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