) * r_{S1}))) * sin((k * r_{S1}) - (ω * t))) + ((A_2 * exp(((-δ) * r_{S2}))) * sin((k * r_{S2}) - (ω * (t - t_{12})) + Δφ))
z_{max} = Max(Z1)
z_{min} = Min(Z1)
EckpunktR = 0.8 (slider from 0 to 1, step 0.1)
A2_1 = 2
A1_1 = 1
O = Intersect(yAxis, xAxis)
i = (0, 1)
A = (29.96477342729321, -8.35134746550519)
Gx1: (P11 * 5)
U1x1: (4 * Gx1) + A
Gx3: (P13 * 5)
U1x3: (3 * Gx3) + A
Gx2: (P12 * 5)
U1x2: (3 * Gx2) + A
vX = Vector(A, U1x1)
vY = Vector(A, U1x2)
vZ = Vector(A, U1x3)
Koordx1: A + ((-1 * 5), (-1 * 5))
Koordx3: A + (0, 7)
Koordx2: A + (7, 0)
Kstrecke1 = Vector(A, Koordx1)
Kstrecke2 = Vector(A, Koordx2)
Kstrecke3 = Vector(A, Koordx3)
v_α = 0 (slider from 0 to 5, step 0.1)
e = false
m = 243 (slider from 0 to 450, step 1)
g = true
b = false
v_t = 0.05 (slider from 0 to 0.2, step 0.01)
Δa = -0.6
Almax0 = KeepIf(0 ≤ x ≤ L, Sequence[L / 2 + (L / 2 * cc) + (Δφ * λ / ((4 * 3.141592653589793))) - (max * λ / 2), max, -25, 25, 1])
Almin0 = KeepIf(0 ≤ x ≤ L, Sequence[L / 2 + (L / 2 * cc) + (Δφ * λ / ((4 * 3.141592653589793))) - (min * λ / 4), min, -25, 25, 2])
bu4 = false
bu5_1 = false
Deckkraft = 0.4 (slider from 0 to 1, step 0.1)
j_1 = 2 (slider from 1 to 5, step 1)
L0: L / j_1
xxxl2max11 = KeepIf(abs(x) ≤ L0, Sequence[(i * λ / 2) - (L / 2 * cc) - (Δφ * λ / ((4 * 3.141592653589793))), i, -20, 20])
xxxl2min11 = KeepIf(abs(x) ≤ L0, Sequence[(i * λ / 4) - (L / 2 * cc) - (Δφ * λ / ((4 * 3.141592653589793))), i, -21, 21, 2])
zk3: ((N + 1) * (N_y + 1))
N1 = 5 (slider from 1 to 60, step 1)
a = false
NN_{S1}: floor((N1 + 1) / 2 + 1)
NN_{S2}: floor((N1 + 1)^(2) - (N1 + 1) / 2 + 1)
ylPunkten10 = Sequence(((P11 * Element[X1, i]) * s_x) + ((P12 * Element[Y1, i]) * s_y) + (P13 * Element[Z1, i]), i, 1, (N + 1)^(2))
yylPolyy = Sequence(If[i ≟ (ceil(i / (N + 1)) * (N + 1)), Polygon[(0, 0), (0, 0), (0, 0), (0, 0)], Polygon[Element[ylPunkten10, i], Element[ylPunkten10, i + N + 1], Element[ylPunkten10, i + N + 2], Element[ylPunkten10, i + 1]]], i, 1, ((N + 1) * N))
ylPunkten10a = Sequence(((P11 * Element[X1, i]) * s_x) + ((P12 * Element[Y1, i]) * s_y), i, 1, (N + 1)^(2))
yylPolyya = Sequence(If[i ≟ (ceil(i / (N + 1)) * (N + 1)), Polygon[(0, 0), (0, 0), (0, 0), (0, 0)], Polygon[Element[ylPunkten10a, i], Element[ylPunkten10a, i + N + 1], Element[ylPunkten10a, i + N + 2], Element[ylPunkten10a, i + 1]]], i, 1, ((N + 1) * N))
zk3a: ((N + 1) * N)
Ro: (Z1 - z_{min}) / (z_{max} - z_{min})
Bl: (Y1 - y_{min}) / (y_{max} - y_{min})
rl_{S1} = Sequence(sqrt((x(Element[Plxy0, i]) - u_1)^(2) + (y(Element[Plxy0, i]) - v_1)^(2)), i, 1, 6)
Gr: (X1 - x_{min}) / (x_{max} - x_{min})
rl_{S2} = Sequence(sqrt((x(Element[Plxy0, i]) - u_2)^(2) + (y(Element[Plxy0, i]) - v_2)^(2)), i, 1, 6)
Al_1 = Sequence(If[Element[rl_{S1}, i]  <  (c_0 * t), s_z, 0], i, 1, 6)
Al_2 = Sequence(If[t  <  t_{12}, 0, If[Element[rl_{S2}, i] ≤ (c_0 * (t - t_{12})), s_z, 0]], i, 1, 6)
Zl1: ((Al_1 * ℯ^(((-δ) * rl_{S1}))) * sin((k * rl_{S1}) - (ω * t))) + ((Al_2 * ℯ^(((-δ) * rl_{S2}))) * sin((k * rl_{S2}) - (ω * (t - t_{12})) + Δφ))
PlBPz = Zip(((P11 * a) * s_x) + ((P12 * b) * s_y) + ((P13 * c) * s_z), a, x(Plxy0), b, y(Plxy0), c, Zl1)
f = false
h = false
bu5a = false
PlBPz1 = Element(PlBPz, 1)
PlBPz2 = Element(PlBPz, 2)
PlBPz3 = Element(PlBPz, 3)
PlBPz4 = Element(PlBPz, 4)
Pl2_{BP} = Sequence((P11 * x((((-y(Element[Pl1_{BP}, i])) * sqrt(8)), x(Element[Pl1_{BP}, i]) - y(Element[Pl1_{BP}, i])))) + (P12 * y((((-y(Element[Pl1_{BP}, i])) * sqrt(8)), x(Element[Pl1_{BP}, i]) - y(Element[Pl1_{BP}, i])))), i, 1, 6)
PlBPz5 = Element(PlBPz, 5)
PlBPz6 = Element(PlBPz, 6)
PlBPz5a = Element(Pl2_{BP}, 5)
PlBPz6a = Element(Pl2_{BP}, 6)
Sα = Angle(Line[Ps1, Ps2], xAxis)
vi = true
n_a = 1 (slider from 1 to cea + 1, step 1)
n_e = 3 (slider from 1 to cea + 1, step 1)
dN = 90 (slider from 10 to 150, step 10)
nmina1: ((n_a - 1) * dN) + 1
nmina2: ((n_e - 1) * dN) + 1
cea: ceil(zk3 / dN)
o = false
i_1 = true
v_n = 0.5 (slider from 0 to 1, step 0.1)
Textfeld3 = Textfield(N)
Textfeld4 = Textfield(x_{max})
Textfeld5 = Textfield(x_{min})
Textfeld6 = Textfield(y_{max})
Textfeld7 = Textfield(y_{min})
fi1 = false
j = false
inputBox1 = Textfield(dN)
inputBox2 = Textfield(s_z)
inputBox3 = Textfield(δ)
inputBox4 = Textfield(T)
inputBox5 = Textfield(λ)
inputBox6 = Textfield(Δφ)
inputBox7 = Textfield(t)
aΔ = 5.2
zmaxar = Sequence(Hyr[Ps1, Ps2, Element[xxxl2max11, i]], i, 1, Length[xxxl2max11])
zminar = Sequence(Hyr[Ps1, Ps2, Element[xxxl2min11, i]], i, 1, Length[xxxl2min11])
xminEllipse = KeepIf(abs(x) ≥ L0, Sequence[(i * λ / 2) - (L / 2 * cc) - (Δφ * λ / ((4 * π))) + (λ * Mod[t, T] / T), i, 0, 20])
zminE = Sequence(Ellips[Ps1, Ps2, Element[xminEllipse, i]], i, 1, Length[xminEllipse])
bu5 = false
Zdmax1 = Max(Z1)
Zdmin1 = Min(Z1)
Ro1 = Sequence(If[Element[Z1, i] > 0, Element[Z1, i] / Zdmax1, 0], i, 1, (N + 1)^(2))
Bl1 = Sequence(If[Element[Z1, i]  <  0, Element[Z1, i] / Zdmin1, 0], i, 1, (N + 1)^(2))
Gr1 = Sequence(0.2, i, 1, (N + 1)^(2))
caption2_1 = If(g, "3D: Farbe Oberfläche", "2D: XY-Projektion")
Info007: "II. in " + (cea + 1) + " Teilschritten "
Text8: "n_e= " + n_e + ""
Text9: "n_a= " + n_a + ""
fi0 = false
Info005_1: "α=" + (LaTeX[α0]) + "\; \;\; β= " + (LaTeX[β0]) + "  \;\;\; γ=" + (LaTeX[γ0]) + ""
Text10: "z_{min}= " + z_{min} + "
z_{max}= " + z_{max} + ""
Text4: "Z1=" + (LaTeX[Text[Z1, false]]) + ""
text3_1: " d_{S1,S2}= " + L + "   t= " + t + "   s_z= " + s_z + "   δ = " + δ + "   T = " + T + "   λ = " + λ + "    Δφ = " + Δφ + " "
Info001: "A_1 = A_2 = s_z  - maximale Amplitude
s_z·exp(-δ r)   - Amplitude
k = 2 π/λ    - Wellenzahl
ω = 2 π/T   - Kreisfrequenz
c = λ/T = " + c_0 + "  - Phasengeschwindigkeit
Δφ =  " + Δφ + "   - Phasenverschiebung
d_{S1,S2} = " + L + "  - Ortsabstand (Quellen)
t_{12} =  " + t_{12} + "   - Zeitabstand (Quellen)"
ce: (c * e)
text3_2: "-warten\; bis\; n_a=" + (LaTeX[cea + 1]) + ""

Tool: Hyr
S1 S2 dd
Input: S_1, S_2, aΔ
Output: ee3

Sα1 = Angle(Line[S_1, S_2], xAxis)
LLL = Distance(S_1, S_2)
ee3 = CurveCartesian((((-aΔ) * cosh(t)) * cos(Sα1)) - ((sqrt((LLL / 2)^(2) - aΔ^(2)) * sinh(t)) * sin(Sα1)) + x(S_1 + S_2) / 2, ((aΔ * cosh(t)) * sin(Sα1)) - ((sqrt((LLL / 2)^(2) - aΔ^(2)) * sinh(t)) * cos(Sα1)) + y(S_1 + S_2) / 2, t, -4.5, 4.5)

Tool: Ellips
S1 S2 ddd
Input: S_1, S_2, aΔ
Output: ee4

Sα1 = Angle(Line[S_1, S_2], xAxis)
LLL = Distance(S_1, S_2)
ee4 = CurveCartesian((((-aΔ) * cos(t)) * cos(Sα1)) - ((sqrt((-(LLL / 2)^(2)) + aΔ^(2)) * sin(t)) * sin(Sα1)) + x(S_1 + S_2) / 2, ((aΔ * cos(t)) * sin(Sα1)) - ((sqrt((-(LLL / 2)^(2)) + aΔ^(2)) * sin(t)) * cos(Sα1)) + y(S_1 + S_2) / 2, t, -4.5, 4.5)