This article is intended to explain the actual physics of how a bass guitar works. I'll start with explaining some of the theory involved around the movement of the strings, and how different tones, or pitches, are achieved. Next I'll explain how this movement of the strings interact with the pickups to produce a signal that can be transfered to the amplifier. The last bit of the signal chain, from the amp input to the speaker cone, will not be covered in depth, but will get a superficial explanation. I don't assume any advanced physics skills from the reader, so I'll try to keep things as simple as possible.
Let's start with the strings, and how they work to produce the different tones on the bass. The strings are suspended between the bridge and the nut. The length of the open string (L) is equal to the scale length of your bass. This is normally 34" for 4-string basses, 35" for 5 and 6-string basses, and even more for special models. Some manufacturers, like Traben and Schechter use a 35" scale on many of their 4-string basses, but the norm is 34".
The figure above illustrates how the string is suspended between the bridge and the nut. Let's assume you pluck the string to get it vibrating. If you pluck the A-string (and the bass is tuned to pitch), the string will vibrate at a frequency of 440 Hz. That's 440 times per second. (Actually, the string on your bass will vibrate much slower than that, but since an A at 440 Hz is universally known as a reference pitch for stringed instruments, we'll use that as our guide. Your bass still produces an A, but it's frequency is a few octaves below 440 Hz). If we look at the vibration of the string as a wave form, we can imagine a wave crest passing by every 1/440th of a second. If you throw a rock in a pond, you see waves forming in a circle and moving outward from where the rock hit. The number of crests passing a given point each second is the frequency of the "wave train". Since the strings on a bass are fixed at both ends, a "wave train" can't move along the string. Instead, the strings vibrate in place. This means that strings get a "down and up" motion. The number of times the string hits the top of the "up" motion (or the bottom of the "down" motion") during a time of 1 second, is the frequency. This particular phenomenon allows us to change the pitch by pressing down on the different frets, and effectively changing the length of the string. The figure below shows how the length of the string is cut in half by pressing down at the 12th fret, and thus producing another A an octave higher.
Since the tension of the string doesn't change, we double the frequency when pressing down at the 12th fret. Since the length of the string is now half of what it was, the frequency has gone up to 880 Hz. This is the octave, and the note produced is the same as for the open string, an A. Every time you double a frequency, you reach an octave. Same thing goes for cutting the frequency in half. Let's assume your bass suddenly grew to a 68" scale. The second string from the top would still be an A, but an octave lower than for the standard 34" bass. So, when the length of the string is halved, the frequency is doubled, and when the length of the string is doubled the frequency is halved. So far we've only produced one note from our bass, A. To get to the other 11 notes, we have to change the length of the vibrating string using the frets 1-11. We get the exact same notes using the frets 12-24, but an octave higher. So, how do we calculate the frequency of the tones between A440 and A880? There are two ways of doing this. The first way is by using a Just Scale. In this scale the half steps are of unequal size since we base all the calculations on a given reference pitch (A at 440 Hz in our case). This means that every time we change the key, let's say from A to F#, we have to retune our bass. This is because the frequency of all the other notes of the F# scale will be based on the frequency of F# and not that of the previous note in the scale. In western music we use a different scale where all the half steps give an equal change in frequency. This scale is called the Equal Tempered Scale. This scale is much easier to use since we can jump between keys without retuning our bass. The basis of this second way of calculating frequencies is that all the half steps are of equal size, and the octave has a frequency twice that of the reference pitch. So, to get from A to A# we have to do the following:
2^(1/12) = 1.05946 so to get to the next note we just multiply the previous note with that. A# has a frequency of 466.164 Hz, and to get to B we multiply that with 1.05946 and get 493.882 Hz, and so on. When we get to A again we should end up at 880 Hz. You might ask what on earth this has to do with anything, and the truth is, nothing. It won't make you a better bass player, but it will make you understand your instrument a bit better. Other than that, it makes good conversation material if you ever come across an audio-nerd and need to kill some time.
So, on to next part. How does all these different frequencies translate into a signal that your amplifier can use? First we need to cover some very basic principles of electro physics. You might already have noticed that the pickups on your bass have magnets in them. These magnets have a north and a south pole. A magnetic field streams between these two poles. It goes out the top and in at the bottom of the magnet. On a standard J-style pickup, the magnets are cylindrical and are lined up next to each other. Around the magnets there is a coil of very fine wire. This is the wire that transfers the electrical signal to the output jack of your bass (or to the preamp first if you have an active bass). A principle of physics states that if you move a conductive material, like a wire, through a static magnetic field, a current is induced in the wire. On a pickup, both the magnets and the coil of copper wire are fixed in place. How come there is an induced current then? Well, the magnetic field and the conductor only have to move relative to each other to produce a current. Because of this, the current can also be produced by moving the magnetic field radiating from the magnets. This is where the strings themselves come into play. By plucking a string, you disturb the magnetic field since the string is metal. This causes the magnetic field to "move" in relation to the coil. We're not talking several feet or even inches here, but on a very small scale the magnetic field is moved. This causes a very small current to be induced in the coil surrounding the pickups.
The picture above shows a vintage J-style pickup for a 4-string bass. We can see the magnets with the pole pieces sticking up. Around the magnets we can see the coil of very fine wire, in this case Copper. The two wires at the bottom connect the pickup to the rest of your bass' electronics. As we can see from the picture, the coil of wire surrounding the magnets is made up from thousands of windings. Using only one winding would also produce a small electrical current when the string is plucked, but using several windings multiplies the effect. The currents itself does not get any stronger by using several windings, that's why we need to amplify the signal before we can drive speakers with it.
Next in the signal chain is the electronics in your bass. If you have a passive bass (i.e. it lacks a battery), you'll have knobs controlling the volume and the tone. If you have an active bass (i.e. it has a battery), you have EQ controls similar to the ones on your home stereo systems. You can control the amount of bass, mid and treble. Some fancy preamps also have switches to apply preset shapes to the EQ circuit. After the signal leaves the bass itself, it travels through the guitar cord and into the amplifier. More correctly, it goes to the pre-amplifier (preamp) section of your amplifier. The preamp is exactly what the name implies. It's a circuit that your signal passes through before it is amplified. A normal bass amp consists of two main parts. A preamp and a poweramp. The preamp shapes the signal, and the poweramp amplifies it so it can drive the speakers in your cabinet. When these two units are in the same box, you have what is referred to as a head. The Hi-fi people call the same unit an integrated amplifier. Another solution is to have a seperate preamp and poweramp. Whatever you use, the principle is the same. All the tone, EQ and gain controls on your amp work on the preamp section. Usually, the only control that has a direct influence on the poweramp section, is the output level control. Some heads and preamps have built-in compressors or effects, like harmonizers and octavers. These also do their thing before the signal reaches the poweramp.
The poweramp is the last stage before the signal leaves the amp and enters the cab. There are two ways of amplifing the signal that are predominant. The first one is using vacuum tubes. This is stone-age technology, but it's still widely used today because of its rather special characteristics. The other option is a solid state poweramp. A solid state amp uses a transformer (usually a ring model) to beef up the signal. In very simplified terms, a transformer changes the voltage of an electrical current. It works either way, either bringing the voltage up or down. So, on to the last stage, the speaker cabinet.
The speaker cabinet houses the speakers. If your amp and speaker cabinet is one compact unit, you're the proud owner of what is know as a combo. If not, you have a separate amp and cab setup. Whatever you use, things work the same way. The only difference is that you gain some flexibility with using a separate amp and cab. This allows you to change either component regardless. Changing the amp in a combo is much more of a hassle, and is not usually done. The job of the speakers is to convert the electrical signal that started in the pickups, and moved all the way through your amp, to sound. Sound is in essence pressure changes in air, or in other words, moving air. To move the air, we need to have something pushing it. This is what the speaker cones do. They work by moving in and out, and thus creating pressure differences in the air directly in front of them. These pressure waves travel to your ear, and you interpret them as sound. If you crank up the stereo or your bass rig, and play a bass heavy tune, you can actually see the speaker cones moving in and out. How do the electrical signal from my amp move the speaker cones, is there a small motor in there? Actually, no. The principle is the same as what produced the signal in the first place. Remember about the magnetic field being disturbed and creating a current through the coil? With speakers it's the same thing, only in revers. We have a good, strong electrical signal from the amp. This is passed through a coil behind the speaker cone. Inside this coil there is a large magnet. As current passes through the coil (the signal from your amp), the magnet moves in and out carrying the speaker cone with it. The current comes and goes as pulses, and this causes the speaker to move in a "musical" way. Since the signal has a waveform (it looks like a wave if were to look at it with an oscilloscope), it has a positive and a negative part. The positive half of the signal pushes the magnet (and the cone with it) out, and the negative half pulls it back in.
So, that concludes this article. I hope it was useful to you, and don't forget to provide some feedback so I can improve my writing.