Sharon E. Finberg
Bentley College
Dept. of Natural Sciences
175 Forest Street
Waltham, MA
02452
781-891-3493
sfinberg@bentley.edu
Student Sonometer
Abstract (50-75 words)
Plans for an inexpensive and enjoyable student sonometer are presented. Assuming a PC is already available to the students, a sonometer can be constructed for approximately $. Data taken with this sonometer clearly show the inverse dependence on length and the square root dependence on string tension of the fundamental frequency. Additionally, the tuner program clearly shows the harmonic modes of the strings.
Construction of Apparatus:
I usually use standard guitar strings as the string. Not only are they convenient and cheap, but they sound pleasant and the students have a sense what sounds are ÒcorrectÓ. I have also successfully used ice-fishing line, which gives good results with less tension.
When using guitar strings, the ferrule of the guitar string is held in a small slit in the bottom of a short piece of aluminum angle stock and wrapped over the top of this piece. A second piece of Al angle stock is used as a bridge and determines the relevant length of the string. The far end of the string has be twisted and soldered into a loop to hold an ÒsÓ-hook, from which calibrated weights are hung. To conserve money, the weights are simply small sandbags or 1- and 2-L bottles of water (1 L of water masses 1 kg). Also to conserve money, instead of using pulleys the weighted end of the string is run over a fairlead made of two short pieces of PVC pipe glued together. The strings do have a slight tendency to dig into the fairlead, but this can be easily compensated for by picking up and replacing the string whenever the tension is changed.
Figure 1 Sonometer schematic
Not shown in the schematic is the input microphone and PC. On the PC is a shareware program named wtune[1]. This program, designed as a guitar tuner, is easy to use and explicitly free for academic use. As can be seen below, the screen shows the spectrum, a digital frequency meter and the nearest note on a musical staff. In the spectrum ÒA small red cross is drawn on the top of the line relevant to the dominant frequency.Ó [2] The freeze/thaw spectrum key temporarily freezes all meters and allows the student to study the transient sound.
Figure 2 Wtune Screen Shot
Use of Apparatus:
In order to determine the fundamental, the students simply pluck the string while watching the tuner on the screen. With the cheap equipment and noisy environment that I have, the data collection is slightly finicky, but my students quickly learn to recognize ÒgoodÓ data. I have found that it helps to place a finger on the far (unplucked) side of the bridge to dampen any unwanted rattles and for one partner to hit the ÒfreezeÓ button as soon as he/she sees the second person pluck the string (the response time matches well with the ending of the ÒattackÓ). Also, while fig. 2 the fundamental frequency as dominant, I should note that sometime the second or third harmonic is the highest intensity note and recorded as dominant. While this entails re-doing that data point so that the frequency of the fundamental note is shown on the digital meter, the entire data collection is fast enough that this has not been a significant problem for my students or me.
By varying the location of
the bridge, the dependence of the fundamental frequency of a wave on a string
as a function of length can be experimentally determined. As seen in the data on fig. 3 below, the
fit is extremely close to the theoretical 1/L dependence given in the equation
. The wave
velocity of the wave in the string can also be determined from a fit to the
data. 
Figure 3 Graph of fundamental freqency vs. string length
Similarly, by varying the tension of the string (by adjusting the weight hung off the end of the string), the dependence of the fundamental frequency of a wave on a string as a function of tension can be experimentally determined; as seen in fig. 4 data fits are extremely close to the theoretical square root dependence.
Figure 4 Graph of fundamental frequency vs. string tension
Equipment and costs required to construct apparatus:
|
Item |
Source |
Part
number |
Cost |
|
Guitar Strings |
MusicianÕs Friend,
|
100056 |
$5.26 / set of 6 |
|
*Guitar Pick |
MusicianÕs Friend, |
110841 |
$0.99/dozen |
|
*Aluminum angle iron ÒBridgeÓ and endpiece (about 8 inches total) |
Local Hardware Store |
n/a |
$2 |
|
C clamp |
Local Hardware Store |
n/a |
$4 |
|
ÒSÓ hook or clip-on bottle |
Local Hardware Store
QVC |
n/a F10764 |
$0.25
$4 |
|
*PVC tubing fairlead (about 8 inches total PVC pipe plus
cement) |
Local Hardware Store |
n/a |
$5 |
|
*Metric adhesive tape |
Sargent Welch |
|
$10 |
|
Weights - 1- and 2-L bottles of water |
Recycling bin |
n/a |
Free |
|
Input microphone |
Buy.com |
10332747 |
$6 |
|
Digital Tuner |
Wtune[3] |
|
Free for Educational Use |
|
PC computer |
Assumed available |
|
n/a |
Total Cost |
$ 33.50 |
||
Access to a saw capable of cutting both PVC pipe and Al angle iron and to a soldering iron is assumed to be free.
Please note that the starred items are essentially the same price for a classroom set of 12 as for a single sonometer.