Experiment 4: Standing Waves

Introduction

                The purpose of this experiment is to investigate the behavior of standing waves caused by an external force to initiate a transverse wave on a string. Standing wave occurs when an original wave and a reflected wave superpose. Standing waves have constant wave pattern, and the amplitude of the points on the wave are changing. The only points that are not changing in a standing wave are nodes. Node is half of the wavelength. The relationship between the length of the string (L), number of half wavelengths or loops (n), and wavelength (λ) is indicated by

L=nλ/2                                  (1)

 As found in experiment 3, frequency, wavelength, and velocity of a transverse wave in a string is related according to this equation.

ƒ = v/λ                                  (2)

And the velocity of a transverse wave can be obtained by

v=√(T/μ)                                (3)

where v is velocity or wave speed, T represents tension, and μ indicated mass density.

                The experiment was conducted using various apparatus such as variable frequency wave driver, function generator, 200g and 50g weight hanger, table clamps, rod, pendulum clamp, pulley, multimeter, and a meter stick. The mass and length of the string were recorded to determine its mass density, μ. One end of the string was attached to a 200g weight hanger, and the other end was tied to the clamp. The length or distance between each end was measured to be 1.2m. The string wave driver which was connected with the function driver was put 10cm away from the pendulum clamp. The function generator was set to 5.0 voltages, and the frequency was adjusted to get a certain number of loops. Frequencies with respective number of loops were recorded. The procedure was repeated using 50g weight hanger in place of 200g. The wavelength and velocity were computed based on the data. Two graphs were constructed, and the velocity obtained from the graph was compared with that from computation. 

Figure 1: The set-up of the apparatus to measure the frequency of standing waves with various loops  

Figure 2: Wave driver that helps to create transverse waves

Data and Analysis

L=1.2 ± 0.05m, m_rope =1.47 ± 0.005g,

      1)  m_tension = 200 ± 0.5g

      2)  m_tension = 50 ± 0.5g

Table1: Recorded frequency, nodes, and calculated wavelength, velocity for 200g of weight

n # of loops	ƒ1 (Hz)	ƒn = nƒ1 (Hz)	# of nodes	λ (m)	velocity(m/s)
1	18 ± 0.5	18 ± 0.5	2	2.40 ± 0.10	40.0 ± 0.84
2	37 ± 0.5	36 ± 1.0	3	1.20 ± 0.05
3	56 ± 0.5	54 ± 1.5	4	0.800 ± 0.033
4	76 ± 0.5	72 ± 2.0	5	0.600 ± 0.025
5	95 ± 0.5	90 ± 2.5	6	0.480 ± 0.020
6	114 ± 0.5	108 ± 3.0	7	0.400 ± 0.017
7	133 ± 0.5	126 ± 3.5	8	0.343 ± 0.014

Table2: Recorded frequency, nodes, and calculated wavelength, velocity for 50g of weight

n # of loops	ƒ2 (Hz)	ƒn = nƒ1 (Hz)	# of nodes	λ (m)	velocity(m/s)
1	8 ± 0.5	8 ± 0.5	2	2.40 ± 0.10	20.0 ± 0.43
2	19 ± 0.5	16 ± 1.0	3	1.20 ± 0.05
3	28 ± 0.5	24 ± 1.5	4	0.800 ± 0.033
4	38 ± 0.5	32 ± 2.0	5	0.600 ± 0.025
5	48 ± 0.5	40 ± 2.5	6	0.480 ± 0.020
6	58 ± 0.5	48 ± 3.0	7	0.400 ± 0.017
7	70 ± 0.5	56 ± 3.5	8	0.343 ± 0.014

Table 3: Ratio of frequencies from table 1 and 2

n # of loops	1	2	3	4	5	6	7	Average
ƒ1 / ƒ2	2.3	1.9	2.0	2.0	2.0	2.0	1.9	2.0

Table 4: Comparison of ratio of wave speeds from computation and graphing

Methods	v1/v2
Computation	2.00
Graph	1.892

Graph 1: Frequency vs. 1/wavelength (200g)

Graph 2: Frequency vs. 1/wavelength (50g)

Calculations of uncertainties

      1)  λ = 2L/n

uλ = √(dλ/dL x uL)² = √(2/n x uL)²

     = √(2/1 x 0.05)² = 0.10 m

      2)  T = mg

uT = √(dT/dm x um)² = √(g x um)²

     = √(9.8 x 0.0005)² = 0.0049 N

      3)  μ = m/L

uμ = √[(∂μ/∂m x um)² + (∂μ/∂L x uL)²] = √[1/L x um)² + (-m/L² x uL)²]

      = √[1/1.20 x 0.000005)² + (-0.00147/1.2² x 0.05)²] = 0.0000512 kg/m

      4)  v = √(T/μ)

uv = √[(∂v/∂T x uT)² + (∂v/∂μ x uμ)²] = √[(0.5/(μ√(T/μ)) x uT)² + (-0.5T/(μ²√(T/μ)) x uμ)²]

     = √[(0.5/(0.001225√(1.96/0.001225)) x 0.0049)² + (-0.5 x 1.96/(0.001225²√(1.96/0.001225)) x 0.0000512)²]

     = √(0.0025 + 0.69875)

     =0.837 m/s

Conclusion

As shown in table 1 and graph 1, the wave speeds from computation and from the graph were approximately 40.0 m/s and 46.04 m/s. Even though these values did not agree within the uncertainties, these results were very similar. In same manner, the wave speeds were 20.0m/s and 24.34 m/s respectively when 50g hanging mass was used. These results were shown in table 2 and graph 2. Again, these values were not within the uncertainties. In column 2 and 3 of table 1 and 2, the comparison between the recorded frequencies and the computed frequencies were shown. The frequencies on column 3 were a factor of the first recorded frequency, and these should have been the same with the recorded frequencies. However, column 2 and 3 values were not agreed within the uncertainties. As wavelength decreased, larger deviation from the recorded frequencies was observed.

According to table 4, the ratios of wave speed computed based on data and graph were approximately 2. Even though the ratios were not exactly the same, they highly agreed with each other. This result was confirmed by the equation 3 mentioned in the introduction. Since velocity was proportional to square root of tension, velocity had to decrease by a factor of 2 when tension decreased by a factor of 4.  In table 3, it was shown that the ratio of ƒ₁ and ƒ₂ were approximately 2. Even though the ratio changed a little for different loop numbers, the ratio maintained fairly constant. This was reasonable because frequency and velocity were directly proportional according to the equation 2. Therefore, as velocity increased by a factor of 2, so did the frequency.

There were a lot of possible experimental errors that could cause the above compared values not to agree within uncertainties. First, there was an inaccuracy in adjusting frequencies to obtain an exact number of loops. As the frequency was raised to higher value to obtain more nodes, the wave became harder to visible. Another factor influenced the experimental result was that the loop should be counted starting from the wave driver, but the loop was counted from the point of the clamp where the string was tied. This was shown in figure 2. If the number of loops was counted from the wave driver, the frequency could change. Therefore, these factors could contribute to inaccurate values of frequency which then led to inaccurate wave speed. In addition, the hanging mass did not remain stable throughout the experiment. This vibration in hanging mass could also cause a variation of tension in the string. Approximation in reading the length of the string also contributed to these errors. All of these led to inaccuracy in calculations of wavelength and velocities.