Wave on a String

Wave on a String

Introduction

Strings are often used to express the properties of waves. The objective of the exercise was to determine the wavelength of a wave on a string, the effect of damping, and the properties of standing waves. A simulation was used to represent the wave motion of a string. The effect of damping on the wave motion of a string was determined on different pulses. Secondly, the wavelength of the motion was determined at different frequencies. Lastly, standing waves were formed at different frequencies.

Procedure

The length of the string was first measured and recorded using a ruler in the simulation. Secondly, the damping effect was determined under a 1.25cm amplitude, 0.30cm pulse width, high tension, and a fixed end. Pulses were then sent down the string with damping set on the first, second, third, fourth, and fifth lines.  The time for each damping effect was recorded using the timer. A graph of time vs. setting was then plotted. Thirdly, the wavelength was determined for frequencies of 0.75, 1.00, 1.25, 1.5, 1.75, and 2.0. The simulation at each frequency was done at 1.25 amplitude, none damping, medium tension, and fixed end. Lastly, standing waves were formed at fundamental resonance mode, second harmonic, third harmonic, and fourth harmonic.

Data

Length of the string, L = 7.6cm

Finding the Effect of Damping:

Table 1: Damping and Time

Graph 1: Time vs. Setting

Finding the Wavelength:

Table 2: Speed of the Waves

Sample Calculation:

Velocity = wavelength  frequency = 4.8  0.75 = 3.6

Standing Waves:

Fig 1: Fundamental Resonance Mode

Wavelength, λ₁ = 2  7.6 = 15.2 cm

Fig 2: Second-harmonic of a standing wave

Wavelength, λ₂ = 7.6 cm

Velocity = wavelength  frequency

Frequency =  =  = 0.478

Fig 3: Resultant Standing Wave

Fig 4: Standing Wave Settings

Third Harmonic:

The frequency of the third harmonic is thrice the fundamental frequency.

Frequency = 0.243 = 0.72

Fig 5: Third Harmonic

Fig 6: Third Harmonic Settings

Fourth Harmonic:

The frequency of the fourth harmonic is four times the fundamental frequency.

Frequency = 0.244 = 0.96

Fig 7: Fourth Harmonic

Fig 8: Fourth Harmonic Settings

Data Analysis

The first part of the exercise involved the reflection of the wave on a string. Once the wave went off the fixed end, several notable observations showed the disparity between the initial wave and the reflected wave. The reflected wave moved in a direction opposite to the initial wave. Secondly, the reflected wave assumed an inverted positive relative to the initial wave. The initial wave was incident upwards towards the clamp. Once it got off the fixed end, the reflected wave returned as a downward displaced pulse.

The second part of the exercise was based on the damping effect. In physics, the damping effect serves to influence an oscillatory system in a manner that prevents or restricts its oscillations. In this case, different ratios of the damping effect were applied on a wave on a string. Graph 1 shows the relationship between time and the damping effect. The graph shows that as the damping effect increased, the amplitude of the string reduced. The damping effect led to the dissipation of the energy stored in the string. The absorption of the energy reduced the amplitude of the oscillation.

The third part of the exercise involved the effect of frequency on the waves on the string. It was observed that increasing the frequency produced nodes and antinodes. Increasing the frequency led to the formation of successive harmonics. The fundamental mode was formed at 0.24Hz. At this frequency, the wavelength was twice the length of the string. The second harmonic was formed when the fundamental frequency was doubled. The wavelength at this point was equal to the length of the string. The third and the fourth harmonics were formed when the fundamental frequency was multiplied by three and four, respectively.

Standing waves were formed at each of the set frequencies. The standing waves were formed when the nodes and antinodes were observed to alternate with equal spacing. Generally, standing waves occur when wave oscillations have a constant peak amplitude over time, and the different points of the oscillations remain in phase. These properties were observed during the experiment, as shown in Figures 3, 5, and 7.

Conclusion

The purpose of the lab was to demonstrate the wave motion of a string. The experiment showed that damping reduces the amplitude of a wave by dissipating its oscillating energy. Secondly, it also shows that frequency is directly proportional to wavelength. Lastly, it showed that standing waves are formed when nodes and antinodes are observed to alternate with equal spacing. Increasing frequencies increases the number of antinodes on a string. Harmonics depend on the frequency of the waves.