Two glasses

One glass is full of water and the other is empty.

When a knife lightly strikes the top of each glass, which glass will produce a lower pitch?

Empty glass Full glass Both will produce sound of equal pitch

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16 solutions

Brad Morin
Mar 14, 2018

I tested a glass of water. The more water I put in it, the lower the sound. I then tested two more classes of different shapes. Same result. I then set up four glasses and adjusted levels so I could play the first few measures of Jingle Bells. There is no argument that can stand up to Jingle Bells, no matter how erudite the explanation.

I wrote a comment earlier but don't see it now, so this may be a repeat.

Moderator note:

Experiment is the lifeblood of science, as Brad has clearly demonstrated with Jingle Bells.

This question was confusing to many because a glass vessel full of water can serve as a percussion instrument (vibrating itself) or as a woodwind instrument (with the air column within it vibrating). If the air is vibrating (for example, by blowing air next to the opening of a wine bottle) then the longer cavity in an empty glass would produce sound at a lower resonant frequency. But if we strike the glass with a knife as stated in this question, the air isn't vibrating, it's the glass itself.

Striking an empty glass causes vibrations in the glass's mass m m , with a restoring force caused by the rigidity of the glass with a spring constant k k . The resonant frequency of this system if the vibration corresponds to simple harmonic motion is proportional to k / m \sqrt{k/m} . When water is added, the effective mass is increased to m + m w a t e r m+m_{water} , and so the denominator of the resonance frequency is increased, causing a lower frequency.

If you know the mass of the glass and the water you've added to achieve different frequencies of sound for Jingle Bells, you may even be able to estimate the spring constant of your glass!

You are right. I have tried my best to explain the reason in my solution. Hope this helps!

Akhash Raja Raam - 3 years, 2 months ago

Hi, first off, hope all is well with everyone, secondly I didn’t realize that once you delete a post you can’t post a new one, so I’ll have to comment like this and hope it helps (and of course, this is a response to Brad!). I’d like to apologize as my former explanation, though “correct,” did not take all variables into account. After experimentation and further consideration, I’d like to post a conjecture: there are many things at play in these glasses but fundamentally the sound is the result of the glass jiggling back and forth, pushing air molecules which send the sound to your ears as pressure waves. Now though it is true that when you add water, you add tension to the glass, making it stiffer (this was my former answer), it is not the most significant variable at play!

So here is what I believe is going on: when a SMALL amount (in the glass I used I added a quarter inch in height or so) of water is added to the glass, the glass with water will have a higher frequency and thus pitch than the empty glass, but a full glass of water will have a lower pitch. This is because when water is added, (I believe) it adds tension to the glass making it stiffer and thus with a small amount you’ll obtain a higher pitch but when more water is added, the mass causes a decreased acceleration of the water and thus the glass wall. Imagine this as trying to move in water, it’s much harder than moving in air! So the glass experiences this as well, and so the rate of the glass jiggling back and forth will be lower when there is a lot of water. When there is only a little, increased tension of the glass wall causes the effect of a plucked guitar string that has been tightened and so the frequency will be higher. This question and responses demonstrate an important note on the philosophy of science, we are all still learning! There are many variables involved in even simple everyday occurrences and even if some are st play, there may be others whose effects are more significant. So in times like these the only measure of truth is via experimentation itself, the backbone of all scientific development. Please correct anything I’ve said as I have only made a conjecture and am open to criticism. I’d just like to add that the decrease in acceleration is the result of Newton’s force law. When we strike the glass we apply a force which deforms the glass a little, causing it to jiggle, this of course displaces the water within the glass, but this water has a much greater mass per unit volume or density than air. Thus, when the glass pushes on the water, the force equation F = ma tells us that since the m part for water is much greater than for air, the acceleration with the same force (strike on glass wall) is much lower. This means that the glass must necessarily accelerate more slowly since the glass can’t go through the water itself! This is what I believe is really contributing to the lack of frequency in the filled case.

Amun P - 3 years, 2 months ago

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Amun P tldr

Matthew Carreiro - 3 years, 2 months ago

Would one with an oscilliscope be able to test what frequency is the result. One could maybe have the glass above the water level on frequency with higher than what humans can hear. While the glass below the water is on a lower frequency which is audible?

Robin Broad - 3 years, 2 months ago

I did the experiment with beer bottles and found that the empty beer bottle resulted in a lower pitch then the full one.

Matthew Carreiro - 3 years, 2 months ago

Very good for the experiment! ;D

Jairo Díaz - 3 years, 2 months ago

Challenge Master seems to be talking about frequency of the system (glass and water) whereas the question was about frequency of sound which is frequency of the movement of the air ONLY and frequency is of the pressure wave generated primarily by the vibrating portion of the glass in the resonant chamber which is foreshortened by the liquid. The smaller the chamber, the higher the frequency generated, and the higher the tone or pitch. If you simplify it like a simple lump of matter rather than taking into account the complexity of the system, and asked for the frequency of the system or the frequency of the glass and water, you get the answer he gives. Sound is different, Brad has the right answer.

Mark Orton - 3 years, 2 months ago

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Mark, I did the experiment, and it was clear from listening that the (dominant) pitch of the pressure wave generated in the air was lower when the glass was full. That would agree with CM's explanation. Did you do the experiment, and what results did you hear?

(Also, sound can exist in water and other media besides air, but I don't think the question hinges on that issue.)

Lars Huttar - 3 years, 2 months ago

I initially thought the provided answer was wrong and Brad was right: that the fuller glass should produce a higher pitch. My thinking was based on many past experiences pouring water into cups and glasses: as the glass gets fuller, the sound of pouring rises in pitch. But Challenge Master makes the important and correct point (which I had assumed was unimportant): that the question is about striking a glass, which gives a different result.

Challenge Master is mistaken in saying that "the air isn't vibrating." The question asks "which glass will produce a lower pitch," not "which glass is vibrating at a lower frequency?" "Producing" a lower pitch includes effects on the air in and around the glass. CM probably meant that the air column in the glass is not resonating significantly.

I did the experiment, using a knife to strike the glass, to check whether my past experience applied to this specific scenario. To my surprise, CM was right: the full glass produced a lower pitch. Since the experiment agrees with his explanation, which also makes intuitive sense, I'm inclined to believe the explanation. I would only add that air also has mass; when water displaces air in the glass, the effective mass is increased from mass of glass-plus-air to mass of glass-plus-water.

Brad, did you strike the glass with a knife, or produce the sound in some other way? I suspect you overlooked the "striking" aspect of the question, like I did at first.

Lars Huttar - 3 years, 2 months ago

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I struck the glass. The original answer was wrong on brilliant.org. It has since been corrected. The fuller the glass, the lower the sound, which means the lower the pitch.

Brad Morin - 3 years, 2 months ago

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I have just made my own experiment. But for me, the full glass produced a higher frequency/pitch!

庭豐 石 - 2 years, 9 months ago
Laura Gao
Mar 12, 2018

Water is denser than air, which means that there are more particles in the same amount of space. Since there are more particles, they vibrate faster and bump into each other force. Pitch is defined as the frequency of vibration of molecules, and water vibrates faster, so that causes it to have a higher pitch.

the question was about 'lower pitch'

Moshe Waisberg - 3 years, 3 months ago

https://www.youtube.com/watch?v=g5A_PwsIGQg

Marina Longnickel - 3 years, 2 months ago

I mistook the dark 'top' of the glass, as a 'lid', probably plastic, like mine are. The sounds I would expect seem to be a combination of at least 3 notes, with one note predominating in this experiment. My misapprehension would predominate with the air thump, highest pitch from the smallest volume of air, and little getting into the water. I think that each glass contains 2 'volumes', water and air, each having its own resonance. Striking down on the uncovered glass rim strike puts most energy into the glass itself for a consistent pitch clack of a complex rigid object. This must be the mode discussed above. Water being denser, perhaps absorbs and resonates louder, and the bigger the volume, the lower the resonance (like the bigger string), but the speed of sound in water is much faster (~5x), making its effective resonance volume with respect to air, much smaller. That is why the air tone is so much lower than the water tone for the same volume. PS: See the complementary tones demonstrated by 'strumming' the air in the bottles with tuned amounts of water, as in a water-bottle mouth-organ.

J B - 3 years, 2 months ago

Since the water is denser, you need more energy to create the same vibrations because there are more particles in the same amount of space. I would like it if you could recheck your solution.

Akhash Raja Raam - 3 years, 2 months ago

The question is about lower pitch.😕😕😕

Tasneem Khaled - 3 years, 2 months ago

Isn't the question about lower pitch ?

Beyond Zero - 3 years, 2 months ago

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The previous answer was higher pitch. The answer has not been changed since then.

Akhash Raja Raam - 3 years, 2 months ago
Basudeb Jena
Mar 14, 2018

Water is used in hydraulics due it's property to distribute the pressure evenly because the glass container water it absurd a lot of energy and this decrease the vibration of the glass as The pitch is dependent upon the frequency or the three number of vibration per second.

The water dampens the vibration of the glass and lowers its resonant frequency. The more water you add, the more dampening and the lower the pitch.

Derrick N - 3 years, 2 months ago

I tried the experiment with a glass of water. The more water I added, the lower the pitch was. I think the best strategy for this problem is to act it out. I would rate this problem a 3 because it was fairly easy.

Lucia Tiberio - 3 years, 2 months ago
Jogeedae Pl
Mar 13, 2018

The glass is like a string on guitar. And water is like a finger on it.

Poetic? or perhaps reference to a glass harmonica?

J B - 3 years, 2 months ago

Nice analogy. Do you think this is about damping?

Agnishom Chattopadhyay - 3 years, 2 months ago

But then the empty glass would be lower pitch

Jeff Erlich - 3 years, 2 months ago

Correct. The question was about frequency of sound which is frequency of the movement of the air ONLY and frequency is of the pressure wave generated primarily by the vibrating portion of the glass in the resonant chamber which is foreshortened by the liquid. The smaller the chamber, the higher the frequency generated, and the higher the tone or pitch. On a guitar the sound is dominated by the string and the resonant chamber of the instrument. Smaller the string and chamber (ukulele) the higher the tone.

Mark Orton - 3 years, 2 months ago
Eleven Pi
Mar 15, 2018

consider simple harmonic motion .. a spring and a mass .. if you increase the mass, you lower the frequency the water acts as extra mass

alternatively: the resonant cavity is larger for the empty glass .. but this is a minor component .. by far the sound you hear comes from the glass vibrating

Yes! The question was about pitch of sound which is frequency of the movement of the air ONLY and frequency is of the pressure wave generated primarily by the vibrating portion of the glass in the resonant chamber which is foreshortened by the liquid. The smaller the chamber, the higher the frequency generated, and the higher the tone or pitch.

Mark Orton - 3 years, 2 months ago
Laszlo Mihaly
Mar 16, 2018

This problem has been posted with an incorrect answer and solution, based on the length of the air column in the glass. In that solution the shorter air column yielded higher frequency. It has been concluded that the higher frequency comes from the glass with lots of water. That solution has been incorrect. When a "knife lightly strikes the glass" it excited vibrations in the glass, not in the air contained by the glass. The length of air column is irrelevant to the solution.

What is relevant, instead, is the water mass m m involved in the vibrations. When the glass is empty the relevant mass is the mass of the glass and the restoring force is the elasticity of the glass, k k . When water is added, the mass is increased and the restoring force is the same (the water does not contribute to the restoring force, because it does not hod its shape). Since the frequency is proportional to k / m \sqrt{k/m} , the frequency of the glass with the water will be lower.

You are talking about the system, sound is about air movement only and is completely dominated by the shorter resonant chamber, shorter chamber is shorter wavelength which is higher frequency which is higher pitch. The water can vibrate at its low frequency all it likes but you won’t hear it.

Mark Orton - 3 years, 2 months ago

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According to your argument, we would not hear the piano, either.

Laszlo Mihaly - 3 years, 2 months ago
Romeo Neto
Mar 14, 2018

Now you got this right. 😉

Yup. I got an e-mail :)

Kevin Abalos - 3 years, 2 months ago
Fabricio Kolberg
Mar 14, 2018

I literally just tested it out in the kitchen.

David Richner
Mar 14, 2018

Water increases total mass which lowers vibrational frequency which means lower pitch

Sangamesh Davey
Mar 12, 2018

An empty bottle produces a lower pitch because there's lots of air in the bottle to vibrate. Adding water to the bottle decreases the amount of airspace, which means there is less air to vibrate. With less air, the vibrations happen more quickly and produce a higher pitch.

I’m a bit confused on what air has to do with the problem. I just assumed it would be the volume of the glass excluding the water

Raphael Pierard - 3 years, 3 months ago
Finn Murphy
Mar 16, 2018

Have you ever tried to talk under water? Then you know your voice is muffled this means the water is denser than air also causing it to be a lower pitch.

Ayush Stark
Mar 16, 2018

In water the frequency is way more than in free air, so in water filled glass will produce higher pitch

Jacob Phillips
Mar 15, 2018

When you tap it, it creates vibrations in the cup. It's like in a guitar, because water has a higher density, acting like the walls of the guitar. This makes it have less energy. Also, when the sound waves hit the water, that makes the sound waves lose energy, causing it to have a lower pitch.

Akhash Raja Raam
Mar 14, 2018

The case here is if the glass is stroked with a knife. Since the empty glass contains air, the air molecules vibrate freely without borrowing much energy from the vibrating walls of the glasses. The full glass contains water, which is more dense than air. Thus, it takes up more energy from the walls of the glasses to produce the same vibrations an air column did. Since the total energy is same, we can say that the empty glass vibrates with a higher frequency, leading to a higher pitch and the full glass vibrates with a lower frequency, i.e. lower pitch.

From an experiment referred to by @Marina Longnickel ,they have explained the same situation as such: "The science of sound is all about vibrations. When you hit the bottle with the spoon, the glass vibrates, and it’s these vibrations that ultimately make the sound. You discovered that tapping an empty bottle produced a higher-pitched sound than tapping a bottle full of water did. Adding water to the bottle dampens the vibrations created by striking the glass with a spoon. The less water in the bottle, the faster the glass vibrates and the higher the pitch. The more water you add to the bottle, the slower the glass vibrates, creating a lower pitch."

Akhash Raja Raam - 3 years, 2 months ago

Thanks @Marina Longnickel for the video! It helped me put the answer I had into understandable sentences!

Akhash Raja Raam - 3 years, 2 months ago
Rocco Succi
Mar 13, 2018

Because a sound wave moves faster in the water rather than in the air (because the water is denser), therefore the frequency of the wave crossing the full glass will be higher than the empty one, resulting in a higher pitch.

David Weisberg
Mar 12, 2018

A lot of you are talking about air pressure, or the resonant frequency of the glass changing due to higher pressure or tension, but those negligible effects that, although they work in the direction of a higher frequency, aren't the main contribution and are mainly irrelevant. The biggest contributing factor is that the length through which the sound wave travels, reflects off the bottom of the glass, and superposes with the original wave, causing a standing wave to form. Shorter length due to added water causes the frequency of the standing wave to be higher. For example, in a trombone, when you pull the rod, it's not the resonant frequency that's changing, or air pressure, but the length through which the wave travels. Here is an animation of this

http://www.walter-fendt.de/html5/phen/standingwavereflection_en.htm

Also, a good Khan Academy video about this for those of you who seem to be really misguided or ignorant and trying to mislead others and need a good review:

https://www.youtube.com/watch?v=BhQUW9s-R8M

https://www.youtube.com/watch?v=g5A_PwsIGQg

Marina Longnickel - 3 years, 2 months ago

I think the difference is, in your explanation water is a boundary but in others its a medium that results in the lower pitch.

Susan Bikram Thapa - 3 years, 1 month ago

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