The sine (or cosine) function defines what is called a regular wave. In order to specify a regular wave we need its amplitude, a, its wavelength, l, its period, T, and in order to be fully specified. also its propagation direction and phase at a given location and time. All these concepts will be introduced below.

Consider the function of the two variables position, x, and time, t:

Convince yourself that this function has the following properties:

The quantity is called the wavenumber and is usually denoted by the letter k. Similarly, is written w (the Greek letter omega ) and called the angular frequency. The unit for k is rad/m and for w rad/s. Note that f = 1/T is called frequency and is measured in Hertz (Hz= 1/s).

The constant a in front of the sine is called the amplitude of the wave. Note that since , . That is, is never larger than the amplitude.

The basic feature of the wave as defined above is that the whole pattern moves along the x-axis as the time changes. Consider for simplicity the point x = 0, t = 0, where h is equal to 0. If now t starts to increase, the points defined by will have the property that for all t. The point where h is 0, , thus moves with velocity l/T along the x-axis. The last property stated above shows this in general.

An additional angle a in the expression is called a phase term.

Do Exercise 2.1:

Do Exercise 2.2:

The argument of the sine, i.e. , is in general called the phase. The phase is often denoted by the letter f (Greek phi). Since phase differences of any multiple of 2p do not matter at all. The phase of the point will be equal to the phase of the point if

that is,

or,

The point on the x-axis which moves with velocity will therefore experience the same phase for all times. Therefore, the velocity is called the phase velocity associated with the wave.

Let us see what happens if we add two general waves, one travelling to the right and one to the left. We first recall the trigonometric identity

)

Consider

This function is a product of a sine and a cosine; the first with x as argument and the second with t. Figure 2 shows a plot of the functions for different t's.

Note that the function is always 0 for . In this case we therefore do not have a travelling wave. However, since we still have a periodic behaviour both in x and t, it is customary to call this case a standing wave.