Standing Waves on a String Experiment

For the past two class blocks I have been working on an experiment involving sound and string. Understanding the interaction between sound waves and strings is essential in understanding string instruments, such as violin, guitar, or piano, but also illustrates concepts that are important in understanding the physics of music.

In this experiment, with the help of Jon Bretan, I attached a piece of string to a speaker and attached the string to a pole. Using an amplifier, I played sound of different frequencies through the speaker in order to find the natural frequencies of the string. This was visible in the string as it made wave shapes with clear nodes and antinodes. I measured the frequency needed to get the string to vibrate in its fundamental mode, as well as its harmonic modes. If this is hard to conceptualize, here is a video showing the changes in the string's modes as I changed the frequency of the sound playing through the speaker.
The frequencies corresponding to each harmonic were recorded as follows:

Frequency (hz)	Mode #
17.3	Fundamental (1st)
34.6	2nd
69.3	3rd
139	4th
277	5th

As you can see, each frequency is twice the previous frequency. This should be a familiar interval if you have read my post about the relationship between frequency and pitch, because doubling the frequency of a sound increases it by exactly one octave in pitch. The above video is a good example of why this is the case. 

The problem with these initial measurements is that the tension of the string was unknown, as it was simply tied down using a clamp. To remedy this, I collected more data, but used weights to tighten the string. The following data sets correspond to weights of 50, 100, and 200 grams respectively:

50 gram weight.

Frequency (hz)	Mode #
17.3	Fundamental (1st)
34.6	2nd
69.2	4th
138	8th
277	16th

100 gram weight.

Frequency (hz)	Mode #
19.5	Fundamental (1st)
39	2nd
78	4th
156	8th
312	16th

200 gram weight.

Frequency (hz)	Mode #
28.5	Fundamental (1st)
57	2nd
114	4th
228	8th
456	16th

As you can see, when the tension of the string increases, the frequency of the harmonics increases. In the context of musical instruments this makes perfect sense; for example, tightening the string of a guitar increases its pitch.

In addition to manipulating the tension of the string, I changed its thickness. I did this by tying an identical string to it in order to double its mass density. I also tied down this modified string using weights of 50, 100, and 200 grams. The following data corresponds to those trials:

50 gram weight.

Frequency (hz)	Mode #
15.1	Fundamental (1st)
30.2	2nd
60.5	4th
121	8th
242	16th

100 gram weight.

Frequency (hz)	Mode #
16	Fundamental (1st)
32.1	2nd
64.1	4th
128	8th
256	16th

200 gram weight.

Frequency (hz)	Mode #
18.6	Fundamental (1st)
37.1	2nd
74.2	4th
148	8th
297	16th

As expected, increasing the tension of the modified string increased its fundamental frequencies, but what difference was caused by the change in the string's mass density? In looking at the fundamental frequencies we get the following comparisons for the strings with varying tension: 17.3 vs. 15.1; 19.5 vs. 16; and 28.5 vs. 18.6 with the initial string listed first and the modified string listed second. The data appear to show that the more dense string resonates at lower frequencies. Again, this makes sense; the guitar strings that are lower in pitch are thicker. 

Fortunately, there is a handy relationship between frequency and the tension and mass density of a medium.

This shows that frequency is proportional to the root of tension and inversely proportional to the root of the mass density. The data follows this trend, as increased mass density decreased the frequency and increased tension increased the frequency. If the data is hard to conceptualize, thinking about this relationship in the context of a guitar string can be very helpful. 

While the data appear to follow this trend generally, it does not follow the relationship. The fundamental frequencies of the first and second mode were graphed. The data for the original string and modified string are plotted on the same graph, but no trend can be inferred, as there are only two different mass densities. Clicking the images will open them in full size, so that you can read the text.

First Mode:

Second Mode:

As you can see, the lines of best fit are quadratic. This is unexpected because frequency is proportional to the square root of tension and not the square of tension.