Pluck a guitar string and only waves that fit a whole number of half-waves between the ends can survive. Sprinkle sand on a ringing plate and you see the same idea in two dimensions.
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Tonight's idea
Pluck a guitar string. Its two ends are held still, so only some wave shapes can live on it. A shape survives only if a whole number of half-waves fits exactly between the ends. Each such shape is a mode. The points that never move are nodes, or still points. The two ends always count.
On a string of length \(L\), mode \(n\) has \(n\) half-waves. Its wavelength is \(2L \div n\) and its frequency is \(n\) times the lowest one.
Why \(n\) times? All waves on one string travel at one speed, and speed \(=\) frequency \(\times\) wavelength. A wave \(n\) times shorter must swing \(n\) times as often. Frequency is swings each second, in hertz (Hz). This is a model of an even, perfectly bendy string, and real strings come close. A pluck sets off several modes at once.
The same thing happens in two dimensions. Sprinkle sand on a ringing metal plate. It bounces off the moving parts and settles on the still lines, drawing the mode. Ernst Chladni showed this in 1787.
Watch one: a guitar's A string
A guitar's A string is about 65 cm long and its lowest note is 110 Hz. In the picture, dots mark the still points and the dashed curve is the string half a swing later.
Wavelengths. Mode 1 is one half-wave, so one wave is \(2 \times 65 = 130\) cm. Mode 2: \(130 \div 2 = 65\) cm. Mode 3: \(130 \div 3 \approx 43.3\) cm. Frequencies. \(110\), \(2 \times 110 = 220\) and \(3 \times 110 = 330\) Hz.
Check the speed. In metres, \(110 \times 1.3 = 143\) and \(220 \times 0.65 = 143\). Mode 3 gives \(330 \times 1.3 \div 3 = 143\) too. The wave runs along this string at 143 metres each second, whichever mode it is in. Still points. Mode 3 has 4: the two ends, and the points \(65 \div 3 \approx 21.7\) cm and \(43.3\) cm along. These wavelengths are on the string. The sound in the air has the same frequencies.
Your turn
A string is 90 cm long and its lowest note is 100 Hz. (a) Find the wavelengths of modes 1, 2, 3 and 4. (b) Find their frequencies. (c) For each mode, change the wavelength to metres and multiply it by the frequency. What do you notice, and what does that number mean?
Same string. Measure from the left end. (a) Where are the still points of mode 3? Count the ends. (b) How many still points does mode 6 have, and where are they? How many does mode \(n\) have? (c) Which of modes 1 to 6 have a still point exactly in the middle, at 45 cm?
Same string. Touch the middle lightly and pluck. In the model, a mode survives only if it has a still point under your finger. (a) Which of 100, 200, 300, 400, 500 and 600 Hz survive? (b) Double the frequency is called an octave up. Why do you now hear an octave up? Which shorter string has these modes? (c) Now touch 30 cm from the left end. Which of modes 1 to 6 survive, and what is the lowest note?
A square drum skin is held still all round its edge. The pictures show three of its modes from above, in a model of an even, perfectly bendy skin. Dashed lines are still lines, where sand would gather. (a) Count the still lines inside each. Which would you guess has the highest frequency? (b) In C, each small square moves like a half-width copy of A. A skin half as wide, just as tight, rings at twice the frequency. A rings at 100 Hz. What about C? (c) B rings at about 158 Hz. Do A, B and C go \(f\), \(2f\), \(3f\) like a string's modes?
Stretch: why a drum is not a guitar
In a model of a round drum skin, the modes after the lowest ring at about 1.59, 2.14 and 2.30 times its frequency. (a) The A string's modes ring at 110, 220 and 330 Hz. How many swings does each make in \(\tfrac{1}{110}\) of a second? (b) A drum's lowest mode rings at 100 Hz. Find the next three frequencies, and each mode's swings in \(\tfrac{1}{100}\) of a second. (c) Your ear hears a clear pitch when the whole motion repeats many times a second. Why is the string's pitch clearer than the drum's?
Answers
(a) Mode \(n\) has wavelength \(2 \times 90 \div n = 180 \div n\) cm: 180, 90, 60 and 45 cm. (b) \(n\) times 100 Hz: 100, 200, 300 and 400 Hz. (c) \(1.8 \times 100 = 180\), \(0.9 \times 200 = 180\), \(0.6 \times 300 = 180\) and \(0.45 \times 400 = 180\). The product is always 180. It is the speed of the wave along this string, 180 metres each second. With the speed fixed, half the wavelength means twice the frequency.
(a) Three half-waves of \(90 \div 3 = 30\) cm each, so the still points are at 0, 30, 60 and 90 cm. That is 4, counting the ends. (b) Six half-waves of 15 cm each: 0, 15, 30, 45, 60, 75 and 90 cm, which is 7 still points. Mode \(n\) has \(n\) half-waves and so \(n + 1\) still points, counting both ends. (c) Modes 2, 4 and 6. The still points of mode \(n\) are whole multiples of \(90 \div n\) cm, and 45 is such a multiple only when \(n\) is even. For odd \(n\) the middle is where the string swings most.
(a) Modes 2, 4 and 6, so 200, 400 and 600 Hz. Modes 1, 3 and 5 need the middle to move, and your finger stops it. (b) They are 1, 2 and 3 times 200 Hz, the modes of a 45 cm string of the same kind. Each half of the string swings like a string half as long. The lowest note left is 200 Hz, double the old 100 Hz, so you hear the note an octave up. (c) 30 cm must be a whole multiple of \(90 \div n\), so \(n\) must be a multiple of 3. Modes 3 and 6 survive, at 300 and 600 Hz. The lowest note is now 300 Hz, three times the open string's.
(a) A has 0, B has 1 and C has 2, so C is the natural guess. More still lines mean smaller moving patches, and smaller patches usually swing faster. That is a rule of thumb, not a law. (b) \(2 \times 100 = 200\) Hz. (c) No. The frequencies are 100, about 158 and 200 Hz. B is about 1.58 times A, not 2 times, and C is 2 times A, not 3 times. A drum skin's special frequencies depend on its shape, and they are not the whole-number ladder of a string. Real metal plates are stiff and their edges are free, so their sand pictures are curvier, but they too have still lines and special frequencies.
Stretch. (a) 1, 2 and 3 swings, since \(110 \div 110 = 1\), \(220 \div 110 = 2\) and \(330 \div 110 = 3\). Every mode is back where it started, so the whole shape repeats 110 times each second. (b) About 159, 214 and 230 Hz. In \(\tfrac{1}{100}\) of a second the four modes make 1, about 1.59, about 2.14 and about 2.30 swings. (c) After one swing of its lowest mode, the string is exactly back at its start, because every other mode has made a whole number of swings. Your ear hears that steady repeat as one pitch, 110 Hz. On the drum the other modes are part of the way through a swing, so the motion does not repeat in step with the lowest mode. The ear gets a mix of unrelated notes, which sounds more like a thump than a pitch.