Standing Waves

Standing Waves

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:

Set up 

Set Up

 The length and mass of the string were measured in order to find the linear density,μ, of the string. M=0.00409kg, l=4.43m,so μ=mass per unit length=0.000923(kg/m). The system was then set-up. The string was held in two places on the table, one of which had a mass hung on it (0.250kg), Case 1. 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 frequency generator was then adjusted to obtain different harmonics in the string. the frequency and wave length was recorded for each harmonic. The procedure was then repeated with a different hanging mass ( 0.050 kg), Case 2 and thus a different tension.

Data Analysis:

Case 1 ( 0.250 kg)

After recording all the date for Case 1, frequency was graphed vs the inverse of wavelength (λ), in Microsoft Excel. This resulted in a linear graph with a slope equal to the velocity of wave propagation. this was also done for case 2 

Case 2 ( 0.050 kg)

Additionally, wave speed was calculated using :



This resulted in speeds of 51.5(m/s) for case 1 and 23.0(m/s) for case 2.
This was the compared to the slope values from the graphs which were 45.697(m/s) for case 1 and 20.81 (m/s) for case
Using the percent error formula gives an error of 12% for case 1 and the percent error for case 2 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 2.024) 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 minor error in the measurement of the mass and length of the rope that impacted 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.