I thought I'd make a post about some things that I've found interesting over the last couple of years, combining my love for bass guitar and my area of study (structural engineering), which some of you may find interesting. Some of it may actually be really obvious, but hopefully it's still a little interesting to a few of you
One of the first things I noticed when I started playing bass was that the scale length of all the strings was generally different, not exactly 34 inches. And almost all of the basses that I'd played or seen others play had a similar thing in common, that the E string had a longer scale length than the A string, and so on, with the G string having the shortest scale length. Of course, this only differed by a small fraction of an inch but it seemed to be pretty consistent among all basses.
My explanation for this is actually found in a lot of structural engineering applications. The 'effective length' of the string is actually shorter than the scale length because of the small amount of flexural stiffness of the string near its ends, where it touches the bridge/nut. For very thin strings like the G string, this flexural stiffness is pretty negligible (it's very easy to bend a G string into a tight coil) but for thicker strings, the flexural stiffness becomes more noticeable (imagine bending an E or B string into the same radius coil as a G string). The fact that the ends of the string are restrained by the bridge/nut to some extent means that the 'effective length' of the string is shortened, between two points of contraflexure which occur somewhere close to (but not coincident with) the nut and the bridge. And of course it's the effective length of the string which determines its frequency (along with linear mass, etc). The same principle applies to columns in buildings: where a column is rigidly connected on either end to slabs, for example, the points of contraflexure of the columns occur somewhat closer to the centre of the column, and it's buckling behaviour depends on its effective length, rather than its actual length. The greater the degree of flexural stiffness and rigidity at the connection, the smaller the effective length.
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In the image you can see that for the case of a relatively flexible string, the effective length would be close to 1.0 times its actual length, but somewhat less than that due to its end stiffness. If the string were exceptionally stiff and rigidly connected to the nut/bridge, you would have a case like the first. In reality it's somewhere in between.
Another thing I've found interesting is the way tones are formed by the superposition of different frequencies when we pluck a string. Any sort of structure has an infinite number of modes in which it can vibrate, and bass strings are no exception. We are all familiar with the first mode of vibration - this is where the string vibrates up and down along its entire length, much like looking at a skipping rope from the side. The first mode of vibration produces the fundamental frequency, essentially the note that you intended to play. However, it's practically impossible to pluck a string and not induce other modes of vibration, especially since we pluck the strings so close to their ends. We can also hear the second mode of vibration, where the string oscillates with a node at its middle, producing a frequency which is 2x that of the fundamental frequency, and of course this is the next octave up. The reason the second mode of vibration is not as audible is because modes with lower frequencies are generally easier to excite. Higher modes corresponding to different harmonic frequencies are also audible, but to a lesser extent.
The resultant tone that you hear is the contribution of infinitely many modes of vibration, each with a slightly lesser intensity as the frequency increases. And of course, the tone also depends on the position of the pick-ups which are different for different basses, as well as the position of your plucking fingers. This all has a lot of application in structural engineering as well; a building will vibrate according to its natural frequencies, much like a string. The resultant vibration in a building is a superposition of contributions of its infinitely many modes of vibration, however, much like a string, the lower-frequency vibrations are easier to excite and are the most noticeable.
The case of a vibrating string is similar to the simply supported beam on the right of the image, although the beam relies on flexural stiffness as opposed to the string's axial stiffness, however the same modes of vibration occur.
I'd be happy to elaborate on anything, or answer any sort of questions that relate to both bass and structural engineering
One of the first things I noticed when I started playing bass was that the scale length of all the strings was generally different, not exactly 34 inches. And almost all of the basses that I'd played or seen others play had a similar thing in common, that the E string had a longer scale length than the A string, and so on, with the G string having the shortest scale length. Of course, this only differed by a small fraction of an inch but it seemed to be pretty consistent among all basses.
My explanation for this is actually found in a lot of structural engineering applications. The 'effective length' of the string is actually shorter than the scale length because of the small amount of flexural stiffness of the string near its ends, where it touches the bridge/nut. For very thin strings like the G string, this flexural stiffness is pretty negligible (it's very easy to bend a G string into a tight coil) but for thicker strings, the flexural stiffness becomes more noticeable (imagine bending an E or B string into the same radius coil as a G string). The fact that the ends of the string are restrained by the bridge/nut to some extent means that the 'effective length' of the string is shortened, between two points of contraflexure which occur somewhere close to (but not coincident with) the nut and the bridge. And of course it's the effective length of the string which determines its frequency (along with linear mass, etc). The same principle applies to columns in buildings: where a column is rigidly connected on either end to slabs, for example, the points of contraflexure of the columns occur somewhat closer to the centre of the column, and it's buckling behaviour depends on its effective length, rather than its actual length. The greater the degree of flexural stiffness and rigidity at the connection, the smaller the effective length.
[Invalid or Expired Link Removed]
In the image you can see that for the case of a relatively flexible string, the effective length would be close to 1.0 times its actual length, but somewhat less than that due to its end stiffness. If the string were exceptionally stiff and rigidly connected to the nut/bridge, you would have a case like the first. In reality it's somewhere in between.
Another thing I've found interesting is the way tones are formed by the superposition of different frequencies when we pluck a string. Any sort of structure has an infinite number of modes in which it can vibrate, and bass strings are no exception. We are all familiar with the first mode of vibration - this is where the string vibrates up and down along its entire length, much like looking at a skipping rope from the side. The first mode of vibration produces the fundamental frequency, essentially the note that you intended to play. However, it's practically impossible to pluck a string and not induce other modes of vibration, especially since we pluck the strings so close to their ends. We can also hear the second mode of vibration, where the string oscillates with a node at its middle, producing a frequency which is 2x that of the fundamental frequency, and of course this is the next octave up. The reason the second mode of vibration is not as audible is because modes with lower frequencies are generally easier to excite. Higher modes corresponding to different harmonic frequencies are also audible, but to a lesser extent.
The resultant tone that you hear is the contribution of infinitely many modes of vibration, each with a slightly lesser intensity as the frequency increases. And of course, the tone also depends on the position of the pick-ups which are different for different basses, as well as the position of your plucking fingers. This all has a lot of application in structural engineering as well; a building will vibrate according to its natural frequencies, much like a string. The resultant vibration in a building is a superposition of contributions of its infinitely many modes of vibration, however, much like a string, the lower-frequency vibrations are easier to excite and are the most noticeable.
The case of a vibrating string is similar to the simply supported beam on the right of the image, although the beam relies on flexural stiffness as opposed to the string's axial stiffness, however the same modes of vibration occur.
I'd be happy to elaborate on anything, or answer any sort of questions that relate to both bass and structural engineering
