Sine Waves

A lot of what we do depends heavily on a few key properties of sine waves (or sinusoids).

Frequency and Wavenumber
The angular frequency of a sinusoid is defined as $$\omega=2\pi/T$$, where $$T$$ is the period of oscillation. Likewise, the wavenumber (spatial frequency) is defined as $$k=2\pi/\lambda$$, where $$\lambda$$ is the wavelength. In this paper, I’ll use $$k$$ and $$x$$ for all of my sine waves, but you could also use $$k$$ with any other spatial variable ($$y, z, r$$, etc.), or you could use $$\omega$$ and $$t$$.

Amplitude and Phase
All sine waves are of the form:



A\cos(kx+\phi) $$

Yes, I’ve just called a cosine a sine wave on purpose. That’s because the sine and cosine functions are actually just special cases of sine waves. Any sine wave can be represented with a sine or cosine function with a proper amplitude and phase.

Phase describes where something is in a cyclic pattern (for example, the phases of the moon). However, because cyclic patterns don’t typically have an inherent starting point, it’s meaningless to talk about &quot;absolute&quot; phase. Phase only has meaning as a relative quantity.

Superposition
Any sine wave can be represented as a superposition of one sine function and one cosine function with proper amplitudes. This can be expressed as


 * $$A\cos(kx+\phi) = B\cos(kx) + C\sin(kx).$$

For any values of $$A$$ and $$\phi$$, there is a corresponding set of values $$B$$ and $$C$$ that will make this equation valid. One way to think of this is that any sinusoid at a given frequency has two degrees of freedom. These may be represented as amplitude and phase or as sine amplitude and cosine amplitude.