Standing Waves Experiment

Objective: Understand driven standing waves and investigate resonant conditions for a standing wave.

Equipment:

• Pasco Student Function Generator 
• Pasco Varible Frequency Wave Driver with String
• 50g weight hanger and slotted weight set
• 2 table clamps
• a Pulley 
• Short Rod
• Pendulum clamp
• 2 meter stick

Procedure: The length and mass of the string were measured in order to find the density μ. M=0.00409kg, l=4.43m,so μ=mass per unit length=0.000923(kg/m). Then the system was set up. The string was held in two places on the table, one of which had a mass hung on it (0.250kg) and went over the edge of the table via a pulley. The wave driver was attached to the string 2.00 ±0.02m away from the pulley. The wave driver was hooked up to the function generator. The set up looked like this:

The frequency of the generator was set to make the wave be in the fundamental mode. Then the number of nodes,the oscillation frequency, and the wavelength were recorded.





The frequency was increased to get different harmonics and that information was recorded again.

Harmonic	nodes	λ, m	frequency, Hz
1	2	4	12.1
2	3	2	23.7
3	4	1.333333333	34.3
4	5	1	46.3
5	6	0.8	57.5
6	7	0.666666667	68.9
7	8	0.571428571	80.2
8	9	0.5	90.6
9	10	0.444444444	103.8
10	11	0.4	115.4

     Then the tension was changed by changing the hanging mass(to 0.050kg) and the data was recorded again.

Harmonic	nodes	λ, m	frequency, Hz
1	2	4	4.7
2	3	2	10.3
3	4	1.333333333	15.6
4	5	1	20.9
5	6	0.8	26.3
6	7	0.666666667	31.3
7	8	0.571428571	36.7
8	9	0.5	41.6
9	10	0.444444444	46.7
10	11	0.4	51.5

Data Analysis:
     For the first case Microsoft Excel was used to make a graph of frequency vs. the inverse of wavelength (λ).The slope is also the speed of wave propagation which came out to be 45.697(m/s). The speed of propagation can also be found in the following way:

This gave a wave speed of 51.5(m/s) which is very close to the experimental. In fact it gives an error of 12%.

The same was done for the second case where the mass was changed. The experimental value was 20.81 (given from the graph). Then we can use the same method to find the wave speed theoretically which was 23.0(m/s). The percent error here is 10%

The ratio of the experimental speeds for case one and case two is 2.20. When This is compared to the ratio of the calculated values of the wave speeds (which is 20.24) it is evident that the ratios of the two are nearly identical.

For case one when the ratio of the first and second harmonic frequencies is taken it comes to be 1.96 which is nearly double as it should be to follow the pattern f=nf0 where n is the harmonic number. The same follows for case two where the ratios of the first two harmonic frequencies is 2.19.

Error: There was small error in the measurement of the mass and length of the rope that carried through the experiment. Another source of error came from finding the resonant frequencies. At some points (especially the higher harmonics) it was difficult to pinpoint the exact frequency that gave it the greatest amplitude. The values for the experimental wave speeds determined from the slopes of the graphs were very accurate and gave a r-squared value of 0.9997 so for the most point the experimental frequencies followed the pattern.