Objective
Measure the speed of a wave on a given string as a function of its tension, and compare to the theoretical wave speed based on the tension and linear density of the string.
Theory
Pythagoras (ca. 550 BCE) is said to have described a connection between constant tension in a vibrating string, small whole number ratios of length, and sound pitch (frequency).[1]
When tuning a guitar by exercising string harmonics, one modifies pitch in two ways. By momentarily touching one string's midpoint while plucking it, one sets up a standing wave whose frequency is twice that of the string's fundamental. Under these conditions, the length of the wave is the effective length of the string.
Turning another string's key changes the string's tension, but its wavelength remains the same. Increasing the tension increases the string's characteristic frequency - the frequency of a standing wave with two nodes and a single antinode. The length of a wave in such a vibrational mode is twice the string's effective length.
A mathematical model of constant-wavelength vibration relates wave frequency directly to wave speed:
(1) f = A v where A = 1/λ, and λ is the wavelength.
A physical model of string vibration shows that its wave speed is related nonlinearly to its tension:
(2) τ = B v2 where B = μ, and μ is the string's linear density (assumed constant).
One sees that:
· For greater tension on a vibrating string, the string's wave speed will be greater.
· For a constant wavelength, the greater the speed of a wave, the greater its frequency.
These relations explain why the pitch of the tone from a guitar string varies with tension - and indicates that this variation is related both to the constant wavelength of the "open" string ("no-hands" tuning), and to the string's physical composition.
These two relations also suggest that for a given increase in wave speed of a string, caused by shifting its vibrational mode to a higher frequency (decreasing its wavelength under constant tension), one could produce a similar wave speed increase by holding the wavelength constant and increasing tension.
In fact, one could use equation (1) to determine various wave speeds of a vibrating string by measuring its wavelength under varying tension, and use equation (2) to predict or verify the observed differences in wave speed.
Plato (ca. 400 BCE) advised students to study harmonics "for ten years to familiarise the mind with relations that can only be apprehended by thought."[2]
Procedure
We recorded wavelength and frequency of a vibrating string held under varying tension, derived wave speed as a function of tension, and compared our experimental curve with a similar curve calculated simply from the measured linear density of the string.
We found the linear density of a sample of nylon string by dividing its (weighed) mass by its (measured) length. From the same roll, another length of string was attached to the node of a mechanical actuator. This length was "strung" above the surface of a lab bench, draped over a pulley, and tension applied by the hanging weight of an iron disk attached to the free end.
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A sinusoidal waveform of a function generator was applied to the actuator, and monitored by a frequency counter with a digital display. For each of five different hanging weights, we adjusted the input frequency to produce three distinct vibrational modes:
· 4 nodes (3 antinodes)
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· 5 nodes (4 antinodes)
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· 6 nodes (5 antinodes)
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For each stable vibrational mode, we measured the length of a half-wave (distance from one internal node to the next), and recorded that length's double as the wavelength for that mode. We recorded the digital counter reading as the frequency for that mode.
Data and
Observations
We applied hanging masses in the following sequence: 200 g, 300 g, 400 g, 500 g, 100 g, 450 g.
Our attempt using 500 g yielded no data. The string would not vibrate, at least not enough for us to detect. This weight evidently produced string tension sufficient to inhibit the mechanical actuator - possibly by offsetting the actuator node from its bearing.
The 100 g mass, on the other hand, provided barely enough weight to stabilize the string's vibration along one axis. The string flopped around and growled a lot, making life difficult for the person trying to "find" any of the string's vibrational modes. We found it necessary to make our length measurements quickly, before the string could lapse into non-modal behavior.
The 450 g mass gave just enough "slack" for the string to vibrate. The wave amplitude, though, was small enough to again make it difficult for the person on the frequency knob - as well as making it hard to distinguish the center of any node for the purpose of measuring length of a half-wave:
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Where's the
node ???
We switched jobs a few times during the procedure, but were both involved in measuring half-wave lengths. Each length measurement - including that of the string sample (for finding linear density) - required at least two scale readings by two different persons. (Each half-wave, as well as the string sample, was too wide for one person to take both scale readings while keeping a scale reference aligned.) Therefore, all lengths were measured "by committee." Then, the length taken as the difference of two measurements was doubled to give the recorded wavelength (which also doubles the magnitude of estimated error). Moreover, we noticed substantial variation, among the various combinations of tension and frequency, in our ability to distinguish center-node.
A rigorous treatment of error estimation and propagation would have required us to record a literal error estimation for each individual measurement - two for each recorded length, each specified by one person for the specific condition being viewed. Those error estimations would then be combined, using standard rules, to find a good estimate of probable error for each recorded and derived quantity.
Under certain conditions, such detail might be desired, out of caution not to overstate nor understate the precision of our measurements. In this circumstance we agreed to lump these sources of probable error into a single conservative estimate for each recorded wavelength.