It can have but then the modes will not form a standing wave pattern. In simple words, the multiple modes will be very short-lived.

In principle there is nothing that impede the existence of multiple modes of oscillation. 

From the movies, it seems that there is a combination of modes, not a single one: The shape of the waves is not sinusoidal. This happens because the oscillation originates in a non symmetric place.

Another aspect to consider is that, the shape of the oscillation depends on the amplitude of the perturbation, this mean that for large amplitude, the cord is in a non-linear regime.

Moreover, the system is not isolated, but it is coupled with another oscillator, that is the air surrounding the chord. This is not a negligible effect, because the lifetime of oscillation is quite short. 

I think you will find that you are not seeing the actual string vibrations but strobe effects caused by the sample rate of the digital camera.  The strings are not bending in that rapid pattern, but move from side to side in the time it takes the camera to scan a small distance up the string (actually across its own image plane).  Similarly there are not waves moving along the string in the way it looks.  These are all to do with the sampling in the camera - just like wheels turn backwards in some films.  The A string has a frequency of 110 Hz.  The camera only looks at it 30 times a second, it does not quite four vibrations between each frame.  It will be in sync with the camera at different points in the picture for each frame, that's why you see the pattern travel along the string.  The camera lies - especially digital cameras.

I am interested in Malcolm White's answer because I think it might be key.

I am also thinking about this statement from another physicist which seems to contradict the notion of a finite state. "Multiple modes are clearly visible on harpsichord strings, just use a strobe light to see them."

Now, I know these videos are made using a frequency driver so I assume the strobe light is timed at the frequency of the driver.  Then it always catches the string wave at the same phase. The wave should appear standing.

So my question is if the strobe is set to the other frequency multiples, besides the frequency of the fundamental, so that the strobe matches the overtones, then do we expect to see the other modes, too, without changing the frequency that drives the string..

Does the claim that multiple modes are clearly visible on harpsichord strings really mean that the nodes and waves co-exist at the same point on the string at the same time?

I haven't  found any videos showing that.  I think it is impossible. Should be easy to demonstrate, if it really works that way.  If we drive the string at 110 Hz then run the strobe at 220, 330, 440, etc the we will see these imaginary waves?  I am pretty sure that at least 330 won't be there.

The physicist also says to me 'And, do you forget that traveling waves exist that reflect from the end points?"  But something is wrong there because when a wave reflects, the potential engery is high but the kinetic energy goes to zero.  But at the boundary of the string the angular velocity (momentum) and kinetic energy both go to zero. so where is the momentum for the reflected wave to propagate.  It is a standing wave!  The 2 wave cannot be demonstrated in any experiment, can they?  We are just are supposed to believe the reflecting wave are there?  But then there is a mode lower than the fundamental, which is absurd.

For a starter I could say that the videos where recorded using a rolling shutter, therefore what you're seeing is not the actual movement of the string but several position of it at different times instead. Also, depending on the frame rate and the string's main frequencies the effect will be different.

When you excite a string you do not do it in a determined frequency, you either pluck it, bow it, or hit it, then it resembles more to a Delta function in which all frequencies are present for a short period of time, then the string will have traveling waves in both directions from the attacking point. When pulses reaches the rigid end of the string, there is a downward force exerted by the wall according to Newton's third law, which creates a pulse travelling in the opposite direction with opposite phase.  By adding these waves a stationary pattern emerges and the frequencies that will remain in the string are those of the normal modes of vibration (harmonics). In a string there are three main stages: attack, sustain, and decay, for instance, if you perform a spectral analysis of a string, on those stages you'll see that during the attack there are a lot more frequencies than those during the sustain which, by the way, are harmonics. Not every harmonic decay at the same rate.

Personally once I did some test on guitar strings with strobe lights and with a high speed camera. And I can tell that it is possible to see the modes co-existing in the string, in a determined time the position that is a node for a specific mode is a maximum in a different time. If is that what you're looking for.

I'm not sure why you call the modes "imaginary", they are there! Due to the fact that is not easy to have a video with exact frame rate, it will be hard to see them, but you could force the mode of vibration to see it with the proper strobe light or run a simulation maybe. The easiest way, I think, is by spectrum analysis, they will appear. 

Very good answer Hector!

Thank you Malcolm!

Regards!!