Transverse Wave: Frequency of Vibration Sayre J. Bongo, PHY12L/A8 [email protected]
Abstract Waves are a wiggle in space that has the ability to carry energy from one location to another and are caused by a vibration or disturbance. There are two different types of waves; transverse and longitudinal. A transverse wave is when the wave is vibrating perpendicular to the direction the wave is traveling. In this experiment the frequency of vibration of a stretched string was determined, and how the frequency of a vibrating string is affected by tension and linear mass density was studied. To achieve the aforementioned objectives, the experiment was divided into two parts. In the first, the frequency of vibration was determined using a constant linear mass density or, simply put, using only one of the five guitar strings for all five trials. And then for the second part, all five strings were used to see how differing linear mass densities would affect the frequency of vibration or factors thereof. The materials included a string vibrator, sine wave generator, five different guitar strings, a set of masses, meter stick, and the rest of the experimental set-up (see Methodology). The data was gathered and percent error was computed for in both cases. From the experiment conducted, the relationship of velocity, frequency and wavelength was studied. Keywords: wave, transverse wave, frequency of vibration, tension, linear mass density
Introduction Transverse waves assume a sinusoidal wave pattern. This waveform is visually represented by a standing wave, which is a result of superimposition of two traveling waves: the incident and the reflected waves. In this experiment, the standing waves are set up in a stretched guitar string by the oscillations of an electrically-driven string vibrator. By using different sizes of the strings, the dependency of frequency to the linear mass density was determined. Parameters like tension, number of segments formed were also identified. 1. Theory If a string, under tension, is connected to a harmonic oscillator, standing waves are produced by the interference of two traveling waves. The waves travel down the string to the other end and back that results to alternate regions of node and antinode. The node (N) and the antinode (A) refer to the regions of no vibration and maximum vibration, respectively. The distance between two nodes is called a segment. See Fig. 1.
Figure 1. Standing waves in a stretched string. Changing the tension will definitely change the number of segments between the ends of the string. Consequently, a change in the number of segments will alter the wavelength (λ) and the rest of the parameters involved. The relationship between velocity, frequency and wavelength is given by (1), known as the general wave equation: (1) For a transverse wave, the velocity of propagation is a function of the tension and the linear mass density of the medium: √
The frequency of vibration, after combining the two equations, is √ The wavelength must be expressed in terms of other measurable quantities like the number of segments formed (n) and the length of the vibrating string (L). The length of one segment is equal to ½ of wavelength and the length of vibrating string is divided into equal number of segments. Thus, the frequency of vibration is √
Linear mass density µ is mass per unit length of the medium and T is the tension in dynes. Methodology 1. Materials, Apparatus and Set-up The materials used for the experiment were a string vibrator, a sine wave generator, two iron stands with clamp (one to mount the pulley on and the other for the string vibrator and sine wave generator), a pulley, a set of weights, a mass hanger, an extension cord, a
meter stick and five different guitar strings. The equipment was set up in the manner shown in Fig. 2.
Figure 2. Experimental set-up. 2. Procedure To achieve the objectives of this experiment, the experiment was divided into two parts. In the first, the frequency of vibration was determined using a constant linear mass density or, simply put, using only one of the five guitar strings for all five trials. And then for the second part, all five strings were used to see how differing linear mass densities would affect the frequency of vibration or factors thereof. 2.1 Determining the Frequency of Vibration (Constant Linear Mass Density) For the first part of the experiment, we set up the materials and apparatus by connecting the sine wave generator to the string vibrator and then attaching our chosen size of guitar string, diameter = 0.022 in (used for all five trials), to the stylus of the string vibrator (see Fig. 3) before passing it over the pulley. Then, the mass hanger was attached to the end of the guitar string that hung from the pulley.
Figure 3. Attaching guitar string to stylus of the string vibrator using a small nut and bolt. 3
A 50-gram weight was then added to the mass hanger. After which a constant frequency of 110 Hz was set by adjusting the frequency knob of the sine wave generator. The amplitude knob was slowly manipulated to make sure that segment formation was clearly defined. Once this was achieved, the number of distinct segments formed was counted and the length of these measured. To minimize error, the segment near the stylus was not counted. The data was recorded in Table 1 and the experimental value for the frequency of vibration was solved using (3). The same was done four more times but with an increasing weight on the mass hanger. Finally, the average of the experimental values for the frequency of vibration was computed and percent error was solved. 2.2 Determining the Frequency of Vibration (Variable Linear Mass Density) For the second part, the same procedure as in 2.1 was followed but with different sizes of the guitar string. But, the tension was kept constant for all five differently sized guitar strings. The data for this part was recorded in Table 2. Results and Discussion In the first part of the experiment, one string was chosen for all five trials so as to keep the linear mass density constant. The linear mass density of the wire was taken from the given data in the second table of the data sheet. The weight at the end of the string, and thus the tension, however was varied by an increment of +10 grams. Table 1 shows the data for this part. Table 1. Determining the Frequency of Vibration (constant linear mass density) diameter of wire = 0.022 in TRIAL
1 2 3 4 5
53900 dynes 63700 dynes 73500 dynes 83300 dynes 93100 dynes
linear mass density of wire = 0.0184 g/cm length of string with number of frequency of complete number of segments, n vibration, f segments, L 5 38 cm 112.6 Hz 5 43 cm 108.2 Hz 5 44 cm 113.6 Hz 5 48 cm 110.8 Hz 5 52 cm 108.1 Hz average frequency of vibration 110.66 Hz actual value of frequency of vibration 110 % error 0.60 %
It can be seen from the data collected above that as the tension on the string increases so does the length of the string with complete number of clearly formed segments. Another observation that can be made from the data above is that although the tension and length of string with complete number of segments (L) may be directly proportional, there is no clear pattern for the relationship between the tension and the frequency of vibration. This could be due to some errors in the way the measurements of L were made and/or maybe not enough trials were performed to establish a pattern. For the second part, the reverse of the first part was done. The linear mass density was varied by using all five guitar strings, but the tension was constant by using only the same weight at the hanger for all trials. The data is shown in Table 2. 4
Table 2. Determining the Frequency of Vibration (variable linear mass density) TRIAL 1 2 3 4 5
diameter of wire 0.010 in 0.014 in 0.017 in 0.020 in 0.022 in
Linear mass density, µ 0.0039 g/cm 0.0078 g/cm 0.0112 g/cm 0.0150 g/cm 0.0184 g/cm
53900 dynes 5 67 cm 53900 dynes 5 60 cm 53900 dynes 6 58 cm 53900 dynes 6 53 cm 53900 dynes 6 46 cm average frequency of vibration actual value of frequency of vibration % error
f 111.0 Hz 109.5 Hz 113.5 Hz 107.3 Hz 111.6 Hz 110.58 Hz 110 0.53 %
Table 2 shows that as the diameter and thus the linear mass density of the string increases, the number of distinct segments formed (n) increases and the length of the string with clearly defined n number of segments (L) decreases. The pattern for the relationship of the linear mass density to the frequency of vibration seems to be like a ‘wave’ that goes down and up and down again and so on and so forth. However, this cannot really be taken as a definite pattern because, once again, the number of trials performed is not sufficient. The sources of error for this part are similar to the first such as the measuring of the length and the inconsistent counting of the whole segments. Conclusion and Recommendation The results in Table 1 show that the tension is directly proportional to the length of the string with complete number of clearly formed segments but did not show any pattern with regard to frequency. And then Table 2 showed that when tension is held constant while linear mass density is increased, number of segments increases, length of string decreases, while the frequency fluctuates. However, it should have been that as tension was increased while linear mass density was held constant, frequency would also increase; and in part two, as linear mass density was increased while tension was held constant, frequency should have decreased. Sources of error were wrong measurement of the length of the total number of segments since you cannot place the meter stick near the string for it will affect the movement of the wave. In addition, we must count the number of segment after it passes the stylus because the stylus is affected by the clip that connects the string vibrator to the stylus. Also, we must also consider the measurement of the length of string with complete number of segment. Lastly, we must check the different relationship of frequency to the segment, tension, linear mass density and length. References Book  Lab Manual, General Physics 3, Department of Physics, Mapúa Institute of Technology  Paul A., Tipler; Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1 (6th ed.). New York, NY: Worth Publishers. pp. 666–670. ISBN 1-4292-0132-0. URL  Michigan State University. (n.d.). Topics: Waves and Vibrations. Retrieved November 2013, from https://www.msu.edu/~murph250/topics/Waves1.htm 5