Resonance in a tall glass with added water

Colleen Haley: 2006

Introduction

Children (and occasionally more serious musicians) have used wine glasses to produce musical tones for some time.  Glasses are filled to different levels with water. When the glass is stroked around the edge with a wet finger, or tapped on the side, notes of different pitch are formed, allowing music to be made. The sound produced is a series of harmonics produced by the resonating wine glass. 

If a bell is not symmetrical about its axis two closely spaced resonances are formed which beat. The same is true for a champagne glass with a wall of nonuniform thickness.  If the glass is struck in just the right place two modes will be excited, and the tone will contain a beat.
 
Research Question

How does the addition of water affect the frequencies of resonance produced in a champagne glass when tapped or stroked. 
 
Hypothesis

It is predicted that the frequency will be linear function of the volume of water added. It is expected that stroking and tapping of the glass will produce different results because of the difference between continuous and intermittent excitation.

Materials and Methods

A hand-blown champagne glass (figure 1) was chosen which through preliminary examination was revealed to be not of uniform thickness.  This created a beat in the resonance, analogous to the effect of a handle on a cup⁽¹⁾.  A spot was marked on the side of the glass where the double frequencies had been excited throughout the preliminary examination. The resonance was measured using the program SignalScope 1.4.  Signal Scope was set to maximum resolution (0.673 Hz when the frequency range is 2000 Hz) and to collect 10 data averages.  The averaging function was used to allow a frequency spectrum to be collected for continuous excitation; stroking around the rim, and for intermittent excitation; tapping against the side. Beginning with the empty glass, 25 ml of water was added 8 times until the glass was full with a measurement of 200 ml of water. This produced 9 measurements in total. At each interval, the frequency produced first by tapping the side at the designated spot with a pair of scissors and second by stroking the rim to excite resonance was recorded.  For the continuous excitation, the finger was dipped in water other than that filling the glass to avoid changing the water level.  The frequencies were then examined by saving the data as text and importing to Logger Pro. The double frequencies were taken from each graph and analyzed together.

Results

Data

Table 1: The upper and lower values of frequencies collected at various water volumes. 
 
The data shown in Table 1 is plotted in Graph 1: showing both the upper and lower frequencies produced when the glass was tapped. The errors of ±1 Hz are too small to display as error bars on the graph and have been omitted.
 
Graph 2 shows the upper and lower frequencies produced when the rim of the glass was stroked.

Analysis

Graph 3 shows the frequencies in Hz (y-axis) graphed against the Volume in mL. The y-axis is labeled with the various lines graphed to aid understanding. The upper and lower frequencies produced from both stroking and tapping the champagne glass are graphed jointly. HY (red) refers to the lower frequency produced when the glass was hit and HY2 (blue) represents the higher frequency when the glass was tapped.  SY (green) represents the lower frequency produced when the glass was stroked and SY2 (orange) represents the higher frequency produced when stroked. 
 
Discussion
 
As predicted, the frequency became lower as more water (mass) was added because there was more inertia and thus less acceleration of the glass wall.  As shown in Graph 3 the differences in frequencies produced by tapping and stroking the champagne glass are nominal, contrary to what was expected. There were differences in the intensities of the frequencies produced by both tapping and stroking but the frequencies themselves were quite similar. 
 
Graph 3 shows that the frequency-water volume relationship is not a straight line. The relationship is only a straight line (within errors) between the volumes of 125 mL and 175 mL. For small volumes, the mass of water added compared to the mass of the glass was a small percentage. The system mass was little affected. As more water was added, the change in water level became more significant because it became the predominant factor in the mass of the system. This explains the change in slopes.  The slope became steeper as the change in water more heavily influenced the frequencies being produced. The frequencies behave as expected with a negative slope indicating loss of pitch and forming a straight line in the latter half of the graph.  In this region, the mass of the water is the predominant factor in the total mass in the system, so changes in water volume produce large changes in frequency.

The water volumes were accurate to within ±2%.  All other errors (±1 Hz) are due to the accuracy of the equipment used: the computer microphone and SignalScope 1.4.
           
Evaluation and suggestions for further work

For this set of measurements the water added was measured by volume.  The champagne glass was not of uniform width, so the constant values of water added produced slightly irregular depth levels of water. It would be worthwhile to conduct the experiment again with depth as the variable.

The relative intensity of the two dominant modes varied in an unpredictable way and graphs 1 and 2 show a wider split in frequency with a lower volume of water. Further investigation into what could have caused these effects would be worthwhile.

The explanation given - that added water lowers the vibration frequency by adding mass to the system - could be confirmed by using liquids of different densities.

The following sets of measurements are suggested.

1 A plot of the frequency against the density of solutions added to the same depth is expected to be a straight (or nearly straight) line with a negative slope when the glass is filled to the same depth (half full) each time.

2 The frequency is expected to be constant when the a same mass is added added with solutions of different density.(to different depths) when the glass is approximately half full.

That the effect is due to the addition of mass could also be confirmed by by submerging the empty glass in liquids of different depths. It is expected that frequency-depth curves would be very nearly identical with water inside or out, and that if water was both inside and out, at the same level, the drop in frequency would be approximately doubled for a given level.

Reference

⁽¹⁾ Xin Yi Yeap, 2002: ISB Physics CD, IB Labs: Resonance in cups