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Rorshach
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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!