# 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.

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

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