Generators:

Noise

Normally distibuted additive noise with specified dispersion. \(U(0,\sigma)\)

Tone

Single (specified) frequency signal of specified duration. \(\sin(2\pi\nu t)\)

Train

Single (specified) frequency signal of specified duration modulated by Gaussian. \(e^{-5(2t/\Delta t-1)^2}\sin(2\pi\nu t)\)

Whistle

Simulated sound of a whistle (frequency \(\nu=100\)) of a train passing an observer (sound speed \(c=1\); minimal distance \(h\)).
\(V^2(t-\Delta t)^2+h^2=\Delta t^2\); \((1-V^2)\Delta t^2+2V^2t\Delta t-V^2t^2-h^2=0\); \(\Delta t=\frac{-V^2t\pm\sqrt{h^2(1-V^2)+V^2t^2}}{1-V^2}=\frac{-V^2t\pm D}{1-V^2}\gt0\);
\(t'=t-\Delta t=\frac{t\mp D}{1-V^2}\); \(\phi=2\pi\nu t'\)
\((1-V^2)\dd{t'}=\left(1\mp\frac{V^2t}{D}\right)\dd{t}=\frac{\pm\Delta t}{D}\dd{t}\); \(A^2\sim E\sim\left\lvert\ddfs{t'}{t}\right\lvert\frac{1}{\Delta t^2}=\frac{1}{D\Delta t}\)
normalize \(A\) by minimal distance: \(A=\frac{h}{\sqrt{D\Delta t}}\)
the whistle sounds while \(t'\in[-p/V,p/V]\)
if \(h^2\ge(V^2-1)p^2\) then only upper sign must be accounted and \(t\in[\sqrt{h^2+p^2}-p/V,\sqrt{h^2+p^2}+p/V]\)
if \(h^2\lt(V^2-1)p^2\) then both signs must be accounted and \(t\in[h\sqrt{1-1/V^2},\sqrt{h^2+p^2}+p/V]\) for upper sign while \(t\in[h\sqrt{1-1/V^2},\sqrt{h^2+p^2}-p/V]\) for lower sign

cases:

\(V\lt1\)

only upper sign meet \(\Delta t\gt0\) restriction; \(t\in \mathbb{R}\); \(D=\sqrt{V^2t^2+h^2(1-V^2)}\)
\(\Delta t=\frac{-V^2t+D}{1-V^2}\); \(\phi=2\pi\nu\frac{t-D}{1-V^2}\); \(A^2=h^2\frac{1-V^2}{-V^2tD+h^2(1-V^2)+V^2t^2}\)

\(V\gt1\)

both signs must be accounted; \(h^2(1-V^2)+V^2t^2\ge0\implies t\ge h\sqrt{1-1/V^2}\); \(D=\sqrt{V^2t^2-h^2(V^2-1)}\)
\(\Delta t_1=\frac{V^2t-D}{V^2-1}\); \(\phi_1=2\pi\nu\frac{-t+D}{V^2-1}\); \(A_1^2=h^2\frac{V^2-1}{V^2tD+h^2(V^2-1)-V^2t^2}\)
\(\Delta t_2=\frac{V^2t+D}{V^2-1}\); \(\phi_2=2\pi\nu\frac{-t-D}{V^2-1}\); \(A_2^2=h^2\frac{V^2-1}{V^2tD-h^2(V^2-1)+V^2t^2}\)

\(V=1\)

singular case; \(t\gt0\); \(D=t\)
\(\Delta t=\frac{t^2+h^2}{2t}\); \(\phi=2\pi\nu\frac{t^2-h^2}{2t}\); \(A^2=\frac{2h^2}{t^2+h^2}\)

BCscOO

Simulated binary compact semiclassically orbiting objects
Consider two objects (BH or NS) of equal masses \(M\) rotating around each other on the same circular orbit of radius \(R\).
Assume that locally equation \((2\omega_lR)^2=\frac{M}{R-2M}\) is true while distantly observed gravitational waves frequency will be \((\omega_wR)^2=\frac{M}{R}\) (double of rotation frequency and time dilation).
Energy loss is described by \(-\ddfs{E}{t}=\frac{128}{5}M^2\omega_l^6R^4=\frac{2}{5}\frac{M^5}{R^2(R-2M)^3}\) with energy estimation as \(E=M\frac{1-M/2R}{\sqrt{1-3M/4R}}\) which give us \(\ddfs{E}{t}=\frac{M^2}{8R^2}\frac{1-3M/2R}{(1-3M/4R)^{3/2}}\ddfs{R}{t}\) and \(-\ddfs{R}{t}=\frac{16}{5}\frac{M^3}{R^3}\frac{(1-3M/4R)^{3/2}}{(1-3M/2R)(1-2M/R)^3}\approx\frac{16}{5}\frac{M^3}{(R-2M)^3}\)
Finally, \((R-2M)^4=-\frac{64}{5}M^3t\), \(R\approx2M(1+\sqrt[4]{-t/M})\), \(\omega_w=\frac{1}{2\sqrt{2}M(1+\sqrt[4]{-t/M})^{3/2}}\), \(A\sim\sqrt{\ddfs{E}{t}}\approx\frac{1}{8}\frac{1}{(1+\sqrt[4]{-t/M})^{5/2}}\)