Wave length, frequency, tension lab

Data for all of the following graphs can be found here.


In the previous experiment, a relationship was found between wavelength and period. In the following experiment, we establish that this relationship also applies to frequency, and there is a relationship between these variables which depends on the tension in the vibrating medium.

The experimental apparatus can be observed in the following video.

The set up consists of a mechanical oscillator, a function generator, a length of string, a pulley and a weight.

The string is attached on one end to the mechanical oscillator, and across the pulley to the weight on the other end. The weight serves to maintain a nearly constant tension for the duration of the experiment.

The function generator is set to generate sinusoidal voltages of varying frequency. This frequency is adjusted until a given number of anti-nodes is established in the pattern of oscillation.

For a given tension, at least 6 different standing waves were tested by varying the frequency, these standing waves represent a specific wave length as measured by comparing the space between nodes to a meter stick. A table with the relevant data is available at the top of this post.

Observations: A plot of the frequency vs. wavelength results in a plot which resembles functions of the form;
frequency = some constant divided by wavelength.

when plotted against the reciprocal of wavelength, a linear relationship is observed.

when a linear regression is added to the chart, it is evident that in order for the units in the function to make sense, the coefficient of x must have units of meters per second. (owing to the fact that 1/wavelength has units of meters, and frequency has units of 1/seconds.)

Thus, the coefficients from the linear regression can be converted to velocities. It is important to note that the units from the above graph are in cm not meters.
These velocities can be plotted against the corresponding tensions in order to establish a trend. 

In the above graph, it is evident that the blue points, (the plot of velocity vs. tension) very nearly adhere to the function f(x)=30.47x^.48.

The equation which models the velocity of a wave through a medium is:

where v is the longitudinal(parallel to the medium) velocity of the wave, and mu is the linear density of the medium. The linear density of the string was determined to be .00119 kg/m.

Also visible in the graph is a plot of the theoretical velocities which correspond to the given tensions. The function which most closely matches those values is f(x)=28.99x^.50.

The equations are similar in that the coefficient of x differs by approximately 5 percent, and the power of x differs by approximately 4 percent. This indicates a strong relationship between the measured data and the theoretical model.

A comparison of the ratio between the real velocities and the predicted velocities can be observed in the aforementioned spreadsheet.

For each step of the harmonics measured, it was observed that the required frequency was always nearly a whole multiple of the first measured frequency. (15,30,45,60...) This indicates that once the first harmonic of the apparatus is discovered, finding the requisite frequencies for any other harmonic is as simple as multiplying the first frequency by the number of the harmonic desired.