Experiment 4: Standing Waves

Introduction:
This lab was conducted in order to understand the effect that an external force has on a standing wave.

Procedure:

The setup of the lab

Oscillate the string like so

Create oscillations for up to 10 harmonics for case 1, then repeat the procedure for the first 6 harmonics with 1/4 of its value.

Data:

Case 1:

Tension = 200g * 9.8 = 1.96 N

Mass of string = 0.74g

μ=0.000308kg/m

Nodes	Frequency (Hz)	Length (m)	Amplitude	Wavelength	1/λ
1	31	1.326	4.500	2.652	0.377
2	61	0.663	2.250	1.326	0.754
3	92	0.442	1.500	0.884	1.131
4	122	0.332	1.125	0.663	1.508
5	153	0.265	0.900	0.530	1.885
6	184	0.221	0.750	0.442	2.262
7	215	0.189	0.643	0.379	2.640
8	245	0.166	0.563	0.332	3.017
9	276	0.147	0.500	0.295	3.394
10	307	0.133	0.450	0.265	3.771

This shows the relationship between the frequency and wavelength when the string has a tension of 1.96N

Case 2: 

Tension = 50g * 9.8 = 0.49N

Mass of string = 0.74g

μ=0.000308kg/m

Nodes	Frequency (Hz)	Length (cm)	Amplitude	Wavelength	1/λ
1	15	1.326	1.100	2.652	0.377
2	31	0.663	0.550	1.326	0.754
3	46	0.442	0.367	0.884	1.131
4	62	0.332	0.275	0.663	1.508
5	77	0.265	0.220	0.530	1.885
6	93	0.221	0.183	0.442	2.262

This shows the relationship between the frequency and wavelength when the string has a tension of 0.49N

Analysis:

Wave velocity case 1

From graph:

v = 81.39 m/s

From equation:

v = 79.77 m/s

Wave velocity case 2

From graph:

v = 41.22 m/s

From equation:

v = 39.87 m/s

Ratio of wave velocities

From graph:

Ratio = 81.39/41.22 = 1.975

From equation: 

Ratio = 79.77/39/87 = 2.001

The ratios of wave velocities are very similar.

Number of harmonics	nf1 (Hz)
1	31
2	62
3	93
4	124
5	155
6	186
7	217
8	248
9	279
10	310

Ratio of frequencies

Ratios of harmonics	n	Case 1	Case 2	Ratio
	1	31	15	2.1
	2	61	31	2.0
	3	92	46	2.0
	4	122	62	2.0
	5	153	77	2.0
	6	184	93	2.0

The ratios for each comparable frequency are very similar. They have a ratio between 2.0-2.1. This is evident through the experimental results because when the tension was reduced to 1/4 its original value, the velocity doubled. 

This concludes that the velocity of a wave is proportional to the tension on the string. Our error could be attributed to measurements and the reliability of the equipment used.