The following simulation shown in Fig.6.3 uses the basic structure of the previous example.

In this simulation some important formulas of physics of the type $y=f\left(x\right)$ are shown, whose parameters have been chosen in such a way, that the variable $x$ and the adjustable parameters correspond to simple, physical quantities. In the second column of the following table the well known formulas from physics are given and the formulation in the simulation syntax is given in the second line. Calling the function random(n) creates a random number between $0$ and $n$. A random distribution with maximum deviation that is symmetric to zero is obtained as random(n)-n/2.

In the third column the meaning of the corresponding variable $x$ and the parameters used are given. Gaussian

Gaussian | $\frac{1}{\sigma \sqrt{\pi}}exp\left[{\left(\frac{x-{x}_{0}}{\sigma}\right)}^{2}\right]$ | $a=$ standard deviation $\sigma $ |

area normalized to 1 | 1/(a*sqrt(pi))*exp(-((x-b)/a)ˆ2) | $b=$ symmetry variable |

Gaussian | $\frac{1}{\sigma \sqrt{\pi}}exp\left[-{\left(\frac{x-{x}_{0}}{\sigma}\right)}^{2}\right]$
+ noise | $a=$ standard deviation $\sigma $ |

with additive noise | 1/(a*sqrt(pi))*exp(-((x-b)/a)ˆ2) + | $b=$ symmetry variable |

random(c/10) - c/20) | $c\u221510$= maximum added | |

noise | ||

Gaussian | $\frac{1}{\sigma \sqrt{\pi}}exp\left[-{\left(\frac{x-{x}_{0}}{\sigma}\right)}^{2}\right](1+$
noise$)$ | $a=$ standard deviation $\sigma $ |

with multiplicative | 1/(a*sqrt(pi))*exp(-((x-b)/a)ˆ2) *(1 | $b=$ symmetry variable |

noise | random(c/10) - c/20)) | $c\u221510$= maximum |

multiplicative noise | ||

Poisson | ${\left(x+{x}_{0}\right)}^{p}exp\left(-\left(x+{x}_{0}\right)\right)\u2215p!$ | $x+{x}_{0}$: expectation value of $p$ |

distribution | (x+10)p^*exp(-x-10)/faculty(p) | $p$= 1,2,3,$\cdots $ |

amplitude | $sin\left({\omega}_{1}t\right)cos\left({\omega}_{2}t\right)$ | $x=\omega t$ angular frequency |

modulation | a*sin(10*x)*cos(b*x) | $10x$: carrier frequency |

bx:modulating frequency | ||

phase | $sin\left({\omega}_{1}t+cos\left({\omega}_{2}t\right)\right)$ | $x=\omega t$ angular frequency |

modulation | a*sin(10*x)*cos(b*x) | $5x$: carrier frequency |

2bx:modulating frequency | ||

frequency | $sin\left({\omega}_{1}t\cdot cos\left({\omega}_{2}t\right)\right)$ | $x=\omega t$ angular frequency |

modulation | a*sin(5*x*cos(b/10*x)) | $5x$: carrier frequency |

b/10 x:modulating frequency | ||

special theory | $\sqrt{1-{\left(\frac{v}{c}\right)}^{2}}$ | x = $\beta $ = v/c |

of relativity: | sqrt(1-xˆ2) | $v$: velocity |

length change | $c$: speed of light | |

special theory | $\frac{1}{\sqrt{1-{\left(\frac{v}{c}\right)}^{2}}}$ | x = $\beta $ = v/c |

of relativity: | 1/sqrt(1-xˆ2) | $v$: velocity |

mass change | $c$: speed of light | |

Planck’s | $\frac{2\pi h{c}^{2}}{{\lambda}^{5}}\frac{1}{{e}^{\frac{hc}{\lambda kT}}-1}$ | $x+2$ = wavelength $\lambda $ in $\mu $m |

radiation law | a*23340/(x+2)ˆ5/(exp(8.958/((x+2)*b))-1) | $a$: scale factor |

$b$: temperature in $1{0}^{3}$ K | ||

For calculating the factorial $p!$ this file contains some special code; in other simulation files this function cannot be used.

Fig.6.3 shows a normalized Gaussian impulse with additive noise superimposed on it, and its integral, that in spite of the perturbation reaches $1$ quite smoothly and accurately. The formula field can be edited, such that functions can be changed or other functions can be filled in.