Abstract: The purpose of this experiment was to verify the relationship between frequencies, wave length and wave velocity of a transverse wave on a string, as well as the relationship between the spring tension and the number of standing waves formed. Two different strings used in this experiment are white and black in colour; the µ1 value calculated for the white string is 2.73 x 10-3 ±0.00055kg/m with an uncertainty of ±8.2932 x 10-5 kg/m while the µ2 value calculated for the black string is 1.38 x 10-3 kg/m with an uncertainty of ±8.6492 x 10-5 kg/m. However, the actual linear mass density µ0 of the white spring calculated is 2.87x10-3kg/m; compared to the experimental linear mass density µ1, the difference in error was 4.88%, however, the mass of the black spring was too small to be weighted therefore the actual linear mass density µ0 was unable to be calculated, so cannot be compared to the experimental linear mass density µ2.
Introduction:
A wave can be described as a disturbance that carries energy and travels through a medium form one location to another. Waves can be classified into two types, transverse waves and longitudinal waves. This experiment was based on transverse wave and can be described by the position of the particles at a particular time as well as how the position changes with time. Transverse wave can also be considered as a wave that remains at constant position. Wave produced by a vibrator travels down a string and reflects back along the string on reaching the fixed end of string, and a wave is often characterized by its wavelength λ, speed v and frequency of oscillation f which are related by the fundamental equation of wave motion λf = v. The speed v, of a wave depends on the tension F, in the string and its linear mass density µ, and this is given by the equation v = {√F/µ}. The stronger the force produced the greater the acceleration therefore the faster the movement of the wave (Newton’s second law of motion). Also, the greater the tension on a string the greater the pulling force from one particle to the other; therefore ability of the particle on a string to pull the particle next to it is directly proportional to the tension on the string. The speed of a wave is inversely proportional to its linear density, µ, because the mass of the end particle of the string will affect how fast it will respond to the pull from another particle. The maximum amplitude of a cord can only be achieved if the vibration of the frequency f, length of the cord L, and the speed of the wave on the cord v, is related such that a resonance and a stationary/standing wave are set up. Therefore, the natural frequencies fn, of the oscillation that produced the wave can be determined using the equation fn = {n/2L (√F/µ)} where n is the number of loops (segments between nodes).
Procedure:
The experiment was performed as written in the lab manual “refer to manual”.
Data and Analysis:
Fig 1: showing the graph of experimental frequencies vs. the natural frequencies of vibration of the white string.

The above graph represents the measured frequency vs. the natural frequency. From the graph, we can see that the slope is 19.137 ±0.00055 and the regression on the graph is 0.973, therefore demonstrating a straight line. The µ value for this graph was calculated to be 2.73 x 10-3 ±8.2932 x 10-5kg/m

The above graph represents the measured frequency vs. the natural frequency. From the graph, we can see that the slope is 26.954 ±0.00055 and the regression on the graph is 0.9756, therefore demonstrating a straight line. The µ value for this graph was calculated to be 1.38 x 10-3 ±8.6492 x 10-5kg/m.
Discussion and Conclusion: In this experiment, the actual linear mass density µ0, calculated is almost the same as the experimental linear mass density µ1. The value of µ0 is 2.87x10-3 ±0.00055kg/m while that of µ1 is 2.73 x 10-3 ±8.2932 x 10-5kg/m and we can see both from these two values that they are directly proportional to each other. The higher the hanged weight the nearer the values of µ0 to µ1 and the higher the tension in the spring thereby producing a higher amplitude. From the graphs, we can see that the linear density increases with increase in tension (increase in mass increases tension in string). Certain errors experienced in this experiment includes error due to the movement of the string along its support as a result of the vibration of the Mechanical Wave Driver, unclear position of the nodes and amplitude of maximum vibration, and error in weight, measurements and how horizontal the tread is. During the experiment, the vibration of the Mechanical Wave Driver causes the string to move along its support and the support to move along the table due to its light weight, therefore producing a non-constant frequency of vibration in the results. This can be corrected by using a more stable or high weighted support to prevent it from move and a string that would be tight (probably screwed with a knot) on the support. Also, the position of the node and amplitude of maximum vibration was unclear and therefore was estimated; as a result of this estimation in amplitude, the values obtained are not exactly correct. This can be corrected by taking the average of many trials (about 5 trials) on the same amplitude in order to reduce the error on the estimated amplitude. Errors in measurements, weight and how horizontal the string is can either affect the linear density calculated positively or negatively. If the string was not in a horizontal plane as it should be; the vibration of the spring might me affected and therefore will affect where the maximum amplitude of vibration will occur. Also, the measurements of the weight and length are not exactly correct due to the precision of the scale and measuring instrument (wooden ruler) used, and therefore can affect the calculated results in this experiment. These errors can be reduced by using a more précised scale and measuring instruments as well as setting the string at the best horizontal plane as possible. Air resistance can also be a source of error in this experiment, by affecting the vibration of the string and in order reduce this; the experiment can be done in a vacuum to reduce air resistance error. In conclusion, we have seen that the speed of a wave along a string is dependent on both the tension F, and its linear mass density µ; and that a wave’s speed is inversely proportional to its linear density, as well as standing waves been produced only at certain frequencies.
Appendix
Table 1: -showing the measurements, antinodes, frequencies and the natural frequencies of the white string. Length (m) ±0.0005 | Mass (kg) ±0.00005 | Antinode # | Frequency (Hz) ±2 | n/2L × (√mg) | 0.80 | 0.100 | 1 | 9.2 | 0.61871843 | 0.80 | 0.100 | 2 | 27.1 | 1.23743686 | 0.80 | 0.100 | 3 | 46.0 | 1.8561553 | 0.80 | 0.100 | 4 | 62.5 | 2.47487373 | 0.80 | 0.300 | 1 | 23.8 | 1.07165176 | 0.80 | 0.300 | 2 | 46.4 | 2.14330352 | 0.80 | 0.300 | 3 | 69.2 | 3.21495529 | 0.80 | 0.300 | 4 | 93.5 | 4.28660705 | 0.80 | 0.500 | 1 | 29.4 | 1.38349648 | 0.80 | 0.500 | 2 | 58.2 | 2.76699295 | 0.80 | 0.500 | 3 | 86.8 | 4.15048943 | 0.80 | 0.500 | 4 | 114.2 | 5.53398591 | 0.80 | 0.700 | 1 | 36.0 | 1.63697511 | 0.80 | 0.700 | 2 | 67.6 | 3.27395021 | 0.80 | 0.700 | 3 | 100.9 | 4.91092532 | 0.80 | 0.700 | 4 | 138.3 | 6.54790043 | 0.90 | 0.100 | 1 | 13.6 | 0.54997194 | 0.90 | 0.100 | 2 | 27.6 | 1.09994388 | 0.90 | 0.100 | 3 | 42.2 | 1.64991582 | 0.90 | 0.100 | 4 | 56.4 | 2.19988776 | 0.90 | 0.300 | 1 | 20.3 | 0.95257934 | 0.90 | 0.300 | 2 | 41.9 | 1.90515869 | 0.90 | 0.300 | 3 | 61.2 | 2.85773803 | 0.90 | 0.300 | 4 | 83.0 | 3.81031738 | 0.90 | 0.500 | 1 | 27.2 | 1.22977465 | 0.90 | 0.500 | 2 | 51.7 | 2.45954929 | 0.90 | 0.500 | 3 | 76.5 | 3.68932394 | 0.90 | 0.500 | 4 | 102.2 | 4.91909858 | 0.90 | 0.700 | 1 | 29.4 | 1.45508898 | 0.90 | 0.700 | 2 | 59.9 | 2.91017797 | 0.90 | 0.700 | 3 | 88.9 | 4.36526695 | 0.90 | 0.700 | 4 | 122.7 | 5.82035593 | 1.00 | 0.100 | 1 | 9.3 | 0.49497475 | 1.00 | 0.100 | 2 | 21.1 | 0.98994949 | 1.00 | 0.100 | 3 | 38.1 | 1.48492424 | 1.00 | 0.100 | 4 | 51.7 | 1.97989899 | 1.00 | 0.300 | 1 | 18.2 | 0.85732141 | 1.00 | 0.300 | 2 | 36.0 | 1.71464282 | 1.00 | 0.300 | 3 | 56.3 | 2.57196423 | 1.00 | 0.300 | 4 | 74.7 | 3.42928564 | 1.00 | 0.500 | 1 | 22.9 | 1.10679718 | 1.00 | 0.500 | 2 | 46.5 | 2.21359436 | 1.00 | 0.500 | 3 | 69.2 | 3.32039154 | 1.00 | 0.500 | 4 | 92.8 | 4.42718872 | 1.00 | 0.700 | 1 | 26.5 | 1.30958009 | 1.00 | 0.700 | 2 | 54.1 | 2.61916017 | 1.00 | 0.700 | 3 | 82.1 | 3.92874026 | 1.00 | 0.700 | 4 | 111.3 | 5.23832034 | 1.10 | 0.100 | 1 | 9.4 | 0.44997704 | 1.10 | 0.100 | 2 | 18.5 | 0.89995409 | 1.10 | 0.100 | 3 | 29.0 | 1.34993113 | 1.10 | 0.100 | 4 | 39.9 | 1.79990817 | 1.10 | 0.300 | 1 | 18.5 | 0.7793831 | 1.10 | 0.300 | 2 | 36.3 | 1.5587662 | 1.10 | 0.300 | 3 | 50.2 | 2.3381493 | 1.10 | 0.300 | 4 | 64.4 | 3.1175324 | 1.10 | 0.500 | 1 | 19.3 | 1.00617926 | 1.10 | 0.500 | 2 | 41.5 | 2.01235851 | 1.10 | 0.500 | 3 | 62.4 | 3.01853777 | 1.10 | 0.500 | 4 | 85.5 | 4.02471702 | 1.10 | 0.700 | 1 | 24.4 | 1.19052735 | 1.10 | 0.700 | 2 | 49.7 | 2.3810547 | 1.10 | 0.700 | 3 | 74.2 | 3.57158205 | 1.10 | 0.700 | 4 | 99.8 | 4.7621094 |

Table 2: -showing the measurements, antinodes, frequencies and the natural frequencies of the black string. Length(m)±0.0005 | Mass(kg)±0.00005 | Antinode # | Frequency (Hz) ±2 | n/2L × (√mg) | 0.7 | 0.1 | 1 | 23.4 | 0.70710678 | 0.6 | 0.1 | 1 | 29.1 | 0.82495791 | 0.5 | 0.1 | 1 | 34.6 | 0.98994949 | 0.4 | 0.1 | 1 | 41.1 | 1.23743687 | 0.3 | 0.1 | 1 | 49.6 | 1.64991582 |

Error propagation of the black string | Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | | Intercept | 6.399065 | 2.784575 | 2.29804 | 0.105177 | -2.4627 | 15.26083 | -2.4627 | 15.26083 | | X Variable 1 | 26.95411 | 2.458547 | 10.96343 | 0.001625 | 19.12992 | 34.7783 | 19.12992 | 34.7783 | | | | | | | | | | | |

White string | | | | | | | | | Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | | 2.915441 | 0.847061 | 3.441832 | 0.00104 | 1.222191 | 4.608692 | 1.222191 | 4.608692 | | 19.1374 | 0.290613 | 69.29287 | 1.89E-60 | 19.55648 | 20.71833 | 19.55648 | 20.71833 | | | | | | | | | | |
Calculations:-
Calculation for µ0 µ0 = kg/m
= 5.74x10-3 ÷ 2 = 2.87x10-3kg/m µ For white string = 1/slope2 µ1 = 1/slope2
= 1/19.1372
= 2.73 x 10-3kg/m µ For black string = 1/slope2 µ2 = 1/slope2
= 1/26.9562
= 1.38 x 10-3kg/m
Calculation for the propagation error of n/2L×√F
SZ/Z = [√(sx/x)2 + (sy/y)2]
Sz = (0.61871843) {√ (0.0005/0.8)2 + (0.00005/0.08)2}
= ±0.00055
Percent difference between µ1 and µ0
% diff. = (/χ1 – χ2/ ÷ /χ2/) x 100
= (/0.00273 – 0.00287/ ÷ /0.00287/) x 100
=4.88%
Uncertainty on µ for white string
Sµ1 = 2(19.137)-3Sslope
= 2(19.137)-30.290613
= ±8.2932 x 10-5
Uncertainty on µ for black string
Sµ1 = 2(26.956)-3Sslope
= 2(26.956)-30.847061
= ±8.6492 x 10-5