Experiment 4: Standing Waves

Purpose: The purpose of this experiment is to verify standing waves driven by an external force and investigating resonant conditions for standing waves on a string.

Formulas and Derivations:

Procedure:

The apparatus used in the experiment was setup as follows:

Figure 1: The wave driver on one end of the the string

Figure 2: Mass at the other end of the apparatus changed for each case in the experiment:
 Case 1 = 200 g
Case 2 = 50 g

Figure 3: The function generator output the frequency . The frequency shown on the function generator was recorded for every harmonic oscillation.

Figure 4: Example of harmonic oscillation

 DATA & ANALYSIS:


1. 

2. 

3.

This equation is used to determine the theoretical  wave speed.

Error Propagation:

4.

Using the slope of the graph, wave speed was determined

Slope	Wave speed (m/s)	Ratio
Case I	42.109	1.8
CaseII	23.216

Using the equation v=sqrt(TL/m) to find the wave speed

Equation	Wave speed (m/s)	Ratio
Case I	40.647 ± 1.752	2.0
Case II	20.320 ± 0.72

The ratios of between the two wave speeds was fairly close.

	Wave speed error calculation	Percent Error
Case I	(42.109-40.647)/(40.647)	3.6%
Case II	(23.216-20.320)/(20.320)	14.2%

The differences between the the slope and the equation are quite close with only a small percentage in error.

5.

Case I
fn	nf1
13	13
26	26
40	39
53	52
66	65
80	78
93	91
106	104
119	117
134	130

For Case I, the measured values for f_n compare quite closely to theoretical frequencies of nf₁. The first two values are the same, while the 3^rd and 4^th harmonic number are off by 1. The pattern in this relationship shows that with a greater number in the harmonic there is a slight more variation between the theoretical and the measured.  

n	f1/f2
1	2.2
2	1.9
3	2.0
4	2.0
5	1.9
6	2.0

The ratio is about the same for all the harmonic numbers with a constant ratio of about 2.0. The first frequency was off from the other values by 2.2 which is only a slight difference which could have come from experimental error. Nevertheless, the pattern of the ratios with increasing harmonic number is at a constant 2. 

CONCLUSION:

                The wave speed in this experiment was calculated in two ways. In the experimental process, a graph was conducted and the slope gave the wave speeds. Comparing this wave speed (Case I- 42.109 m/s; case II- 23.216 m/s) to a theoretical,  the equation v=sqrt(T/μ) was used. Although the uncertainty did not agree with the experimental results, the experimental error was quite low with a 3.6% difference in case I and a 14.2% difference in case II. Based on the tables and ratios analyzed in the experiment, it can be concluded that tension in the system affects the frequency and wave speed.  According to the graphs the velocity and frequency between case I and case II showed a difference of a factor of 2. The relationship showed that with a decrease in tension there was a decrease in both frequency and velocity. More specifically, as the tension was decreased to ¼ of its value from case I, both the frequency and velocity decreased by a factor of 2. It also verified that the ratio was consistent through every nth harmonic, which consistently showed a factor of 2 in the ratio of f₁/f₂. This can also be seen from the equation that was used to find the theoretical values for the wave speed. Because the equation has a radical present the factor is always decreased or increased by a square or radical

There are many experimental sources of error that need to be considered in the experiment. One of which was in the inaccuracy of finding the right frequency when trying to the nth order of a harmonic. It didn’t seem to be a problem for the first 5 or 6 harmonics, but when moving to higher harmonics it was more difficult to find the number of loops the string had produced. Therefore there could have been more inaccuracies in the higher order of the harmonics, which explains the higher difference between the theoretical as the frequency had increased. Another experimental error could have risen from the decreased mass of case II. With the decreasing tension, it was more difficult to determine the number of loops accurately which led to a more inaccurate way of measuring frequency. Reading the length of the string was also an error in the experiment because a more reliable and precise measurement tool should have been considered since it was dependent on the theoretical value of the experiment as well as the wavelength.