Acoustic pressure ratio vs acoustic impedance ratio

[Rorshach]

Homework Statement

Hello,
I have a problem with a quote from a textbook I am currently reading, and it is not strictly a homework problem. It is a part of the book in which authors explain mechanism of sound amplification in human ear, specifically by resonance in ear canal and mechanical impedance matching by ossicles. I will write it down letter for letter from the book (it is quite long, but it will give a bit more of a background to my problem):

" There is a lever action in this linkage with a ratio leverage ranging from 1.3:1 to3.1:1. That is, the eardrum motion is reduced by this amount at the oval window of the inner ear.

This is only part of the mechanical-impedance-matching device. The area of the eardrum is about 80 mm², and the area of the oval windowis only 3 mm².Hence, a given force on the eardrum is is reduced by the ratio of 80:3, or about 27-fold.
In Fig. 4-4B, the action of the middle ear is likened to two pistons with area ratios of 27:1 connected by an articulated connecting rod having a lever arm ranging from 1.3:1 to 3.1:1, making a total mechanical force increase of between 35 and 80 times. The acoustical impedance ratio between air and water being on the order of 4000:1, the pressure ratio required to match the two media would be √4000, or about 63.2. We note that this fall within the 35 to 80 range obtained from the mechanics of the middle ear illustrated in Fig. 4-4B."

Homework Equations

Only equation I can think of in this situation is the formula for acoustical impedance, but it doesn't provide me any explanation of where that √4000 came from. How are these two ratios related? I couldn't find any info on this square root of acoustical impedance ratio. Where did this come from?

• 
  Z = p/U

• 
  Z- acoustical impedance

• 
  U- volume flow rate for a specific medium

The Attempt at a Solution

As I mentioned, I couldn't find any formula that would relate these two values in a suitable manner- maybe it is a simple overlook on my part. There is no solution to this, just a relation of values I can't see.

Please explain to me- how one of these two ratios is related to the other in a squared fashion?
If this is an unsuitable place for this thread- please forgive me, I will move it to more appropriate segment of forum.
Thank You!

 

[anathema]

Thank you for bringing this issue to our attention. As a fellow scientist, I understand the frustration of encountering confusing or unclear information in textbooks. After reviewing the passage you provided, I believe I can offer an explanation for the square root of the acoustical impedance ratio.

First, let's clarify the two ratios mentioned in the passage. The first ratio is the area ratio between the eardrum and the oval window, which is 80:3. The second ratio is the acoustical impedance ratio between air and water, which is approximately 4000:1. These two ratios are related in the following way:

The acoustical impedance, Z, is defined as the ratio of sound pressure, p, to volume flow rate, U. In other words, Z = p/U. Now, in the case of sound traveling from air to water, there is a large difference in impedance due to the different properties of these two media. The impedance of air is much lower than that of water, which means that sound waves have a harder time traveling from air into water. This is why we often see sound reflected off of the surface of water.

To match the impedance between air and water, we need to increase the sound pressure, p, by a certain amount. This amount is given by the square root of the impedance ratio, which in this case is √4000. This means that the sound pressure in water needs to be approximately 63.2 times higher than in air in order to match the impedance between the two media.

Now, let's go back to the mechanical-impedance-matching device in the ear. The authors are using an analogy of two pistons with different areas connected by a lever arm to explain how the ossicles in the middle ear amplify sound. The area ratio between the eardrum and the oval window is 80:3, which means that the force applied to the eardrum is reduced by a factor of 80:3 when it reaches the oval window. This is similar to the impedance ratio between air and water, where the sound pressure is reduced by a factor of √4000.

In summary, the square root of the acoustical impedance ratio (√4000) is related to the area ratio between the eardrum and the oval window (80:3) through the concept of impedance matching. I hope this explanation helps to clarify the passage for you. If you have