According to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. (End of quote.)
For instance, when two specific hammers were struck together they produced consonance and when two different hammers were struck together they produced dissonance.
Two hammers sounding well together, blending well, fitting well, and (more accurately from an etymological point of view) “binding” well are harmonic.
Standing Wave on a string and the notion of harmonics
View in the following video what happens when a specific string is moved up and down about a dozen times per second. https://www.youtube.com/watch?v=oZ38Y0K8e-Y
The mechanical oscillator which moves up and down creates a wave that is propagated on the left, reaches the end of the string and is then propagated on the right. At the same time new wave activity is coming from the right. A wave superposition will occur. When one wave meets the other on a specific string molecule or point, what will this do? If one wave tells it to rise by x and the other wave tells it to rise by x, it will rise by 2x. For specific conditions, for instance in this case, at the frequency of 11 Hz, a specific pattern is created as shown in the picture below, corresponding to what seems to be a vibrating loop: the string is in resonance. The word resonance means actually “resounding”; in this case the string responds to the specific frequency by this loop pattern. This pattern is termed a standing wave.
What happens if the frequency is doubled? At 22Hz we see two loops. What if the frequency is tripled? At 33 Hz we see three loops. And so on and so forth.
The first frequency at which resonance is achieved (11Hz) the one at which the overall length of the string vibrates is called first harmonic or fundamental frequency.
The second frequency at which resonance is achieved is its double (22Hz) and is called second harmonic, the third or its triple (33Hz) is called third harmonic etc.
We notice that there are points where no displacement occurs. These points (which seem like knots that have been tied are remain still) are called nodes. Conversely, the points of maximal displacement are called antinodes.
A nice tutorial explaining how a standing wave is created, using the software that had been used on this site, is available at this link: https://www.youtube.com/watch?v=mh3o8gUu4AE
An animation is available here: http://www.physicsclassroom.com/mmedia/waves/swf.cfm
The blue wave and the green wave added together in accordance with the principle of superposition create a new wave pattern (in black) termed a standing wave.
Standing wave in tubes, cavities, air columns
Why do we hear a sound when we blow in a bottle or a tube?
As mentioned at this link http://newt.phys.unsw.edu.au/jw/Helmholtz.html, when you blow into the bottle you push “a small packet” of air in the bottle and this goes down a bit and presses onto the air that is in the bottle; the latter resists and throws it back out. As it is going out with force, the bottle neighbouring air molecules that had been compressed are "dragged along" and a spring-like motion is initiated. A vibration or oscillation will occur for a few cycles (some back and forth movements) at the neck of the bottle.
Can we find the wavelength of the sound that is produced?
Let us consider the movement behavior of the air molecules inside the bottle. Those that are close to the top/opening are free to move around with maximal displacement. Those that are at the bottom cannot move. It was mentioned at string resonance that points that move with maximal displacement are called antinodes.
If we want to represent how far away right and left can move the air molecules at different locations in the tube, we could refer to this figure from this video.
Below is a graphical representation of the amplitude of the vibration of the molecules (from the same video).
What is the relationship with the wavelength? As the opening corresponds to the first point of maximal displacement it corresponds to one quarter of the full cycle of the wave or to one quarter of the wavelegth.
Let us assume that the length of the tube is L=25cm.
The wavelength of the sound we hear is:
At ambient temperature the sound travels at v=340m/s
Since v= λ* f
f=340m/s / 100m= 340 Hz
Therefore the frequency of the sound we hear is 340Hz.
This can be verified as mentioned in this video using an oscilloscope.
In the same video, a demonstration for finding higher harmonics of a tube is presented. A tube with two open ends has been placed in a bucket of water and and has been filled with water at a level above that of the bucket. A diapason that was struck is providing a specific frequency on the top of the tube. The tube is lifted progressively and at three specific intervals we hear three different resonance frequencies.
The resonances correspons to antinodes, or points of maximal displacement. Here is how this works out. (Refer to the video for more details).
Here is a video presenting sounds linked to different harmonics: https://www.youtube.com/watch?v=Ew0fZh9INbQ
Here is another interesting video:
The shape of the bottle affects the tone because it causes energy to be transferred into harmonics of the fundamental frequency.