Experiment 4: Standing waves

Purpose:
The purpose of this experiment is to deeply gain the knowledge and understanding of the standing waves. We will examine the behavior of the standing waves depend on the external force. Moreover, the resonant conditions for standing waves on string will be investigated.

Procedure:

1. Obtain a string and measure its mass and length. m = 2.11 +/- 0.01g and l = 1.5 +/- 0.01 m.
2. Attract one end of the string to the hanging mass of 200g and other end to the table clamp.
3. Attract the wave driver to string approximate 10 cm from the clamp.
4. Set up the function generator to attract the wave driver.
5. Adjust the amplitude of the function generator to 5.00 volts and start adjusting the frequency until the string oscillates in its fundamental mode. Record the frequency, the number of nodes, total length of string.
6. Repeat step 5 for second, third, ... harmonic.

Data:

Harmonic	Frequency (Hz)	# nodes	length (m)
1	18	2	1.1
2	38	3	1.1
3	57	4	1.1
4	71.1	5	1.1
5	79.2	6	1.47
6	95.1	7	1.47
7	111.6	8	1.47

8. Reduce the mass of the hanging mass to 50g.
9. Repeat step 5.

Data:

Harmonic	Frequency (Hz)	# nodes	length (m)	Amplitude
1	8	2	1.47	10
2	16	3	1.47	5
3	23.9	4	1.47	5
4	32.3	5	1.47	5
5	40.8	6	1.47	5

  Analysis Data:

 Lambda λ = 2L/n = 2*1.1/1 = 2.2 m

Case 1: 

lambda λ (m)
2.2
1.1
0.733333
0.55
0.588
0.49
0.42

 The graph of the frequency vs. 1/λ

From the equation (5) : v = sqr (T/mu) = sqr (0.2*9.81 / (2.11*10^-3 / 1.5) = 37.3 m/s

% error = ( v1 - v2)/ (v1 + v2)/2 = (47.977 - 37.3)/(47.977 + 37.3)/ 2 = 6.3%

Case 2:

Lambda λ (m)

2.94

1.47

0.98

0.735

0.588

 The graph of the frequency vs. 1/λ

From equation (5) v = sqr (T/mu) = sqr (0.05*9.8/(2.11*10^-3 / 1.5)) = 18.7 m/s
% error = (v1 - v2)/(v1 + v2)/2 = (24.079 - 18.7 ) / (24.079+18.7)/2 = 6.3 %

Ratio of the wave speeds from the graph: R = v1/v2 = 47.977/24.079 = 2

Ratio of the wave speeds from equation (5) : R = v1/v2 = 37.3/18.7 = 2
From the two ratios, the ratios of wave speeds from two case is the factor of 2.

The measured frequencies for case 1 are most likely equal to n*f_1, where n is the number of harmonic.
The ratio of frequency of the second harmonic for case 1 compared to case 2.
R = f1/f2 = 38/ 16 = 2.38

Ratio for the third, fourth, fifth harmonic.

Harmonic	ratio
3rd	2.38
4th	2.20
5th	2.37

The ratios of the frequency of the third, fourth, fifth harmonic for case 1 and 2 are closed to each other.


Summary:

Based on the purpose of this experiment, the results come out for two cases are reasonable if the sources error are concerned. The % error for the wave speeds are the same for each case which mean there are the same error occur in this experiment. The errors may from  the friction of the pulley, the uncertainty of the mass and length of the string, and the uncertainty of the function generator. In comparison the frequencies for two cases, the ratios come out are very closed to each other. Therefore, we can assume that the ratio of the frequency of the standing wave on string for different tension is a constant. Theoretically, the ratio is equal to the square root of the tension ratio. However, the ratio in this experiment is more that ratio. There are some source errors in this experiment.