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Vibrating String Simulation

Vibrating String Simulation
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2023-03-09 17:54
FunctionTrigonometryCoordinate Geometry
This example demonstrates the wave patterns generated by a vibrating string and its varying behaviors. Core math concepts include functions, trigonometry, coordinate geometry, and parametric equations. Construction highlights involve using the Function command to plot functions, controlling parameters like frequency and amplitude via Slider Animation, and utilizing objects such as Locus and Geometric Construction. Suitable for high school and university students studying wave phenomena, function transformations, and the effects of parameters on graphs, as well as for physics or mathematics courses.

GeoGebra Commands & Parameters

  1. Function((((A1 * k) * sin((pi * (x / 18 + 1 / 2)))) * sin((((2 * pi) * t) * v))), -9, 9) → a1
  2. Function(((((-A2) * k / 2) * sin(((2 * pi) * (x / 18 + 1)))) * sin((((4 * pi) * t) * v))), -9, 9) → a2
  3. Function((((A3^(2) * k / 6) * sin(((3 * pi) * (x / 18 + 3 / 2)))) * sin((((6 * pi) * t) * v))), -9, 9) → a3
  4. Function(((((-A4^(4)) * k / 24) * sin(((4 * pi) * (x / 18 + 2)))) * sin((((8 * pi) * t) * v))), -9, 9) → a4
  5. Function((((A5^(5) * k / 120) * sin(((5 * pi) * (x / 18 + 5 / 2)))) * sin((((10 * pi) * t) * v))), -9, 9) → a5
  6. Function(a1(x) + a2(x) + a3(x) + a4(x) + a5(x), -9, 9) → f
  7. Function(a1(x) + a2(x) + a3(x) + a4(x) + a5(x), -9, 9) → fR
  8. Function((((A1 * k) * sin((pi * (x / 18 + 1 / 2)))) * sin((((2 * pi) * t) * v))), -9, 9) → a1R
  9. Function(((((-A2) * k / 2) * sin(((2 * pi) * (x / 18 + 1)))) * sin((((4 * pi) * t) * v))), -9, 9) → a2R
  10. Function((((A3^(2) * k / 6) * sin(((3 * pi) * (x / 18 + 3 / 2)))) * sin((((6 * pi) * t) * v))), -9, 9) → a3R
  11. Function(((((-A4^(4)) * k / 24) * sin(((4 * pi) * (x / 18 + 2)))) * sin((((8 * pi) * t) * v))), -9, 9) → a4R
  12. Function((((A5^(5) * k / 120) * sin(((5 * pi) * (x / 18 + 5 / 2)))) * sin((((10 * pi) * t) * v))), -9, 9) → a5R
  13. Function((b1(x) + b2(x) + b3(x) + b4(x) + b5(x)) / 2.5 + 5.5, -10, 5.1) → g1

AI Drawing Prompt Reference

Draw a set of time-varying “vibrating string” displacement wave curves over x in [-9, 9]. Define the main displacement as the sum of five components, each component being a product of two sine terms: one sine in x with different multiples of the period (using x/18 plus phase shifts 1/2, 1, 3/2, 2, 5/2) and a corresponding amplitude scaling involving powers of adjustable parameters with alternating signs; the other sine depends on time t with frequency multiples (2πt·v, 4πt·v, 6πt·v, 8πt·v, 10πt·v). Plot two total curves using the same component structure, and also plot one additional reference curve formed from the sum of the same five components with a global vertical shift and scaling, over x in [-10, 5.1].
The prompt above was compiled from the original drawing commands and is for reference only.