Hooke’s Law is a principle of physics – a description of how certain interactions in the physical world work. The principle is not as inviolable as the word “law” implies, but since Robert Hooke first articulated it in the seventeenth century, it has proven broadly applicable and useful in many areas. These include fields as diverse as mechanical engineering, seismology, clock making, and our own area of interest, acoustics. Hooke’s law says that the greater the force applied to an elastic material, the more the material will be deformed, and the amount of deformation is proportional to the strength of the applied force. The classic example is that of a spring: the more forcefully you press on a coil spring, the more it will squash down, and the amount of squashing is in direct proportion to how hard you press. Implicit in this also is the converse: the more an elastic material is deformed, the proportionally greater the force with which it pushes back against whatever is deforming it. Thus, the more squashed down the spring is, the more spring-back force it exerts against whatever is pushing it down.

Continuing with the spring example: If after squashing a spring you release it, the spring-back force – which we can call a restoring force – causes the spring to spring back, as if it were seeking to return to its unstressed rest position. But of course it won’t just return to its rest position and stop. Left to do its own thing, it will overshoot as its momentum carries it beyond rest position. Being then deformed in the opposite direction, a restoring force now in the pull-back direction will cause it to slow down, reverse and head back once again toward rest position. Naturally it will overshoot again, and then reverse again. It will repeat the process until its energy has been dissipated through internal friction and other damping factors and it settles into quietude.

This doesn’t only apply with springs. A similar sort of springy overshooting and pulling back underlies most of the vibration that goes on in our highly vibratory universe. Speaking only of musical contexts, both Hooke’s Law and the springy mechanics of vibration just described apply with pretty good accuracy to vibrating strings, drum membranes, marimba bars, kalimba tines, and to the springy way that a column of air behaves within the pipe of a flute or clarinet or saxophone.

In the description above, a key word was proportional. This word indicates that the strength of the displacing force, and correspondingly, the restoring force, increase in a predictable and linear manner as the displacement increases: a doubling of the force corresponds to a doubling of the displacement, and vice versa. This leads to a wonderful consequence of Hooke’s Law, and to see this consequence in action musically we will switch our example now from springs to musical strings. Much like springs, if a stretched string is displaced to one side and released, it will start vibrating in a series of overshoots and returns as just described. And here’s the above-mentioned consequence of Hooke’s Law: No matter how much you initially displace the string (within reason), and no matter how large its resulting vibration is when you release it, it takes the same amount of time to complete one back-and-forth cycle. Perhaps you can picture this: If you pluck the string hard – which is to say, you give it a large initial displacement and then release – then it travels further with each back-and-forth excursion than it would with a softer pluck/smaller initial displacement. But because the restoring force for the larger displacement is proportionally greater, the string also travels faster. As a result it completes each round trip in the same amount of time as it would with a softer pluck. To repeat: harder pluck = greater travel distance and higher speed; softer pluck = lesser distance and lower speed; the two factors compensate for one another and the round trip time is the same. A noteworthy feature of this is that as the vibration dies away and each round trip becomes smaller, the restoring force decreases proportionally; the speed correspondingly decreases, and the round-trip time continues the same. Let me bring in a couple of key words here: the frequency of the vibration is the number of round trips (vibratory cycles) per second; that frequency is a function of the round-trip time. Amplitude is the “size” of the vibration – how large the displacement is and how far from its rest position the string travels with each excursion, corresponding to distance travelled. Using these terms we can restate the thesis by saying that the frequency of the vibrating string remains constant, even as the amplitude diminishes. Frequency, as you probably know, corresponds to musical pitch. So, translated into musical terms, we can say that the vibrating string that started out producing any given note will continue to produce the same note as it sustains and slowly dies away.

That’s not news: obviously, when you pluck a guitar string, it produces one pitch and stays with it as it dies away. But think about this for a moment. For people who are interested in music, this seemingly obscure fact about the way the physical world behaves is huge. If displacement and force didn’t happen to be proportional in most cases, as Hooke’s Law posits, we would live in a seriously weird musical universe. If the relationship were slightly different in one way, then whenever you plucked a musical string, its pitch would gradually rise as its amplitude diminished and the tone died away. If out of balance the other way, the pitch of each plucked note would drop. The whole concept of scales would be thrown into question, because the string tones you produced would never stay settled at a steady pitch; they would constantly be bending up or down. Likewise for marimba bars and kalimba tines. On the violin, the pitch you got from any particular fingering would depend on how loud you happened to play the note, and any time you attempted a crescendo or decrescendo the pitch would slide off on in one direction or the other. In wind instruments too, any change in volume would mean a change in pitch.

It’s fun to speculate what sorts of musics we humans might have developed had we been born into such a universe.

Important to note before closing: In the real world Hooke’s Law doesn’t always apply; not all situations involving force and resulting deformation are Hookean (yes, that word is sometimes used). Stress-and-strain relationships in which Hooke’s proportionality doesn’t rule the day can be called non-linear. Here are a few musical situations in which nonlinear behavior may arise: Strings under very low tension, when plucked vigorously, show noticeable pitch-drop as their amplitude diminishes. Free reeds such as those in harmonicas are known for going flat when amplitude increases – that is, when they are heavily overblown (a fact that blues harmonica players often use to excellent effect). Some tom toms and similar drums drop in pitch after striking.

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