Vibrational Behavior of an Empty Beer Bottle

The Experiment In response to a question regarding the sound made by clinking two beer bottles together I became curious and wondered what the vibrational mode shapes of a glass beer bottle look like. First I made a recording of the sound resulting from striking the neck of a beer bottle with a hard object. (I didn't want to clink two bottles together because then the sound would involve the vibrations of both bottles together). The frequency spectrum at right represents the sound obtained by holding an empty glass beer bottle lightly at the base, striking the neck with a small metal hammer, and recording the sound using a microphone. The almost imperceptible peak at 192 Hz is due to the Helmholtz mode, and is the same frequency that one would produce by blowing over the opening of the empty bottle. The peak at 1152 Hz is the next higher order acoustic mode inside the cavity. But, the next three prominent peaks (at 3200 Hz, 4448 Hz, and 5632 Hz), which appear to dominate the "clink" sound, are the lowest frequency vibrational modes of the bottle, as we will see below.

I used experimental modal analysis to determine the vibrational mode shapes and frequencies for the bottle. I marked the bottle with 106 points forming a grid representative of the shape of the bottle. There are points every 0.5-inch along the length and every 1cm around the bottle. The figures at left show the grid compared to the actual bottle. The bottle was suspended from rubber bands - which allow the bottle to vibrate freely without significantly altering the damping or vibration amplitudes. I attached a small (0.5g) accelerometer to one of the locations near the base of the barrel portion of the bottle, and used a special force hammer to tap the bottle at each of the 106 locations along the length and around the circumference. At each location I recorded a Frequency Response Function, consisting of the ratio of acceleration to force, using a 2-channel frequency analyzer. I then used the modal analysis software program STAR Modal to curve fit the frequency response functions in order to obtain the natural frequencies and the animated mode shapes shown below.

The Results In order to catalog the important mode shapes, I am using the modal designation which is most commonly used to describe the vibration of systems with cylindrical symmetry (cylinders, bells, drums, baseball bats). Each mode shape is designated by a pair of integers, (n,m) where n represents the number of nodal diameters and m represents the number of nodal circles. A node a location (a line on a 2-D surface, or a plane in a 3-D volume) where there is no vibration for a particular mode shape. A nodal diameter indicates that there are two lines down the length of the bottle (on opposite sides of the bottle) which do not move while the rest of the bottle is vibrating. So, for a mode shape with n=2 if you traced a path around the circumference of the bottle you would encounter 4=2n lines (2 diameters = 2 pairs of lines on opposite sites of the bottle) where no vibration occurs. On one side of a nodal line the bottle will be vibrating outwards, while on the other side of the nodal line the bottle is vibrating inwards. The m integers representing nodal circles are evidenced by a line around the circumference - (a circle) at a fixed location along the length of the bottle - that does not move while the bottle is vibrating. Actually, the integer m counts the number of anti-nodes (locations of maximum amplitude) so that the number of nodal circles is really m-1. Looking at the measured mode shapes below might help. The mode shapes identified in red correspond to the most prominent peaks in the frequency spectrum above.
(n=2,m=1) - 3200 Hz The first prominent vibrational mode of the bottle is essentially a breathing mode, similar to what is observed in a bell, wineglass, or baseball bat. The mode shape designation (n=2,m=1) means that there are two nodal diameters (moving around the circumference you would find 2n=4 lines with no motion) and as you moved along the length of the bottle you would find zero (m-1) nodal circles. As the bottle vibrates it looks like it has two regions which alternate between bulging outwards and compressing inwards.	(n=3,m=1) - 4910 Hz This mode, which has three nodal diameters (6 nodal lines around the circumference) and zero nodal circles, does not radiate sound nearly as well as the n=2 modes. It shows up as a peak right near 5000 Hz in the frequency spectrum at the top of the page, but has an amplitude about 25dB lower than the two most prominent peaks. As the bottle vibrates it looks like it has three regions which alternate between bulging outwards and compressing inwards.	(n=4,m=1) - 8520 Hz This mode has four nodal diameters (8 nodal lines around the circumference) and zero nodal circles. As the bottle vibrates it looks like it has four regions which alternate between bulging outwards and compressing inwards. This mode is a very poor radiator of sound, and thus does not show up prominently in the microphone frequency spectrum.	(n=5,m=1) - 13,500 Hz This mode has five nodal diameters (10 nodal lines around the circumference) and zero nodal circles. As the bottle vibrates it looks like it has five regions which alternate between bulging outwards and compressing inwards. This mode is an extremely poor radiator of sound, and does not figure prominently in the sound spectrum.
(n=2,m=2) - 5360 Hz The second prominent vibrational mode of the bottle is the second member of the n=2 family, with two nodal diameters and one nodal circle located about half-way up the bottle. Because this mode involves significant motion of the neck, it shows up very strongly in the sound spectrum when the bottle is struck on the neck (as would happen when clinking two bottles together to make a toast).	(n=3,m=2) - 7437 Hz This mode shape is the second member of the n=3 family, with three nodal diameters and one nodal circle (about 3 inches or 7 dots from the bottom of the barrel).	(n=4,m=2) - 10,330 Hz This mode shape is the second member of the n=4 family, with four nodal diameters and one nodal circle (about 3 inches or 7 dots from the bottom of the barrel).	(n=5,m=2) This mode shape is the second member of the n=5 family, with five nodal diameters and one nodal circle (about 3 inches or between the 7th and 8th dot from the bottom of the barrel).
1st Bending - 4440 Hz This mode is essentially a flexural bending vibration in which the entire bottle behaves like a free-free beam. Because this mode involves significant motion of the neck, it shows up very strongly in the sound spectrum when the bottle is struck on the neck (as would happen when clinking two bottles together to make a toast).	(n=3,m=3) - 11,610 Hz This mode is the third member of the n=3 family and has three nodal diameters and two nodal circles, one about 2-inches (5 dots) from the bottom and the other about 4-inches (8 dots) from the bottom.

What happens when the bottle contains liquid?

The experimentally measured mode shapes above are for an empty glass beer bottle. However, when two people clink their bottles together, the bottles usually contain liquid, and are most often quite full. So, one could ask the question: "How does the vibration of a bottle filled with liquid compare with an empty bottle?" I haven't done an experimental modal analysis of a full bottle yet (I didn't have any full, capped bottles lying around the lab when I did this experiment) but I did take the empty bottle used in this experiment, fill is with water, and compare the sounds recorded when striking the neck of the bottle. The frequency spectra at right show the comparison. The plot on the bottom is for the empty bottle, and shows the three strong vibrational modes at roughly 3200Hz, 4440Hz, and 5600Hz, as identified from the modal analysis. The top plot shows the result for the bottle filled with water up to the beginning of the taper into the neck. The sound was quite different, and died away much faster (water damps the vibration quickly). But, it is interesting to note that the water-filled bottle also has three strong peaks in the 3000-6000 Hz range. Without doing a more complete modal test, I can't identify whether it is the same three modes, or whether other higher-frequency modes have shifted downwards due to the mass loading of the liquid. That may be a future experiment.