This exercise looks at travelling waves, superposition and standing waves in strings of different sorts. There will also be sound waves, but we shall not study these quantitatively. Standing waves in strings are important in musical instruments and in cables used as structural elements. Further, they are interesting to study as examples of waves in general because they illustrate many general properties but have the advantage that they are easy to see, to measure and to feel.

Background. The wave speed in a string is given by

Velocity of wave on a string.

where F is the tension in the string and m is the mass per unit length of the string. Consider a string of length L, fixed at both ends. If a pulse or wave travels one complete round trip, it takes a time T = 2L/v, and so its frequency f = 1/T = v/2L. Its wavelength is l = v/f. In the diagram below, you can see that one half of a complete wavelength 'fits' onto the string, and satisfies the conditions that the ends are fixed (called boundary conditions). However, it is also possible to satisfy these conditions by fitting one complete wavelength (two half wavelengths) into L: this is called the second harmonic. Similarly, n half wavelengths fit into L to give the nth harmonic, as shown below.

Combining the results above, the frequency fn of the nth harmonic of a stretched string of length L is given by

Frequencies of harmonics on a string.

For further background on waves and harmonics, see www.phys.unsw.edu.au/music/strings.html
For information about how the bows work, see www.phys.unsw.edu.au/~jw/Bows.html

Homework Question 1. The third string on a classical guitar has a diameter 1.0 mm and is made of nylon with density 1100 kg.m^-3. The free vibrating length of the string (between the fixed ends) is 650 mm. Calculate the tension F that must be applied to tune it to the note G3 (196 Hz)¹ . (Hint: the string is shaped like a cylinder and density is defined as r + m/V).
¹To preempt the questions from guitarists: yes, this note is written as G4. This is because guitar music is usually transposed an octave higher than it sounds, so that it may be written in the treble clef rather than the tenor clef.

Homework Question 2. A circus performer balances at the midpoint of a rope. The ends are secured a distance L apart. The weight W of the performer and her balance pole causes the centre of the rope to be lower than the support points by a distance h. Derive an expression for the tension F in the rope in terms of h, L and W. q is exaggerated in the sketch. Assume q<< 1.

Homework Question 3. If you cut a spring in half (make it half as long), by what factor does its stiffness k change? Does it increase, decrease, or stay the same? The stiffness or spring constant k is defined by the equation F = -kx, where x is the extension of the spring and F is the force produced by that extension. (Hint: it may be useful to imagine two springs, each of length L/2, joined together and subject to the same tension F.)

Homework Question 4. As an assignment in PHYS1000, physics students at UNSW were asked to find novel ways to measure the speed of sound, preferably using simple equipment. One team did it thus (true story): Stand a distance L from a large flat wall. Clap your hands and listen for the echo. Clap again, hear the echo again, etc. Adjust the rate of clapping so that the echoes fall exactly halfway between the claps. (One of them was an experienced musician.) Then either count the number of sounds (each corresponds to two round-trips) in a minute², or set a metronome to match time or estimate the 'beat' rate (some musicians, especially conductors, can do this with precision). Below are two representations of a measurement of 124 sounds and echoes per minute. (Hint: this is much easier than it looks.)
²In practice, the count rate is fast, so it's easier to think of the sounds in groups of four, as shown in the right hand figure, and count the number of bars rather than the number of beats.

Take the speed of sound as 344 m.s?1. Calculate the L required to give 124 sounds per minute.

Homework Question 5 (optional). What happens when you add harmonics? The example below shows a fundamental (frequency f, amplitude a1) and a second harmonic (frequency 2f, amplitude a2). (Their phases are both zero.) Their sum is also shown. At right, you can see the spectrum of this wave: a plot of amplitude (of each component) as a function of frequency.

Now it's your turn. The spectrum (below right) shows that only the first and third harmonics are present, and these have been drawn in the time display (below left) as well. Your job is to sketch (approximately) the waveform produced by this spectrum.

What you have just done is called harmonic synthesis. Its reverse is called spectrum analysis, which you will do in this exercise. For more about spectra, see www.phys.unsw.edu.au/~jw/sound.spectrum.html

Experimental Exercise

Equipment:

• "slinky" plastic spring.
• Ruler.
• Tape measure.
• Stopwatch.
• Electric guitar.
• Lead and adaptor.
• Pad to put under guitar.
• Spring balance.
• Violin bow.
• Computer with oscilloscope and frequency analysis software.
• A long fingernail gives a slight advantage, but it's probably too late to start growing one now.

Marking scheme: Marks awarded for the measurement on p6, and for answers in the boxes. Marks deducted for playing Smoke on the Water, Bourée in E minor Stairway to Heaven or part thereof.

Part 1. Transverse waves in a "Slinky" spring.
Slinky parameters. The spring constant varies among slinkies—one of the reasons is that it changes with use, so be careful not to stretch yours too much. The total mass Mt and total number of turns Nt also varies. Further, you will need to hold on to a small number of turns at either end, so let's say that you use N turns (N < Nt) having a mass M. Let's define three different masses of interest in these experiments.

M_t is the total mass
m = M_t/N_t is the mass per turn
M = Nm is the mass being used in your experiment

Reflection at a fixed boundary. Two students should each hold about three turns of the spring. Stretch the remaining coils to a length between 1.5 and 2.0 m. Send a pulse down the spring by pulling a section aside and releasing it. i) For a the pulse that is initially to the left, in which direction is it after reflection?

ii) At the reflection described above, in which direction does the spring exert a force on the hand at which the reflection occurs? (You can feel this force when the pulse arrives.)

iii) Briefly explain your answer to (ii) in terms of your answer to (i) and Newtons laws.

Determine k from the wave speed.
Using all of the slinky except the two or three turns held fixed by a person at each end, stretch it over a length L. Do not stretch beyond about 2 m (beyond this, Hooke's law fails and the spring may be damaged). On the other hand, at L < 1 m, the spring will sag so much that it is hard to estimate its length. Measure the time for several round trips. Hint: do not start when you release the pulse. Instead, count zero and click the stopwatch when the pulse hits one hand, then stop the watch when the pulse hits the same hand after several trips. For each of three different lengths, make five measurements.

At this stage, you should weigh the spring using one of the balances, and count the number of turns. Then use these data and your values of T to determine the spring constant k for the spring.

From our measurements k =

One of the problems with the preceding method of measuring v is that the pulse has a sharp corner, and its spectrum therefore has lots of high harmonics. The high harmonics lose energy rapidly, both at reflections and in internal losses in the spring. The result is that the kink loses shape quickly and you are left with a smooth curve, whose arrival time is difficult to estimate exactly. We can avoid this problem by starting with a smooth curve, and having only one harmonic present. You can also count more cycles.

Determine k from the frequency of standing waves.
Again, using all of the slinky except the two or three turns held fixed by a hand at each end, stretch it over a length L. (The longest L you used above is probably a good choice.) Now grasp the spring about 20 cm from one end and move it backwards and forwards with your hand. If you try a range of frequencies around 1 per second, you will probably find a resonance at the fundamental frequency, as sketched below. Do not excite the resonance too strongly (you don't want the spring to be much longer when stationary than it is when straight): an amplitude of 30 cm is plenty. Once you have excited it, stop shaking it and keep the two ends fixed. Now count several complete cycles of oscillation. Again, remember to start counting from zero.

Period T = _____ . How many wavelengths are there in the length L in this case? Use this and the other data you have gathered to determine the wave speed and thus k.

From our measurements k =

On this diagram, indicate the displacement nodes and antinodes (N_d and AN_d). Now think about the shape of the spring at one extremum of the motion. From this shape, where would you expect the nodes and antinodes of force (N_f and AN_f) to be. Think also about the transverse force on your hand. Mark N_f and AN_f on your diagram .

Repeat the previous experiment and count (quietly) ("1, 2, 3, 4") in time with the cycles. Then, between the counts, insert the word "and" ("1 and 2 and 3 and 4"). For your next trick, you will shake the spring at this doubled frequency—ie push it on "1" and "and" and "2" etc. This will excite the second harmonic. If you have trouble, go back to the fundamental to remind yourself of its frequency.

From our measurements k =

Can you manage the third harmonic? Musicians count this "1 and a 2 and a ...". Skip it if it's too difficult. Conversely, if you manage it, try for the fourth.

You now have several different measurements of the spring constant k for your length of N turns of the spring. Take an average, or choose what you think to be your most reliable measurement. What would k be for N/6 turns of the spring?

N =        . kN =         . kN/6 = .

When you are satisfied with your answers, call a demonstrator to mark your work. Here's how it will go: the demonstrator will bring a test mass mt and attach it to your spring with N/6 coils, hanging vertically. S/he will excite simple harmonic motion. The frequency of the motion of a mass on a spring is f = (1/2?)k/mt . Your mark will depend on the agreement between your result and this calculation: the demonstrator will tell you the mass, you will calculate f, and then you will measure it together. Have your calculators and stopwatch ready. Can you think of any reasons why the prediction might be in error?

Part 2. Transverse waves in an electric guitar.
Some information about the functioning of electric guitars is given in Appendix 1. For our immediate purposes, we have a set of strings. (Please don't tune them.) We use just the thickest, lowest pitch string (called "E") for most of the experiment. The pickup produces an electrical signal that is related to the motion of the portion of the string located directly above the pickup. The signal is proportional to the velocity of the string, not the displacement, because of Faraday's law (Appendix 1).

Open "Mac the Scope" on a computer to obtain an oscilloscope and spectrum analyser. See Appendix 2 for details, but briefly:

If it hasn't already been done, plug the guitar into the microphone input of the computer⁴ . Turn the volume and both tone controls up to at least 10 (hey, it is an electric guitar, isn't it?). Use the pickup switch to select pickup 'c' as indicated in the diagram.

Period, frequency, tension and linear density of the lowest string.
Pluck the lowest string (with a fingernail or with a fingertip) somewhere near where a guitarist would do so. Watch the signal and how it changes in time. Of course it becomes smaller as the energy you put into the string is gradually lost (via internal losses in the string and as sound). However, you will notice that the shape changes. From your knowledge of harmonics (homework question 5), comment on how quickly the energy is lost from the higher harmonics, compared to the fundamental. Hint: you can hit the 'pause' key at any time to 'freeze' the display. (Note: the 'pause' control disables several functions. To 'unfreeze' the display, click on 'START'.)
With one team member ready to hit the key, pluck the string and pause immediately. You can now measure the period. To do this precisely, use the mouse to position a cursor over the screen on some identifiable point of the wave, such as a sharp maximum. Then position it on the same point in the next cycle of the wave, which should occur one period T later. (You can drag from one point to the next to get the period directly.) When you are satisfied, write down your answer. Then calculate the frequency f.

T =           f =

Now you can calculate the tension, using a method from homework question 2. Use the spring balance to lift the string gently at the midpoint. (The midpoint is near to the 12^th fret.) Remember to convert to newtons. Measure the displacement from the undisturbed position with the ruler. Do this for at least three different displacements and calculate F for that string. Then measure the string length. Use these data and the frequency to calculate the mass per unit length of this string. (While you have a spring balance in your hand, you might want to remind yourself what a force of one newton or five newtons feels like.)

F =                     L =                        m =

The harmonic series on a string.
For this exercise, you will use a violin bow. Two important notes:

1. Do not touch the hair of the bow with your fingers. Rosin on the bow gives it the friction properties that allow it to work, grease from fingers stops it working.
2. Do not over-tighten violin bows. The bow should always be curved so that it is closer to the hair at the middle than at the ends. The hair should be ~11 mm from the bow at midpoint.

The bow allows a player to put energy into the instrument continuously. By reversing direction, one can maintain a note almost continuously. It gives rise to a motion that has an interesting waveform, as we shall see. Because the guitar body gets in the way, bow it near the head (the small end of the guitar). It takes a little practice to get the downwards force and the bow speed adjusted so as to produce both a steady tone and a steady trace on the oscilloscope. Use the "c" pickup, furthest from the bridge.

First bow a note at the fundamental frequency. Touch the string only with the bow. Sketch here the waveform V(t) that you observe. Remember that the voltage V(t) is proportional to the transverse speed v(t) of the string. There's a complication: the sound card may have an offset of a few mV. Rest assured that the true average v is zero, and that the string will remain on the guitar! So you can add the v=0 line to your sketch. Then, from your knowledge of displacement-time graphs, use v(t) to sketch the displacement y(t) as well. Only worry about the main features: don't concentrate on the fine details.

Next, bow a note at the second harmonic. To do this, touch the string at a position L/2 (measure with a tape). Touch it very lightly: enough to stop it vibrating, but not enough to bend it at all. This will stop the fundamental (and all odd harmonics). Measure the period T₂ and calculate the frequency f₂.

T₂ =                     f₂ =

Higher harmonics. Try to produce some of the higher harmonics, too. Touch at L/n to produce the nth harmonic. 7^th, 5^th and 4^th frets are at approximately L/3, L/4 and L/5. Listen to the pitch: each time you increase n by 1, you add f1 to the frequency, but the pitch increases become successively smaller. This is because your hearing responds logarithmically. Musical notation (right) recognises this.
(More detail at www.phys.unsw.edu.au/music)

Natural harmonics on a guitar E string.

Effect of pickup position.
Would you expect any harmonics not to be present in the signal from a particular pickup? Explain why.

At which pickup would the low harmonics have relatively little input? Use the pickup selector switch to test your answer. Describe your observation and explain it.

Optional exercises.
Repeat the measurements of period T, frequency f, tension F and linear density ? for the thinnest string.

Investigate the effect of different bowing and plucking positions.

Investigate the effect of the tone controls, which are RC filters.

If you feel comfortable using the oscilloscope, you can convert it to a spectrum analyser by clicking on "FREQ". Now you can see the harmonics individually, and watch as they fade away. Using this technique, you can give a more complete answer to some of the questions asked above.

Appendix 1. The electric guitar.

The body of an acoustic guitar serves to transmit vibrations from the string into vibrations of the top plate and the air inside the body, and thus produce a loud radiated sound. In contrast, the electric guitar makes very little sound without an amplifier, and the body of an electric guitar has only a small acoustic effect. Its body needs to be rigid and should be heavy enough so that the guitar hangs in a position that makes the right hand position comfortable. And of course it has to have an attractive/sexy/violent etc shape.

The pickups use Faraday's law, which gives the induced emf:

The coils in questions are wound on the poles of a permanent magnet. When a ferromagnetic material (here a section of the steel string) passes near the pole, it distorts the magnetic field and changes the flux. The rate of change is almost proportional to the velocity of the string.

Schematic diagram of a simple pickup

Appendix 2. The computer as oscilloscope.
The sound card in a computer contains an analogue-digital converter (see the digital electronics display in the main corridor of the OMB to see how this works). For audio frequencies, and a voltage range up to about 1 V_RMS, it can be used as an oscilloscope⁵ . The computers for this exercise run software called Mac the Scope. This screen dump shows the screen and the five controls that we shall use most often.

Further reading
The Acoustics Group at UNSW does research on musical instruments and the voice. Their web site has explanations, sounds, animations etc: www.phys.unsw.edu.au/music . Click on the 'basics' bar for pages about waves, strings, bows, spectra, decibels, hearing etc. Or look at the sites for different instruments. For filters, see www.phys.unsw.edu.au/~jw/RCfilters.html

This page has animations of a range of different waves: longitudinal, transverse waves, surface water waves, Rayleigh waves: www.kettering.edu/~drussell/Demos/waves/wavemotion.html

This page concentrates on superposition of reflecting waves. It has animations in which you get to set the parameters: home.austin.rr.com/jmjensen/JeffString.html

There are some animations of standing waves in two dimensions (eg drum heads) at www.falstad.com/mathphysics.html