Click here to purchase the entire book in PDF format. Angular frequency For this section, it's important to remember two things. As we saw in Section 1.5, a sound wave is essentially just a ``side view'' of a rotating wheel. Therefore the higher the frequency, the more revolutions per second the wheel turns. Angles can be measured in radians instead of degrees. Also, that this means that we're measuring the angle in terms of the radius of the circle. We now know that the frequency of a sinusoidal sound wave is a measure of how many times a second the wave repeats itself. However, if we think of the wave as a rotating wheel, then this means that the wheel makes a full revolution the same number of times per second. We also know that one full revolution of the wheel is 360$^{\circ }$ or 2$\pi$ radians. Consequently, if we multiply the frequency of the sound wave by 2$\pi$, we get the number of radians the wheel turns each second. This value is called the angular frequency or the radian frequency and is abbreviated $\omega$. $$\omega = 2 \pi f$$ (4.5) The angular frequency can also be used to determine the phase of the signal at any given moment in time. Let's say for a moment that we have a sine wave with a frequency of 1 Hz, therefore $\omega = 2 \pi$. If it's really a sine wave (meaning that it started out heading positive with a value of 0 at time 0 or $t = 0$), then we know that the time in seconds, multiplied by the angular frequency will give us the phase of the sine wave because we rotate $2\pi$ radians every second. This is true for any frequency, so if we know the time $t$ in seconds, then we can find the instantaneous phase using Equation 3.6. $$\varphi = \omega t$$ (4.6) Usually, you'll just see this notated as $\omega t$ as in $\sin ( \omega t )$.