16.4: Graphing the Wave Function
Consider the wave equation for a sinusoidal wave moving in the positive x-direction. The wave equation is a function of both position and time. From the wave equation, two different graphs can be plotted.
If a specific time is taken, say t = 0, it means a "snapshot" of the wave is taken, and the obtained graph is the shape of the wave at t=0. This graph is called the displacement versus position graph and represents the displacement of the particle from its equilibrium position as a function of the position. The wavelength can be deduced from this graph. The highest point of the wave from the equilibrium position is known as the crest, and the lowest point is known as the trough. The distance between two consecutive troughs or crests with the same height and the same slope is the wavelength of a wave. Considering the case of a transverse wave on a string, the graph represents the actual shape of the string at an instant in time.
On the other hand, when a specific coordinate is chosen, say x = 0, graphing the wave equation results in a displacement versus time graph. This graph gives the displacement of the particle as a function of time. The period of the wave can be obtained from the graph. The time taken for the particle for one complete oscillation is the wave's period.
In the wave equation, the argument of the cosine function is called the phase of the wave. It is an angular quantity and is measured in radians. The value of phase for any values of x and t determine which part of the sinusoidal cycle is occurring at a particular point and time. For a crest, when the cosine function has a value of 1, the phase could be 0, 2π, 4π, 6π, etc. Conversely, for a trough, when the cosine function has a value of −1, the phase could be π, 3π, 5π, 7π, etc. The phase velocity is the speed at which the wave moves when keeping the phase constant. The expression for the phase velocity is given as follows:
