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The Doppler effect is the shift in the frequency of a wave observed when the source of the wave (or the medium through which the wave travels) is moving relative to the observer. We’re most familiar with the Doppler effect as it appears in sound waves traveling through air (i.e., pressure waves) – think of how the pitch of a siren drops as an emergency vehicle passes you. The first GIF shows a stationary source and the second shows a source moving to the right at 40% the speed of sound. Notice in the second GIF how the wavefronts are closer together in front of the source (producing a higher frequency) and further apart behind it (producing a lower frequency). The Doppler effect is interesting in its own right, but things get much more exciting when the source travels at speeds greater than or equal to the speed of sound. When the source travels at the speed of sound (GIF 3) the source will always be at the leading edge of the waves it produces, and when traveling faster than the speed of sound (GIF 4), the source will always be in front of the waves it produces. In both of these cases, notice how the waves overlap with each other. The high pressure areas of each wave constructively interfere and produce a region of extremely high pressure (much higher than in the surrounding areas). This rapid rise in air pressure is a shock wave, and the sound associated with it is a sonic boom. In each of the GIFs above we see the radiating wavefronts on the left, and the pressure distribution and interference of the waves on the right. Mathematica code posted here. Additional source not linked above. ...

A chaotic system is one in which infinitesimal differences in the starting conditions lead to drastically different results as the system evolves. Summarized by mathematician Edward Lorenz, “Chaos [is] when the present determines the future, but the approximate present does not approximately determine the future.” There’s an important distinction to make between a chaotic system and a random system. Given the starting conditions, a chaotic system is entirely deterministic. A random system, on the other hand, is entirely non-deterministic, even when the starting conditions are known. That is, with enough information, the evolution of a chaotic system is entirely predictable, but in a random system there’s no amount of information that would be enough to predict the system’s evolution. The simulations above show two slightly different initial conditions for a double pendulum — an example of a chaotic system. In the left animation both pendulums begin horizontally, and in the right animation the red pendulum begins horizontally and the blue is rotated by 0.1 radians (≈ 5.73°) above the positive x-axis. In both simulations, all of the pendulums begin from rest. For more information on how to solve for the motion of a double pendulum, check out my video here. Mathematica code posted here. ...

A couple of weeks ago I made a post about the classical three-body problem, which involves determining the motion of three masses interacting via gravity over time. In that simulation, the three masses were restricted to a plane. Even though we live in three spatial dimensions, a two-dimensional model for celestial orbits isn’t such a bad approximation – the orbits of masses in celestial system are often within a few degrees of the same plane. In the simulation above I’ve built in a third spatial dimension and modeled a system in which the masses do not stay within the same plane. In this simulation, the green and red bodies are 9 and 13 times more massive than the blue body, respectively. The Mathematica code for this one was a bit long, so I’ve posted it here. ...

An orientable surface is a surface on which it’s possible to make a consistent definition of direction. Most surfaces we encounter – like spheres, planes, and tori (doughnut shapes) – are orientable. When visualized in three dimensions, orientable surfaces have two distinct sides. Non-orientable surfaces, on the other hand, have only one side. From Wikipedia, “The essence of one-sidedness is that [an] ant can crawl from one side of the surface to the ‘other’ without going through the surface or flipping over an edge, but simply by crawling far enough.” At any point on a non-orientable surface it’s impossible to uniquely define, for example, the “clockwise” direction. The GIFs above show two examples of non-orientable surfaces: a Klein bottle and a Möbius strip. Mathematica code [Klein Bottle]: xk[u_, v_] := (-2/15)*Cos[u]*(3*Cos[v] - 30*Sin[u] + 90*Cos[u]^4*Sin[u] - 60*Cos[u]^6*Sin[u] + 5*Cos[u]*Cos[v]*Sin[u]) yk[u_, v_] := (-15^(-1))*Sin[u]*(3*Cos[v] - 3*Cos[u]^2*Cos[v] - 48*Cos[u]^4*Cos[v] + 48*Cos[u]^6*Cos[v] - 60*Sin[u] + 5*Cos[u]*Cos[v]*Sin[u] - 5*Cos[u]^3*Cos[v]*Sin[u] - 80*Cos[u]^5*Cos[v]*Sin[u] + 80*Cos[u]^7*Cos[v]*Sin[u]) zk[u_, v_] := (2/15)*(3 + 5*Cos[u]*Sin[u])*Sin[v] kb[u_, v_] := {xk[u, v], yk[u, v], zk[u, v]} Manipulate[ParametricPlot3D[kb[u, v], {u, 0, umax}, {v, 0, 2*Pi}, PlotRange -> {{-1.8, 2}, {0, 4.5}, {-0.75, 0.75}}, Axes -> False, Boxed -> False, PlotStyle -> {Opacity[0.65]}, Mesh -> {20, 11}, MeshStyle -> Directive[Gray, Opacity[0.65], Thickness[0.003]]], {umax, 0.001, Pi}] Mathematica code [Möbius Strip]: xm[u_, v_] := (1 + (v/2)*Cos[u/2])*Cos[u] ym[u_, v_] := (1 + (v/2)*Cos[u/2])*Sin[u] zm[u_, v_] := (v/2)*Sin[u/2] ms[u_, v_] := {xm[u, v], ym[u, v], zm[u, v]} Manipulate[ParametricPlot3D[ ms[u, v], {u, 0, umax}, {v, -1, 1}, PlotRange -> {{-1.1, 1.5}, {-1.5, 1.5}, {-0.5, 0.5}}, PlotStyle -> {Opacity[0.65]}, Axes -> False, Boxed -> False, Mesh -> {20, 5}, MeshStyle -> Directive[Gray, Opacity[0.65], Thickness[0.003]]], {umax, 0.001, 2*Pi}] ...

Given the starting positions, velocities, and masses of three objects interacting via gravity, the classical three-body problem involves determining the motions of the three particles throughout time. What’s cool about the three-body system is that it’s impossible to solve for the motions of the objects exactly. That is, we can’t write down an equation that describes the system. Instead of finding an exact solution, we solve the system numerically, which amounts to finding an accurate approximation. The three-body problem is an example of a chaotic system, meaning that even a slight change in the starting conditions drastically changes the time-evolution of the system. The GIF above shows a planar (i.e., two-dimensional) three-body system. Mathematica code: G = 1; time = 30; mA = 1; xA0 = 0; yA0 = 0; vxA0 = 0; vyA0 = 0; mB = 1; xB0 = 1; yB0 = 0; vxB0 = 0; vyB0 = 0; mC = 1; xC0 = 0; yC0 = 0.8; vxC0 = 0; vyC0 = 0; soln1 = NDSolve[ {mA*Derivative[2][xA][t] == -((G*mA*mB*(xA[t] - xB[t]))/((xA[t] - xB[t])^2 + (yA[t] - yB[t])^2)^(3/2)) - (G*mA*mC*(xA[t] - xC[t]))/((xA[t] - xC[t])^2 + (yA[t] - yC[t])^2)^(3/2), mA*Derivative[2][yA][t] == -((G*mA*mB*(yA[t] - yB[t]))/((xA[t] - xB[t])^2 + (yA[t] - yB[t])^2)^(3/2)) - (G*mA*mC*(yA[t] - yC[t]))/((xA[t] - xC[t])^2 + (yA[t] - yC[t])^2)^(3/2), mB*Derivative[2][xB][t] == -((G*mB*mC*(xB[t] - xC[t]))/((xB[t] - xC[t])^2 + (yB[t] - yC[t])^2)^(3/2)) - (G*mB*mA*(xB[t] - xA[t]))/((xB[t] - xA[t])^2 + (yB[t] - yA[t])^2)^(3/2), mB*Derivative[2][yB][t] == -((G*mB*mC*(yB[t] - yC[t]))/((xB[t] - xC[t])^2 + (yB[t] - yC[t])^2)^(3/2)) - (G*mB*mA*(yB[t] - yA[t]))/((xB[t] - xA[t])^2 + (yB[t] - yA[t])^2)^(3/2), mC*Derivative[2][xC][t] == -((G*mC*mA*(xC[t] - xA[t]))/((xC[t] - xA[t])^2 + (yC[t] - yA[t])^2)^(3/2)) - (G*mC*mB*(xC[t] - xB[t]))/((xC[t] - xB[t])^2 + (yC[t] - yB[t])^2)^(3/2), mC*Derivative[2][yC][t] == -((G*mC*mA*(yC[t] - yA[t]))/((xC[t] - xA[t])^2 + (yC[t] - yA[t])^2)^(3/2)) - (G*mC*mB*(yC[t] - yB[t]))/((xC[t] - xB[t])^2 + (yC[t] - yB[t])^2)^(3/2), xA[0] == xA0, yA[0] == yA0, Derivative[1][xA][0] == vxA0, Derivative[1][yA][0] == vyA0, xB[0] == xB0, yB[0] == yB0, Derivative[1][xB][0] == vxB0, Derivative[1][yB][0] == vyB0, xC[0] == xC0, yC[0] == yC0, Derivative[1][xC][0] == vxC0, Derivative[1][yC][0] == vyC0 }, {xA, yA, xB, yB, xC, yC},{t, 0, time}, MaxSteps -> 100000] x1[t_] := Evaluate[xA[t] /. soln1[[1,1]]] y1[t_] := Evaluate[yA[t] /. soln1[[1,2]]] x2[t_] := Evaluate[xB[t] /. soln1[[1,3]]] y2[t_] := Evaluate[yB[t] /. soln1[[1,4]]] x3[t_] := Evaluate[xC[t] /. soln1[[1,5]]] y3[t_] := Evaluate[yC[t] /. soln1[[1,6]]] Manipulate[Show[ {ParametricPlot[ {{x1[t], y1[t]}, {x2[t], y2[t]}, {x3[t], y3[t]}}, {t, tmax - 0.5, tmax}, Axes -> False, PlotRange -> {{-0.55, 1.45}, {-0.55, 1.08}}, PlotStyle -> {Red, Green, Blue}, GridLines -> {Table[0.25*x + 0.07, {x, -100, 100}], Table[0.25*y + 0.01, {y, -100, 100}]}, GridLinesStyle -> Directive[LightGray]]}, {Graphics[{Opacity[0.7], EdgeForm[Directive[Black]], Red, Disk[{x1[tmax], y1[tmax]}, 0.03], Green, Disk[{x2[tmax], y2[tmax]}, 0.03], Blue, Disk[{x3[tmax], y3[tmax]}, 0.03]}]}, ImageSize -> 600], {tmax, 6.05, 16.05}] ...