Wine Glass Resonance Frequency

[BobDylan22]

Homework Statement

Question 1: When analysing AP French's Formula for Frequency of Wine glasses, what is the direct relationship between the Frequency heard (at different levels - f_d) and the level of water from the top of the glass (d). I know that as the level of water from the glass decreases (meaning the water level is higher), the lower the frequency. However, is this relationship linear or quadratic, etc and why.

Question 2: If you graph (f₀/f_d)² on the x-axis and (1-(d/H*)⁴ on the y axis, is this relationship going to be linear or something else.

Homework Equations

The equation is in the picture below and above (Link is http://imgur.com/a0pQVdN)

The Attempt at a Solution

I believe that the answer to question 1 is quadratic and the answer to question 2 is linear. Please help

 

[BobDylan22]

(f0/fd)^2

 

[gneill]

Hi BobDylan22, Welcome to Physics Forums.

It's better to upload and insert in-thread any images that are critical to the problem. I've inserted a copy into your original post. I also fixed the equations in your text to make them easier to parse. Note that you can use the x₂ and x² icons in the editing panel to insert subscripts and superscripts for equations (or use LaTeX notation).

You'll need to provide more explanation for your attempt at a solution. Describe your thinking behind your conclusions in more detail.

 

[rude man]

Homework Statement

Question 1: When analysing AP French's Formula for Frequency of Wine glasses, what is the direct relationship between the Frequency heard (at different levels - f_d) and the level of water from the top of the glass (d). I know that as the level of water from the glass decreases (meaning the water level is higher), the lower the frequency. However, is this relationship linear or quadratic, etc and why.

Question 2: If you graph (f₀/f_d)² on the x-axis and (1-(d/H*)⁴ on the y axis, is this relationship going to be linear or something else.

If you let x = (1-(d/H*)⁴ and
y = (f₀/f_d)²
then if the other coefficients are constants, what do you get for your equation? Should be apparent if the relationship between y and x is linear or not.

Also, if you let y = (f₀/f_d) and
x = √{(1-(d/H*)⁴}, is that quadratic or not?

BTW it may interest you that a portion of the vibration has an inertial component, i.e. if you set up the vibration nodes at 0 and 180 degrees, then turn the glass say 90 degrees, the vibrations will follow the glass only about 2/3 of the way, i.e. ~60 deg. The so-called "Hemispherical Resonator Gyroscope (HRG)" is based on this phenomenon.

Cf. https://en.wikipedia.org/wiki/Hemispherical_resonator_gyroscope#Advantages

 

[BobDylan22]

If you let x = (1-(d/H*)⁴ and
y = (f₀/f_d)²
then if the other coefficients are constants, what do you get for your equation? Should be apparent if the relationship between y and x is linear or not.

Also, if you let y = (f₀/f_d) and
x = √{(1-(d/H*)⁴}, is that quadratic or not?

BTW it may interest you that a portion of the vibration has an inertial component, i.e. if you set up the vibration nodes at 0 and 180 degrees, then turn the glass say 90 degrees, the vibrations will follow the glass only about 2/3 of the way, i.e. ~60 deg. The so-called "Hemispherical Resonator Gyroscope (HRG)" is based on this phenomenon.

Cf. https://en.wikipedia.org/wiki/Hemispherical_resonator_gyroscope#Advantages

I know that if you graph x = (1-(d/H*)4 and y = (f0/fd)2, you get a linear line,
but what if you graph fd and d?

 

[rude man]

I know that if you graph x = (1-(d/H*)4 and y = (f0/fd)2, you get a linear line,
but what if you graph fd and d?

OK, we've done question 2.

Question 1: let y = f_d and x = d.
Then [f₀/y]² = 1 + a(1 - x/H)⁴
where a is the given constant.
In order for the relationship between y and x to be a quadratic, you have to be able to come up with constants α, β and γ such that
y = α + βx + γx².
Considering just the binomial expansion of (1 - x/H)⁴, does that seem likely ?

 

[BobDylan22]

OK, we've done question 2.

Question 1: let y = f_d and x = d.
Then [f₀/y]² = 1 + a(1 - x/H)⁴
where a is the given constant.
In order for the relationship between y and x to be a quadratic, you have to be able to come up with constants α, β and γ such that
y = α + βx + γx².
Considering just the binomial expansion of (1 - x/H)⁴, does that seem likely ?

Ok, but what I don't get is that as the distance from the top of glass to the top of water level decreases (becoming fuller), the frequency decreases. But this formula does not show that relationship. It shows the opposite.

 

[gneill]

Ok, but what I don't get is that as the distance from the top of glass to the top of water level decreases (becoming fuller), the frequency decreases. But this formula does not show that relationship. It shows the opposite.

Did you try plotting it just to get an idea of the shape of the curve? Assign some arbitrary values to the constants and plot f_d versus d. Note that the way the given function is defined the f_d is in the denominator on the left hand side. So if you rearrange to isolate it you'll get something of the form:

$$f(d) = \frac{f_o}{\sqrt{1 + a\left(1 - \frac{d}{h} \right)^4 }}$$
where f_o, a, and h are chosen suitably.

 

[BobDylan22]

this is my actual data, and my plotting of it http://imgur.com/aAHgNDC

 

[gneill]

I can't see how your spreadsheet does its calculations, but your results are certainly different from what I see plotting a function of the form I showed in my previous post. Here's a snip from a Mathcad worksheet:

The constants f_o, a, and h were chosen arbitrarily just to get a feel for the form of the function.

 

[BobDylan22]

I can't see how your spreadsheet does its calculations, but your results are certainly different from what I see plotting a function of the form I showed in my previous post. Here's a snip from a Mathcad worksheet:
View attachment 98231The constants f_o, a, and h were chosen arbitrarily just to get a feel for the form of the function.

I was graphing fd and d

 

[rude man]

Ok, but what I don't get is that as the distance from the top of glass to the top of water level decreases (becoming fuller), the frequency decreases. But this formula does not show that relationship. It shows the opposite.

No way! As d goes down, f₀/f_d goes up but notice that f_d is in the denominator on the left-hand side, so they both go down & up together.

 

[BobDylan22]

No way! As d goes down, f₀/f_d goes up but notice that f_d is in the denominator on the left-hand side, so they both go down & up together.

Fo will always be the same. As d decreases, that means that the level of water in the glass increases, meaning lower frequency.

 

[rude man]

Fo will always be the same. As d decreases, that means that the level of water in the glass increases, meaning lower frequency.

Right, contradicting your post #7!

 

[BobDylan22]

No it doesn't. As the distance between the top of glass decreases (that means d decreases), which means the glass is more full, hence the frequency decreases.

 

[BobDylan22]

Right, contradicting your post #7!

No it doesn't. As the distance between the top of glass decreases (that means d decreases), which means the glass is more full, hence the frequency decreases.

 

[rude man]

My last post on this thread: look again at your post 7. here is what you said: "Ok, but what I don't get is that as the distance from the top of glass to the top of water level decreases (becoming fuller), the frequency decreases. But this formula does not show that relationship. It shows the opposite."

The formula does NOT " ... show the opposite"!

 

[BobDylan22]

My last post on this thread: look again at your post 7. here is what you said: "Ok, but what I don't get is that as the distance from the top of glass to the top of water level decreases (becoming fuller), the frequency decreases. But this formula does not show that relationship. It shows the opposite."

The formula does NOT " ... show the opposite"!

So you are saying that the formula shows the correct relation? Can you please show me

 

[rude man]

So you are saying that the formula shows the correct relation? Can you please show me

See my post 12. Read it carefully.

 

[BobDylan22]

I can't see how your spreadsheet does its calculations, but your results are certainly different from what I see plotting a function of the form I showed in my previous post. Here's a snip from a Mathcad worksheet:
View attachment 98231The constants f_o, a, and h were chosen arbitrarily just to get a feel for the form of the function.

Thankyou, but is the relationship quadratic?

 

[gneill]

Thankyou, but is the relationship quadratic?

Are you addressing your Question 2?

Question 2: If you graph (f0/fd)2 on the x-axis and (1-(d/H*)4 on the y axis, is this relationship going to be linear or something else.

 

[rude man]

Just intuitively, disregarding the formula altogether - one would expect the frequency to increase as the water level rises. That's because the dry-glass part gets shorter. Typically if a taut string or whatever is shortened the frequency increases (Mersenne's law: one would think that tension and density don 't change much).