Standing Waves Lab

The purpose of this lab was to study the basic idea of standing waves produced by a source of energy. Having a standing wave, one can study various elements that build up the wave: frequency, wavelength, wave speed, etc. Wave motion is governed by the wave equation:

y = A sin (kx ± ωt)

where two functions take positive and negative signs in the sine parameter to address a particle traveling to the left (positive) and to the right (left). Combining both functions yields a less complex expression: 

y = 2 A sin (kx) cos (ωt)

Setting y=0 and focusing on the sin component (x-dependent), we can find the number of nodes;

kx = πn (the sin (kx) needs to be zero to make y=0, and have a constant that also yields zero), or 

2πx/λ = nπ

This equation can then be rearranged to find the wavelength based on the number of nodes, n, and having the length of the string:

2L/λ = n

λ = 2L/n

Furthermore, knowing that v = λf, we can interpret the above equation as v = (2L/n)f to find the frequency f;

f = vn/2L

which is dependent of the number of nodes, n and the length of the string, L.

The equation above has the parameter v (wave velocity), which can be obtained through the equation:

where T is the tension of the string, or just T = mg, and μ is the linear density of the string.

The apparatus was set by tying one end of a string to a wave driver, and a mass at the free end. The mass hanged from the edge by a pulley. The driver was then connected to a function generator whose frequency would be controlled.

Mass of string, (kg) = .00237 ± .000005 kg

Length of the string, (m) = 1.98 ± .005 m

Linear density μ, (kg/m) =  0.001197 ± 0.0000039 kg/m

Case A) A total mass of 200 grams was hanged from the free end, and the function generator ran at  3 Volts. 

 

Frequency , f (Hz)	17.900	19.700	39.100	63.100	84.200	62.200	42.200	59.900	104.500
Number of nodes, n	2	2	3	4	5	4	3	4	6
Length of string, (m)	0.92±.005	1.00±.005	1.02±.005	0.96±.005	0.96±.005	1.00±.005	0.98±.005	1.02±.005	1.02±.005
Distance between nodes, (m)	0.92±.005	1.00±.005	0.51±.005	0.32±.005	0.24±.005	0.330±.005	0.49±.005	0.34±.005	0.20±.005
Wavelength λ, (m)	0.92±.005	1.00±.005	0.340±.0033	0.640±.0033	0.384±.0020	0.500±.0025	0.653±.0033	0.510±.0025	0.340±.0017

Tension T, (N) = mg = (.200 ± .0005 )(9.81) =  1.962 ± 0.0049  N

 Wave speed v, (m/s) = 40.48 ± 0.083 m/s

Case B) A mass of 50 grams was hanged from the free end, and the function generator was run at 5 Volts.

Frequency , f (Hz)	21.000	32.000	45.000	45.900	20.400	33.100	46.300	42.400
Number of nodes, n	3	4	3	5	3	4	5	5
Length of string, (m)	1.04±.005	1.02±.005	0.98±.005	1.02±.005	1.03±.005	0.94±.005	0.90±.005	0.96±.005
Distance between nodes, (m)	0.52±.005	0.34±.005	.49±.005	.25±.005	.51±.005	.31±.005	.22±.005	.24±.005
Wavelength λ, (m)	0.693±.0033	0.510±.0025	0.653±.0033	0.408±.0020	0.687±.0033	0.470±.0025	0.360±.0020	0.384±.0020

Tension T, (N) = mg = (.050 ± .0005 )(9.81) =  0.491 ± 0.0049  N

Wave speed v, (m/s) = 20.2 ± 0.11 m/s

Analysis of Data

v  =  λf      and,     λ =  v/f 

n = 2L/ λ

Case A):

 v =  40.48 ± 0.083 m/s

Frequency, f (Hz)	17.900	19.700	39.100	63.100	84.200	62.200	42.200	59.900	104.500
Wavelength of wave, λ (m)	2.261±.0046	2.055±.0042	1.035±.0021	0.642±.0013	0.4808±.00099	0.651±.0013	0.959±.0020	0.676±.0014	0.3874±.00079
Value of n	2.00±.015	2.00±.014	6.00±.065	3.00±.022	5.00±.037	4.00±.028	3.00±.022	4.00±.028	6.00±.042

Case B): 

v = 20.2 ± 0.11 m/s

Frequency, f (Hz)	21.000	32.000	45.000	45.900	20.400	33.100	46.300	42.400
Wavelength of wave, λ (m)	0.962±.0052	0.631±.0034	0.449±.0024	0.440±.0024	0.990±.0054	0.610±.0033	.436±.0024	0.476±.0026
Value of n	3.00±.020	4.00±.028	3.00±.022	5.00±.035	3.00±.020	4.00±.030	5.00±.039	5.00±.037

Experimental Wave speed: v = 40.493 m/s
Calculated Wavespeed:  v =  40.48 ± 0.083 m/s
Percent Error: 0.032%

 

Experimental Wave speed: v = 20.189 m/s
Calculated Wave speed: v = 20.2 ± 0.11 m/s
Percent Error: 0.054% 

Wave ratios.- Case A : Case B (Experimental):

v_A / v_b = 40.493 / 20.189 = 2.0057

 

 Wave ratios.- Case A : Case B (Calculated):

v_A / v_b = 40.48 ± 0.083 /  20.2 ± 0.11 = 2.00 ± .012

The ratios of both experimental and calculated wave speeds turned out to be the same, with an insignificant difference in the order of hundredths.

Case A

Length of string, (m)	0.92±.005	1.00±.005	1.02±.005	0.96±.005	0.96±.005	1.00±.005	0.98±.005	1.02±.005	1.02±.005
Number of nodes, n	2	2	3	4	5	4	3	4	6
f1	22.0±.13	20.3±.11	19.9±.11	21.1±.12	21.1±.12	20.3±.11	20.7±.12	19.9±.11	19.9±.11
fn	44.0±.26	40.6±.22	59.7±.33	84.4±.48	105.5±.60	81.2±.44	62.1±.36	79.6±.44	119.4±.66

The measured frequencies and fn were not the same because they all depended on the number of nodes, n.

There were some results that might have been incorrect, most specificallly when analyzing data from case A. This is due to the string not being well set up. We were having trouble getting a consistent standing wave, so that is why the first wave lenghts of Case A were very off.