Wave Functions in 1D

wave motion is defined as the movement of a distortion of a material or medium, where the individual parts or elements of the material only move back-and-forth, up-and-down, or in a cyclical pattern.

Characteristics of waves: wavelength (), amplitude (A), velocity (C or V), and frequency (f).

Wavelength ():

the distance from one crest (or maximum of the wave) to the next crest or maximum.

Waveform showing wavelength and amplitude 

Amplitude (A):

The height of the wave is called its amplitude. Some areas consider the middle of the wave to its peak as the amplitude, while others consider peak-to-peak as the amplitude.

Velocity (C) or (V):

The velocity of the wave is the measurement of how fast a crest is moving from a fixed point. For example, the velocity of water waves can be measured as their speed in a given direction with respect to the land.

Frequency (f):

The frequency of waves is the rate the crests or peaks pass a given point. Frequency is the wavelength divided by the velocity and is designated as cycles (or peaks) per second. Cycles per second is also called Hertz.

Frequency =  Velocity / Wavelength

Another way of writing that is:

Wave Equation in 1D

Here's a wave moving to the right. (Applying the notion scaling and shifting in the wave motions) If the wave shape at time zero has the form y=f(x), then a moving wave will have a shape at later time y=f(x-Vt).

It's the same f, so it's the same shape, just located at a shifted value of x. Left moving waves will have the shape y=f(x+Vt). Any point on the wave travels at (x-Vt=constant), i.e. with velocity V. This is called the "phase velocity", because if you think of f as a sin wave, then this is the velocity that any given phase (like e.g., Pi/2, the peak) moves.

So the simplest form of a wave in 1D is the "harmonic wave",i.e. sinusoidal:

The travelling wave is then

(A is the amplitude.)

We can also use cosine instead of sin (this is the SAME function, just phase shifted).