Understanding the Basics of Waves

At its core, a wave is a disturbance that transmits energy. A single disturbance is a pulse, while a series of periodic pulses is a wave.

Modelling their motion

Waves travel in sinusoidal motion, meaning they can be expressed by cosine. This is because waves cause particles to move in the opposite direction to which they are accelerating, $$ \Delta x \propto -a$$

Defining Variables

$$\omega \text{ is the speed of the wave in } \frac{rad}{s}$$ $$x_0 \text{ is the initial displacement and amplitude in } m$$ $$k \text{ is the spring constant in } \frac{N}{m}$$ $$T \text{ is the time between the period of the wave}$$ $$f \text{ is the frequency between time intervals } T \text{ in Hertz (Hz)}$$

Deriving Formulas

$$\Delta x = x_0 \cos (\omega t)$$ $$\omega = 2 \pi f$$ $${\omega}^2 = \frac{k}{m}$$ $$T = \frac{1}{f} \rightarrow f = \frac{1}{T}$$

Further understanding

Applying calculus to the previous equations as well as prior concepts such as energy, we can derive a whole lot more functions to model the nature of waves.

$$a = -{\omega}^2x$$ $$a_{max} = -{\omega}x_0$$ $$v_{max} = \omega x_0$$

$$v(t) = -x_0 \omega \sin (\omega t)$$ $$a(t) = x_0 \cos (\omega t)$$ $$KE(t) = \frac{1}{2}mx_0^2 {\omega}^2 \sin (\omega t)$$ $${PE}_{elastic}(t) = \frac{1}{2}kx_0^2 {\cos}^2 (\omega t)$$