Experiment 3: Relationship between wave speed, frequency, and wavelength

Introduction

The purpose of this experiment is to determine how the frequency and wavelength are related by measuring the time taken for a fixed number of waves to pass through a point using a spring with different wavelengths.

In this experiment, a long spring with a wavelength of 6.0m was created by wiggling the spring, and the time taken for 10 waves to pass through a point was recorded. The same procedure was repeated for a wavelength of 3.0m and 2.0m by creating 2 waves and 3 waves in the spring. The time taken was divided by 10 in order to get the period. Then the respected frequency and speed were computed for each trial by using following equations:

ƒ  = 1/T and v = λ/T

where ƒ is the frequency, T is period, v represents speed, and λ indicated wavelength. The speed resulted from the calculation was compared with the speed obtained from the graph. The graph was constructed using Logger Pro.

Figure 1: A spring forming a wavelength of 3.0m wave 

Figure 2: Brief procedure and recorded data

The time recorded was the time taken for 10 waves to pass through a point. Hence, frequency should be 10 times larger than what was written on the board.

Data and Analysis

Table 1: Recorded wavelength and period, along with calculated frequency and velocity

λ(m)	time(s)	T(s)	ƒ(Hz)	v(m/s)
6.0 ± 0.05	9.30 ± 0.050	0.930 ± 0.0050	1.08 ± 0.006	6.45 ± 0.064
3.0 ± 0.05	4.40 ± 0.050	0.440 ± 0.0050	2.27 ± 0.026	6.82 ± 0.138
2.0 ± 0.05	2.56 ± 0.050	0.256 ± 0.0050	3.91 ± 0.076	7.81 ± 0.248

Table 2: Comparison of two wave speeds from computation and graphing

Wave speeds	Average vdata(m/s)	vgraph(m/s)
	7.03 ± 0.150	7.509 ± 0.4662
Largest possible	7.18	7.975
Smallest possible	6.88	7.043

Graph 1: Frequency vs. wavelength

Calculations of uncertainties

         1)   ƒ = 1/T

uƒ = √[dƒ/dT x uT]² = √[(-1/T²) x uT]² = √(-1/0.930²) x 0.0050)² = 0.0006 Hz

         2)   v = λ/T

uv = √[( ∂v/∂T x uT)² + (∂v/∂λ x uλ)²] = √[(-λ/T² x uT)² + (1/T x uλ)²] = √[(-6.0/0.930² x 0.0050)² + (1/0.930 x 0.05)²] = 0.064 m/s

Discussion

According the graph 1, it was observed that the relationship between frequency and wavelength of the spring were inversely proportional. In this graph, y is frequency, x represents wavelength, and A indicates speed since frequency times wavelength results in a unit of m/s. Therefore, the relationship between ƒ, λ, and v could be presented by ƒ =v/ λ. This was reasonable because as wavelength decreases, more waves could pass through a point in a certain amount of time. Therefore, frequency increases. This relationship was also confirmed by the equations mentioned in introduction. However, as the graph shown, the auto fit line was not smooth. This was because very few data points were recorded, and there were also some errors in recorded time and wavelength. Yet all of these errors were covered by the uncertainties.

As shown in tabl-2, the wave speed computed from the recorded data was within the uncertainty of the speed given by the graph. For instance, the largest possible computed wave speed 7.18 m/s fell within the range of the graph wave speed which was 7.403 m/s to 7.975 m/s. This slight difference in speeds was possibly due to rounding errors and the tendency to auto fit the data by Logger Pro.

Conclusion

                The frequency and the wavelength of a spring were inversely proportional, and the constant in the graph was found to be the speed of the wave. This speed was computed to be approximately 7.03 m/s according the recorded data and 7.509 m/s according to the graph.