Simple harmonic functions

A function $f$ is called a simple harmonic function (or, harmonic oscillation) if

$\displaystyle f(x) = a \sin \omega(x - \phi)$    or $\displaystyle \quad f(x) = a \cos \omega(x - \phi)$ (1-1)

with $ \omega > 0$.

(S) $ y = a \sin \omega(x - \phi)$


(C) $ y = a \sin \omega(x - \phi)$


Amplitude, period, and phase. The absolute value $ \vert a\vert$ of $a$ represents the largest value (in absolute value) attained by the function $f$, and is called the amplitude of $f$. The interval  $ \left[\phi,\phi+\dfrac{2\pi}{\omega}\right]$ represents one cycle (or, one complete oscillation), and $ \dfrac{2\pi}{\omega}$ is called period of $f$. Note that if $ \phi = 0$ then $f(x)$ is a periodic function with period  $ \dfrac{2\pi}{\omega}$. $ \phi$ is called the phase shift (or, the initial phase) of $f$.

Graph. Each cycle repeats itself for every interval

$\displaystyle \left[\phi+\dfrac{2\pi}{\omega}n
,\phi+\dfrac{2\pi}{\omega}(n+1)\right]$    for $ n = 0,\pm 1, \pm 2, \ldots$.

The graph of one cycle is given by the simple harmonic function (1-1) with phase shift $ \phi = 0$.

Department of Mathematics
Last modified: 2005-09-29