User:Chad.hogan/method of stationary phase

The method of stationary phase is an approximation to the integration of oscillatory integrals.

The Main Idea
The main idea of stationary phase methods rely on the cancellation of sinusoids with rapidly-varying phase. If many sinusoids have the same phase and they are added together, they will add constructively. If, however, these same sinusoids have phases which change rapidly as the frequency changes, they will add destructively.

An Example
Consider a function

$$f(x,t) = \frac{1}{2\pi} \int_{\mathbb{R}} F(\omega) e^{i(kx - \omega t)} d\omega$$

The phase term in this function, $$\phi = kx - \omega t$$ is "stationary" when

$$\frac{d}{d\omega}(kx - \omega t) \approx 0$$

or equivalently,

$$\frac{d\omega}{dk} \approx \frac{x}{t}$$

Solutions to this equation yield dominant frequencies $$\omega_{dom}(x, t)$$ for a given $$x$$ and $$t$$. If we expand $$\phi$$ in a Taylor series about $$\omega_{dom}$$ and neglect terms of order higher than $$(\omega - \omega_{dom})^2$$,

$$\phi \sim k(\omega_{dom})x - \omega_{dom} t + \frac{x}{2}\frac{d^2k}{d\omega^2}(\omega-\omega_{dom})^2$$

When $$x$$ is relatively large, even a small difference $$\omega-\omega_{dom}$$ will generate rapid oscillations within the integral, leading to cancellation. Therefore we can extend the limits of integration beyond the limit for a Taylor expansion. If we double the real contribution from the positive frequencies of the transform to account for the negative frequencies,

$$f(x, t) = \frac{1}{2\pi} 2 \mbox{Re}\left\{ \exp\left[i\left[k(\omega_{dom})x-\omega_{dom}t\right]\right] \left|F(\omega_{dom})\right| \int_{\mathbb{R}}\exp\left[i\frac{x}{2}\frac{d^2k}{d\omega^2}(\omega-\omega_{dom})^2\right]d\omega\right\}$$

This integrates to

$$f(x, t) \sim \frac{\left|F(\omega_{dom})\right|}{\pi} \sqrt{ \frac{2\pi}{x\left|\frac{d^2k}{d\omega^2}\right|}} \cos\left[ k(\omega_{dom})x - \omega_{dom}t \pm \frac{\pi}{4}\right]$$