Mechanical Waves: Standing Waves

          The purpose of this lab is to explore standing waves, the properties of these waves, and how the properties relate. The experiment consist of a string that is fixed at two ends. The tension is known in the string. Then string then has a wave driver attached which in turn is attached to a function generator. The function generator is turned on and the frequency is adjusted. At certain frequencies, the string will oscillate at it's fundamental node or a multiple of that fundamental node. At these frequencies which cause harmonic oscillations, we will measure frequency, wavelength, and the number of loops. The following data was acquired during this process along with three other groups of data obtained from peers conducting the same lab.

Due to technical difficulties with the camera, this picture is not of my personal lab set up. This is a picture of a peer's lab. source

Linear density of the string used = 0.001839 kg/m

T = tension in string

L = total length of the string

Loops = number of loops visible at a given frequency

T(N)	L(m)	Loops	Frequency (hz)	Wavelength (m)
3.924	2	1	15	4
	2	2	29.4	2
	2	3	44	1.33
	2	4	58.4	1
	2	5	72.3	0.8
	2	6	100.3	0.666
1.959	1.33	1	16	2.66
	1.33	2	32	1.33
	1.33	3	46	0.887
	1.33	4	63	0.665
	1.33	5	76	0.532
0.981	1.31	3	34	0.87
	1.31	4	46	0.655
	1.31	5	56	0.524
	1.31	6	61	0.437
	1.31	7	77	0.374
	1.31	8	93	0.327
1.962	1.31	3	46	0.87
	1.31	4	64	0.655
	1.31	5	79	0.524
	1.31	6	96	0.437
	1.31	7	110	0.374
	1.31	8	126	0.327

            This data was then used to construct a plot of f vs. 1/wavelength. The resulting curve fits yielded 1/v which could be used to find v (the wave speed). The following is the curve fit and the wave speeds derived from the graph, experimentally from the data, and predicted from known parameters.

 

Wave Speed "Graph" (m/s)	Wave Speed Predicted (m/s)	Wave Speeds Experimental (m/s)
66.3	53.15123	60
		58.8
		58.52
		58.4
		57.84
		66.7998
38.2	37.55485	42.56
		42.56
		40.802
		41.895
		40.432
43.4	26.57561	29.58
		30.13
		29.344
		26.657
		28.798
		30.411
43.4	37.5836	40.02
		41.92
		41.396
		41.952
		41.14
		41.202

           After computing those values for wave speed, the experimental average, standard deviation, and the percent difference between experimental average and predicted values was computed. The table below shows these values.

Experimental Average (m/s)	Standard Deviation (m/s)	%Dif.
60.06	3.379	13
41.65	0.9898	10.9
29.15	1.35	9.7
41.27	0.7053	9.81

          The overall data from my group and other groups was relatively consistent with computing values for n (harmonic number) and wave speed. When using the equation that relates the harmonic number to wavelength, all the values computed for N matched the observed values of N loops on the string. The waves peed values for the graph, experimental, and predicted all fell within a reasonable range of each other considering the standard deviation and the limitations on the equipment. The limitations in the equipment can be seen in the consistency of the percent difference between predicted and average wave speed values. Things such as variances in frequency, fine tuning the harmonic frequency, correctly measured wavelength, non ideal string and tension slightly oscillating can cause the 10.8% average error seem in the experiment.