## 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).