TABLE OF CONTENTS One of the most important features of linear problems is that solutions can be superimposed: If A and B each are solutions then A + B and A - B are also solutions to the problem. This is explored further in this section. However, adding solutions that are waves of simple sinusoidal shape can lead to solutions of any shape (as in Fourier series). Therefore, a corrolary to this is that the equations for linear waves we have solved supports waves of any shape. Or in other words, the linear wave theory cannot predict the shape of the waves.
This fundamental fact is often overlooked. In order to extract information about the wave motion that includes results about the wave shape (that is the phase variation), it is necessary to include the nonlinear terms we disregarded to obtain the linear form of the equations. This is discussed extensively in Chapters 5 - 8.
Adding even just two sinusoidal wave motions (not to speak of several) of different amplitude and frequency very quickly leads to complicated solutions in terms of the phase variation for the waves. Usually, a computer is required to analyze e.g. the time or space variation of such general solutions. In the following, we only focus on two important examples: standing waves and wave groups. In addition, a brief overview is given of the problem formulation and theory for arbitrarily many wave components (wave spectra).
