It is easy.
We need an elastic rope or string of a few meters. (Sewing rubber band is fine).
Someone else holds one end steady.
We hold the other end at a light tension and we oscillate our hand sideways.
We find the first reasonance which is the fundamental mode.
We then oscillate our hand at roughly double the frequency in order to see the 2nd mode of vibration or 2nd harmonic or 2nd overtone.
At roughly 3 x (fundamentally frequency) we see the 3rd overtone etc.
Both partners in the experiment can feel the wave bouncing back and forth. As it finds impedance mismatching (our hand), it is reflected back and the back and forth reflections create the vibration of the string.
Tension is light. The speed of travel of the wave in rubber band at light tension is slow.
So we can see and feel everything.
If fundamental is A, 2nd overtone is roughly A octave above. 3rd overtone is E octave above A. 4th overtone is A double octave. Fifth overtone is C sharp above double octave...
When we play A on a piano all these permissible modes of vibration take part and we hear a rich tone. Each component of this complex waveform decays at a different rate.
Additional information:
The word roughly was used in this post. The wave may travel at a slightly different speed inside the string for different frequency. Why should the speed be the same? For example steel strings are elastic but also have stiffness. The second overtone of a piano string may consequently be a higher frequency than x2 fundamental.
At the same time the human brain wants a little more than x2 fundamental for a pleasing result.
If this "Inharmonicity" between string and human brain is matched for every note by the corect scaling of a piano we have a great piano. An evolution of hundreds of years leading to the great Steinway, Pleyel, Bechstein etc.
Further reading:
Musical Acoustics - Donald Hall
A Level Physics - Roger Muncaster - 3rd edition
Conceptual Physics - Hewitt
The Pleiades tuning - euroelectron
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