Question

ip standing waves in the human ear the human ear canal is much like an organ pipe that is closed at one end (at the tympanic membrane or eardrum) and open at the other. a typical ear canal has a length of about 2.4 cm.

part e
suppose a person has an ear canal that is shorter than 2.4 cm. is the fundamental frequency of that persons ear canal greater than, less than, or the same as the value found in part a? explain.

Explanation:

Express the fundamental frequency formula

$$ f_1 = \frac{v}{4L} $$

Analyze the relationship between frequency and length

$$ f_1 \propto \frac{1}{L} $$

Determine the effect of a shorter length

$$ L' < L \implies f_1' > f_1 $$

Answer:

The fundamental frequency of that person's ear canal is greater than the value found in part A.

According to the fundamental frequency formula for a pipe closed at one end, \(f_1 = \frac{v}{4L}\), the frequency is inversely proportional to the length \(L\) of the canal. Therefore, a shorter ear canal (\(L < 2.4\text{ cm}\)) results in a higher (greater) fundamental frequency.