# Calculating τ by knowing I is proportional to A^2

• Cc518
In summary,A vibrating standing wave on a string radiates a sound wave with intensity proportional to the square of the standing-wave amplitude. When a piano key is struck and held down, so that the string continues to vibrate, the sound level decreases by 8.0 dB in 1.0 s.The string's damping time constant τ is 1.08 s.
Cc518

## Homework Statement

A vibrating standing wave on a string radiates a sound wave with intensity proportional to the square of the standing-wave amplitude. When a piano key is struck and held down, so that the string continues to vibrate, the sound level decreases by 8.0 dB in 1.0 s.

What is the string's damping time constant τ ?

## Homework Equations

I∝( 2asin(kx))^2
B=10log(I/1*10^-12)

## The Attempt at a Solution

From 8dB, I got change in sound intensity is 6.31*10^-12 w/m2
Since the intensity is proportional to the the square of amplitude, the amplitude will decrease by (6.31*10^-12)^1/2
So I got

(6.31*10^-12)^1/2 = 2a - 2a•e^-t/τ =
2a (1- e^-t/τ)

I assumed sin(kx)=1 because we are looking at the greatest amplitude.
t=1s
I don’t know where to go from here as I don’t know what a is.

Since I don’t know what the original sound intensity is, I won’t be able to know the percentage the sound intensity has decreased in order to calculate the percentage the amplitude has decreased.

Any help is appreciated:)

"Decibels" in this context is not an absolute measure of sound intensity (as in " a noise of 100 dB"), but a relative measure. 1 bel (10dB) is an intensity difference of a factor of 10. A decrease of 8 dB means a decrease by a factor of 100.8.

So does that mean amplitude will decrease by a factor of √10^0.8 ?
If so, then e^-t/τ =√10^0.8,
t=1, I got τ=1.08s which is not the right answer:(
Can anyone tell me where I went wrong?
Thank you!

mjc123 said:
... 1 bel (10dB) is an intensity difference of a factor of 10. A decrease of 8 dB means a decrease by a factor of 100.8.
What is not quite so clear from this, is that the 8dB is the intensity rather than the amplitude.
As you noted yourself, intensity ∝ square of amplitude. So the amplitude2 decreases by 8 dB, which means the amplitude decreases by ... ?

Cc518 said:
So does that mean amplitude will decrease by a factor of √10^0.8 ?
So it means A2 decreases by a factor of √10^0.8 (Though it may be easier to calculate it slightly differently, depending on your math preferences.)

Merlin3189 said:
So the amplitude2 decreases by 8 dB, which means the amplitude decreases by
Amplitude will decrease by √8dB, right?
Then why did you say
Merlin3189 said:
it means A2 decreases by a factor of √10^0.8
?

No. Not √8 dB.
If A is the ratio of amplitudes and the ratio of intensities is 8 dB,
$$8 = 10 log( A^2 ) \ \ ⇒ \ \ 8 = 20 log( A )$$
$$so A^2 = 10^{0.8} \ \ and \ \ A = 10^{0.4}$$
$$so A^2 = 6.3 \ \ and \ \ A= 2.5 = \sqrt {6.3}$$

Cc518 said:
So does that mean amplitude will decrease by a factor of √10^0.8 ?
Yes. You can easily simplify that.
Cc518 said:
If so, then e^-t/τ =√10^0.8,
It decreases by that factor. Is that factor greater or less than 1?

haruspex said:
Yes. You can easily simplify that.

It decreases by that factor. Is that factor greater or less than 1?
The factor is less than 1?
A0/At=√10^0.8 and At=e-t/τ A0
Then e-t/τ = 1/√10^0.8
t=1,
I got τ=1.08s which is twice the answer, 0.54s, though, but I don’t see why I have to divide 1.08 by 2.

Cc518 said:
Then e-t/τ = 1/√10^0.8
yes, that's better
Cc518 said:
I got τ=1.08s which is twice the answer
Hmmm.. so do I. Looks like a confusion between amplitude and intensity.

## 1. How do you calculate τ when given that I is proportional to A^2?

To calculate τ, you can use the formula τ = kA^2, where k is the constant of proportionality. This means that as A increases, I will also increase by a factor of A^2.

## 2. What is the purpose of knowing that I is proportional to A^2 when calculating τ?

Knowing that I is proportional to A^2 allows us to determine the relationship between the two variables and use it to calculate τ. It also helps us understand how changes in A will affect the value of τ.

## 3. Can you explain the concept of proportionality in this context?

In this context, proportionality means that as one variable (A) changes, the other variable (I) changes in proportion to A^2. This means that the ratio of I to A^2 remains constant.

## 4. How is the constant of proportionality (k) determined in this scenario?

The constant of proportionality can be determined by conducting experiments and collecting data for different values of A and I. It can also be calculated by finding the slope of the graph of I vs. A^2.

## 5. Can this relationship between I and A^2 be applied to other scenarios in science?

Yes, the concept of proportionality and the formula τ = kA^2 can be applied to other scenarios in science where a variable is directly proportional to the square of another variable. This relationship is commonly seen in physics, chemistry, and engineering.

• Introductory Physics Homework Help
Replies
16
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
1K
• Introductory Physics Homework Help
Replies
5
Views
2K
• Introductory Physics Homework Help
Replies
11
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
2K
• Introductory Physics Homework Help
Replies
20
Views
2K
• Introductory Physics Homework Help
Replies
5
Views
1K
• Introductory Physics Homework Help
Replies
17
Views
521
• Mechanics
Replies
5
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
1K