Standing Waves

The purpose of the experiment is to investigate the ideas of standing waves emanating from an external force. The wave speed v, wavelength, period T will be determined for two distinct cases.  For the first case there will be a hanging mass of 0.200 +/- 0.0005 kg. For the second case, there will a hanging mass of 0.050 +/- 0.0005 kg.  The first seven harmonics will observed for both cases.

The equipment is set up such that one end of the string tied down to the wave driver and the other runs over a pulley where there is a hanging mass attached to it.  

Note: It should be noted that the mass of the string is 0.00286 +/- 0.000005 kg and the length of the string is 2.48 +/- 0.005 M while the length of the string oscillating is 1.60+/- 0.005 M.  





Data for .250 kg hanging mass:


Harmonic   frequency f (hz) wavelength (m)        Period T (s)     1/wavelength (m^-1)
1               13 +/- .00005         3.2 +/- .00005     .077+/-.00005    .3125+/-.00005
2               26+/- .00005         1.6 +/- .00005     .038+/-.00005       .6250+/-.00005
3               39 +/- .00005       1.06 +/- .00005     .026+/-.00005   .9434+/-.00005
4              51.5 +/- .00005  .8 +/- .00005     .019+/-.00005     1.250+/-.00005
5              64 +/- .00005        .64 +/- .00005     .016+/-.00005      1.5625+/-.00005
6             77.2 +/- .00005      .53 +/- .00005     .013+/-.00005   1.8867+/-.00005
7            90.1 +/- .00005     .457+/- .00005     .011+/-.00005    2.1882+/-.00005




The slope of the graph represents the experimental velocity, 40.9 m/s.
The theoretical velocity v is equal to (T/mu)^(1/2) = 46.11 m/s.
This yields a percent error of 12.7 %.






Data for .050 kg hanging mass:

Harmonic       frequency f (hz) wavelength (m) Period T (s)             1/wavelength (m^-1) 1               6+/-.00005   3.2+/-.00005        0.1667+/-.00005 0.3125+/-.00005 2               12+/-.00005 1.6+/-.00005        0.0833+/-.00005 0.625+/-.00005 3               18+/-.00005 1.07+/-.00005      0.0556+/-.00005 0.935+/-.00005 4               24+/-.00005 0.8+/-.00005        0.0417+/-.00005 1.25+/-.00005 5               32.2+/-.00005 0.64+/-.00005      0.0311+/-.00005 1.5625+/-.00005 6               39.2+/-.00005 0.53+/-.00005       0.0255+/-.00005 1.8868+/-.00005 7               46+/-.00005 0.457+/-.00005      0.0217+/-.00005 2.1882+/-.00005




The slope of the graph represents the experimental velocity, 21.5 m/s.

The theoretical velocity v is equal to (T/mu)^(1/2) =20.62 m/s
This yields a percent error of  4.10%

Boiled down to its purest form, the ratio of the experimental wave speeds is roughly 2:1. This inevitably is also the case for theoretical wave speeds--2:1.  Moreover, this lab supports that the nth harmonic frequency is equal to the product nf, where f is the fundamental frequency.  One should consider case one.  For case one, the third harmonic frequency was measured to be 39 hz.  Similarly, applying f-n = nf where n is equal to three since it is the third harmonic and f= 13, one also gets 39 hz.  For the seventh harmonic in the same case, the frequency was found to be 90.1 hz meanwhile the product nf--n is seven this time--yields 91 hz. These numbers do not perfectly emulate one another but this is the small error pronounced in the experiment rooted in the lil-bit-faulty equipment.  Nonetheless, the datable for the second case--the 0.050 kg case--also  supports f-n = nf.