# Can anyone please give some explanation in terms of the frequency domain of the time evolution?

This might be a silly question. But I was puzzled for a long time, even some comments are greatly appreciated.

Is it possible to claim that "All the time domain evolution can be thought of starting from zero frequency"?

Assume every variable (in phase space?) is starting from zero. At $$t=0$$ poles in pairs lie on the horizontal axis, then we claim that $$\omega=0$$ and no oscillatory motion can be excited. But the system has somehow disturbances which pushed the poles away from the horizontal axis (as a complex conjugate pair).

Especially, if someone can provide more insights or perspective in understanding the famous paper "W. E. Cummins, 1962. The Impulse Response Function and Ship Motions. Schiffstechnik, 9, 101-109. Reprinted as David Taylor Model Basin Report 1661"

EDIT: I will try my best to provide some formulations of a specific example later. But for now, let's look at the typical impulse response function. $$h(t)=0$$ for $$t<0$$, so the Fourier transform of $$h(t)$$ becomes \begin{align} H(f)=\int_0^\infty h(t)e^{-i2\pi ft}dt \end{align}

If $$h(t)$$ is sufficiently smooth and converges sufficiently rapidly to zero as $$t\to\infty$$, we can integrate by parts $$k$$ times to obtain \begin{align} H(f)&=\sum_{k=0}^n\frac{h^{(k)}(0)}{(i2\pi f)^{k+1}}+\frac{1}{(i2\pi f)^{n+1}}\int_0^\infty h^{(n+1)}(t)e^{-i2\pi ft}dt\\ &=\frac{h(0)}{i2\pi f}+\frac{h'(0)}{(i2\pi f)^2}+\frac{h''(0)}{(i2\pi f)^3}+\frac{1}{(i2\pi f)^3}\int_0^\infty h'''(t)e^{-i2\pi ft}dt \end{align} Take a simplest linear system with mass $$m$$, damping $$b$$ and stiffness $$k$$ is excited by the external force $$f(t)$$. The equation of motion can be written as \begin{align} m\ddot{x}+b\dot{x}+kx=f(t). \end{align}

• Can you include more details, equations and context? Commented Apr 23 at 11:27
• the Taylor expansion you have for $H(f)$ is a so-called asymptotic expansion, it is not valid for $f=0,$ instead it is true only asymptotically as $f\to \infty$. You can read about this very interesting topic here en.wikipedia.org/wiki/Asymptotic_expansion Commented Apr 23 at 12:35
• Thanks a lot for the comments. Commented Apr 24 at 8:36