E1: Run the default Gaussian and understand the observation as the solution of the wave equation with two oppositely running waves. Consider that the string is reflecting at its ends.
E2: Choose a = 0.3; now pulses are clearly separated.
E3: Choose the sine function and the base mode with w = 1.
Increase w in integers. The harmonics will appear as standing waves, deflecting perpendicular to the string axis.
E4: Choose a non integer w. Now you will recognize the oppositely running waves. (The axis of the string may be askew). Approach an integer in small steps.
E5: Try the other functions and consider what will be decisive for an interesting tone quality, with overtones and traveling excitation.
E6: In reality a string will be damped by acoustic radiation and by friction. For the long term impression of tone quality it is important how different harmonics will be damped. Normally high harmonics will be damped much stronger than low ones. Thus a single tone may start with a brilliant, overtone rich spectrum and fade to a soft base harmonic.
E7: In the piano for each tone (in the middle and higher range) three strings are hit simultaneously, which are nearly but not exacly tuned to the same frequency. Each one oscillates in two transverse directions; all three are strongly coupled by the air and by the frame. At the same time all other strings will become softly excited at their suiting harmonics, depending on the degree of damping. Consider how complicated the real behavior wil become in time and space. For this reason it is practically impossible to simulate a grand piano by electronic synthesis (an interesting trial is the V-Piano). The common way to simulate it is to copy the sound of a real grand piano by sampling.