A wave is a periodic disturbance that travels through a medium. Each
particle in the medium oscillates back and forth but does not change position permanently.
In this way, energy is transfered from particle to particle through the medium.
Definition of waves
Examples of waves:
We can model a wave, frozen at some particular time, as a graph with the following shape:
Figure 2.1 


In this graph, the horizontal direction indicates position in the medium and the verticle direction indicates the displacement of particles (the amount of disturbance) at that position.
The dashed line drawn through the center of the diagram represents the equilibrium or rest position of the string. This is the position that the string would assume if there were no disturbance moving through it. Once a disturbance is introduced into the string, the particles of the string begin to vibrate upwards and downwards. At any given moment in time, a particle on the medium could be above or below the rest position. Points 1 and 2 on the diagram represent two of the crests of this wave. A crest of a wave is a point on the medium which exhibits the maximum amount of positive displacement from the rest positon. Points 3 and 4 on the diagram represent two of the troughs of this wave. A trough of a wave is a point on the medium which exhibits the maximum amount of negative displacement from the rest positon.
Amplitude 
Wavelength 
In the diagram above, the wavelength is the distance from 1 to 2, or the distance from 3 to 4. Any one of these distance measurements would suffice in determining the wavelength of this wave. Wavelength is commonly designated by the greek letter λ.
Frequency 
The frequency of a wave refers to how often the particles of the medium vibrate when a wave passes through the medium. In other words, frequency is the number of complete vibrational cycles of a medium per a given amount of time. Given this definition, it is reasonable that the quantity frequency would have units of cycles/second, waves/second, vibrations/second, or something/second. Another unit for frequency is the Hertz (abbreviated Hz) where 1 Hz is equivalent to 1 cycle/second. In our formulas, we will designate the symbol ω to mean frequency.
Speed 
Period 
Relationships between properties 
Wave equation 
To model waves, we start with the equation y = cos(x). The cosine function has a wavelength of 2Π and an amplitude of 1 as the following figure indicates.
Figure 2.2 


But we want to scale it horizontally to have any desired wavelength. To do this we must first scale by 1/2Π to reduce its wavelength to 1; then we scale by λ where λ is the desired wavelength. The combined effect of these two operations is to scale by λ/2Π. From Part I we know that scaling f(x) horizontally by a will transform the function into f(x/a). We can now write our function as:
y = cos(2Π/λ * x).
A wave is a function that moves in time. Recall from Part I that to displace any function f(x) to the right, you just change it's argument from x to xa, where a is a positive number. If we let a = ct, where c is positive and t is time, then the displacement will increase with time. Hence cos(2Π/λ * (x  ct)) represents a forward propagating wave. Similarly cos(2Π/λ * (x + ct)) represents a backward propagating wave. Since c defines how much our wave is moved for each t, we can call c our speed. For simplicity, we will allow c to be negative as well as positive, so that we can describe waves moving in either direction using only the first formula. This new c is velocity since it expresses direction as well as speed. Our function so far is:
cos(2Π/λ * (x  ct))
Finally, we will take into account amplitude. Recall that the amplitude of a wave is the magnitude of its maximum displacement from a mean value. To increase the amplitude, we would draw our wave as having higher crests and lower troughs, or simply scale our equation vertically by some amount greater than 1. To scale a function f(x), we multiply it by some amount a (the amount we are scaling by) to get af(x). Since the amplitude of the equation we are starting with is 1, we need only scale it by the desired amplitude A. Applying this to our equation we get:
A * cos(2Π/λ * (x  ct)).
The resulting graph will look like:
Figure 2.3