Standing Wave Explorer

Pluck a guitar string and it gives you one note. Not a smear of pitches, not whatever frequency the finger happened to supply, but a single clear tone with a family of fainter ones above it. The finger did nothing musical. It pulled the string sideways and let go. The string chose the note. More exactly, the two points where the string is held chose it, and the same is true of the air inside a flute, an organ pipe or a bottle. To see why, it helps to stop thinking of the vibration as a shape and think of it as traffic.

Two waves that meet and stand still

Send a kink down a stretched rope tied to a wall and it comes back upside down. The wall cannot move, so the rope’s own momentum carries it through and out the other side, and the returning kink is the mirror image of the one that arrived. A string on an instrument is held at both ends, so every disturbance on it is trapped, running back and forth and flipping at each end.

Now make the disturbance a steady wave rather than a single kink. The wave heading right meets its own reflection heading left, with the same frequency and the same size. Add the two together and something odd appears. At certain points the two always cancel, and the string there never moves at all. Halfway between those points the two always reinforce, and the string swings as far as it ever will.

The still points are nodes and the busy ones are antinodes. Nodes sit half a wavelength apart. Written out, the sum of the two travelling waves is a fixed shape, a sine curve along the string, multiplied by a single rise and fall in time. Nothing travels any more. The pattern breathes in place, which is where the name standing wave comes from, even though it is built entirely from motion.

This is the idea worth holding onto. A standing wave is not a different kind of wave. It is the ordinary travelling kind, caught between two reflections and added to itself.

The ends decide

A string fixed at both ends must have a node at each end. That one requirement throws away almost every wavelength there is. The longest wave that fits has half a wavelength between the supports, so its wavelength is twice the string’s length. The next has a whole wavelength, then one and a half, then two. In general the wavelength is twice the length divided by a whole number, and nothing in between is allowed.

Frequency is speed divided by wavelength, and every wave on a given string travels at the same speed. So the allowed frequencies are the lowest one, called the fundamental, and exact whole-number multiples of it. Musicians call this ladder the harmonic series. Mode number four has nodes at the ends and at one quarter, one half and three quarters of the length, which is exactly where a player touches a string lightly to coax that harmonic out of it.

The speed itself comes from two properties of the string. It is the square root of the tension divided by the mass per unit length. Tighten the string and waves run faster, so the pitch rises. Use a heavier string and they run slower. Shorten the vibrating length and the same speed is divided by a shorter wavelength. Those are the three handles on every stringed instrument: tuning pegs change tension, string gauge changes mass, and fingers change length.

A worked case makes it concrete. Take a string 0.65 m long, stretched to 73 newtons, with a mass of 1.14 grams per meter. Tension over density is 73 divided by 0.00114, and the square root of that is 253.05 m/s. The fundamental wavelength is 1.3 m, so the fundamental is 253.05 divided by 1.3, or 194.66 Hz. The next three modes are 389.31, 583.97 and 778.62 Hz.

That fundamental is 11.9 cents below the note G3 when A4 is set to 440 Hz, the usual convention, and the octaves above it land the same 11.9 cents low, because an octave is exactly a doubling. The other modes do not line up so neatly. Equal temperament divides the octave into twelve equal semitones, each a ratio of the twelfth root of two, and whole-number multiples only roughly agree with that grid. The seventh mode of this string, 1362.6 Hz, falls 43.1 cents under the nearest note.

Air in a tube, and the missing even harmonics

Air in a tube plays the same game with one complication: there are two things to watch. The air moves back and forth along the tube, and the pressure rises and falls, and the two do not peak in the same places. Where the air piles in from both sides it barely moves, and the pressure swings the most. Where it sloshes freely the pressure hardly changes. A pressure node is a displacement antinode, and the reverse.

A curve drawn inside a pipe therefore means nothing until it says which of the two quantities it shows, because a node of one is an antinode of the other. So state the rule for each end in both languages. At a closed end the air has nowhere to go: the displacement has a node and the pressure has an antinode. At an open end the air moves freely and the pressure is pinned to that of the room: the displacement has an antinode and the pressure a node.

A tube open at both ends is therefore the string’s twin. Its pressure has a node at each end, the longest wave that fits is twice the tube’s length, and every whole-number multiple of the fundamental is allowed. A modern flute behaves roughly like this. Its lowest note is middle C, 261.6 Hz when A4 is 440 Hz. With sound travelling at about 343 m/s, an ideal open tube sounds that note when it is 343 divided by twice 261.6, or about 0.66 m long.

Close one end and the bookkeeping changes. Now one end needs a pressure antinode and the other a pressure node, and the longest wave that satisfies both has only a quarter of a wavelength inside the tube. The fundamental wavelength is four times the length rather than twice, so the note is an octave lower from the same tube. A clarinet, with its fixed reed and nearly cylindrical bore, behaves approximately like a cylindrical pipe closed at one end, and it plays far lower than an open pipe of its length would for this reason.

The next shape that fits has three quarters of a wavelength in the tube, then five quarters, then seven. There is no way to fit two quarters or four quarters with a node at one end and an antinode at the other. So a tube closed at one end sounds harmonics 1, 3, 5 and so on, and the even ones are simply not there. This is why a clarinet’s second register sits a twelfth above the first, three times the frequency, where a flute overblows at the octave.

Put numbers on it. At 20 C the speed of sound in air is about 343 m/s. A tube one meter long and open at both ends has a fundamental of 171.6 Hz, then 343.2 and 514.9 Hz. Close one end of the same tube and the series becomes 85.8, 257.4 and 429.0 Hz. A tube closed at both ends returns to the full series, though such a pipe is rarely used on purpose in music.

Temperature matters because the speed of sound in a gas goes as the square root of its absolute temperature. It is about 331 m/s at 0 C. A wind instrument therefore plays flatter in cold air, and its pitch rises as the air inside it warms. A string does not care about the air’s temperature in this way. Its pitch is set by its own tension and mass.

Why the plucking point changes the tone

A real note is hardly ever one mode. It is a mix, and the recipe depends on how the motion began. A plucked string starts as a triangle: two straight segments meeting where the finger held it. That triangle can be written as a sum of the allowed sine shapes, each with its own strength, and those strengths are what the ear hears as tone color. The strength of mode n is proportional to one over n squared, times the sine of n times pi times the plucking position as a fraction of the length. The first factor weakens the higher modes. The second becomes zero whenever the plucking point is a node of that mode.

Pluck at the exact middle and every even mode vanishes, since each of them has a node there. You cannot set a mode going by pulling at a place where it does not move. Pluck one third of the way along and modes 3 and 6 disappear instead. In general a pluck at one nth of the length removes the nth harmonic and its multiples. Near the end of the string the sine factor grows almost in step with n, so by this formula the higher modes come out relatively stronger than they do for a pluck near the middle.

It takes a surprising number of terms to rebuild the sharp corner. With forty terms and a pluck at the middle, the sum still misses the peak of the triangle by about one percent of its height. Corners are expensive. On a real string those high modes also lose their energy first, which is why a plucked note turns mellower within a second or so.

What the tidy picture leaves out

All of this describes ideal objects. The ideal string is perfectly flexible, and a real one resists bending. That stiffness lets waves of higher frequency travel slightly faster than the rest, so the upper modes of a real string run sharp of exact whole-number multiples. Thick, stiff strings show it most. The formulas above know nothing about it.

The ideal tube is a perfect cylinder whose open end is a perfect pressure node. A real open end is not quite one. The wave has to push a little air just outside the tube, and the effect is roughly that of a tube longer by about 0.6 times the radius at each open end. A closed end needs no such correction. On a one meter tube of 20 mm radius, open at both ends, that lowers the fundamental from 171.6 to 167.6 Hz, about 41 cents, so a fat pipe sounds flatter than a thin one of the same length.

Real instruments depart further still. A clarinet has a flared bell and tone holes, and its bore is only nearly cylindrical, so its upper notes do not follow the pure odd series exactly. Conical bores follow different rules altogether. The speed of sound used here assumes dry air treated as an ideal gas, and humid air is slightly different. None of the formulas says how loud a mode will be, how long it will ring, or how the player keeps it going.

What the simple model does give is the reason. A string or a column of air is a place where waves are trapped between two reflections, and only the wavelengths that fit the two ends survive. Change an end from open to closed and you change which notes can exist at all.