Question

The length of the auditory canal in humans averages about \(2.5 \mathrm{cm} .\) What are the lowest three standing wave frequencies for a pipe of this length open at one end? What effect might resonance have on the sensitivity of the ear at various frequencies? (Refer to Fig. 12.12 Note that frequencies critical to specch recognition are in the range 2 to \(5 \mathrm{kHz}\) )

Short answer

The lowest three standing wave frequencies are approximately 3430 Hz, 10290 Hz, and 17150 Hz. Resonance may enhance sensitivity to sounds in the speech frequency range.

Step-by-step solution

Understanding the problem

We need to find the lowest three standing wave frequencies for a pipe of length 2.5 cm open at one end. These frequencies are related to the natural vibration modes of the pipe.

Using the formula for pipe open at one end

The fundamental frequency of a pipe open at one end is given by \(f_1 = \frac{v}{4L}\), where \(v\) is the speed of sound in air (approximately 343 m/s) and \(L\) is the length of the pipe.

Calculating the fundamental frequency

Substitute the known values into the formula: \(f_1 = \frac{343}{4 \times 0.025}\). Calculate to find \(f_1\).

Understanding harmonics for a pipe open at one end

For a pipe open at one end, the harmonics are odd multiples of the fundamental frequency: \(f_3 = 3f_1\) and \(f_5 = 5f_1\).

Calculating the second and third lowest frequencies (3rd and 5th harmonics)

Calculate \(f_3 = 3f_1\) and \(f_5 = 5f_1\) using the fundamental frequency calculated in the previous step.

Interpreting the effect of resonance

Resonance can amplify certain frequencies. If these frequencies are in the speech recognition range (2 to 5 kHz), it might aid in hearing certain speech sounds more clearly.

Key concepts

Auditory Canal

The auditory canal is a crucial part of the human ear that plays a key role in hearing. It's a tube-like structure that connects the outer ear to the eardrum. On average, it is about 2.5 cm long. This length helps determine which sound frequencies are naturally amplified because it acts like a pipe open at one end.
The auditory canal enhances sound through a process called resonance, where certain frequencies are made louder. The length of the auditory canal influences which frequencies are increased. Typically, these are the standing wave frequencies unique to its dimension.

• Acts like a pipe open at one end
• Averages about 2.5 cm in length
• Plays a major role in amplifying sound

Knowing about the auditory canal helps us understand how we naturally heard some sounds louder than others.

Harmonics

Harmonics in acoustics refer to the natural frequencies that a system vibrates at. In the context of the auditory canal, harmonics are standing wave frequencies. Since the canal is like a pipe open at one end, the harmonics are odd multiples of the fundamental frequency.
This means if you find a fundamental frequency, the harmonics would be at 3, 5 times this frequency, and so forth. This creates a set of tones that are related in a simple, mathematical way.

• Standing wave frequencies involved in sound structures
• For a pipe open at one end, harmonics are at odd multiples
• Enhances certain sound frequencies

Understanding harmonics helps in explaining why certain sounds are more prominent than others, particularly for those sounds within the speech range.

Resonance

Resonance occurs when a system is able to absorb more energy at its natural frequency. For the auditory canal, resonance means some sound frequencies will sound more intense. This is because they are in harmony with the natural frequency of the sound pipe. With the auditory canal having resonance characteristics, those frequencies that match its natural resonance are amplified.
In simple terms, resonance helps increase the volume of specific sounds, making them easier to hear. This is particularly useful for frequencies that are important for speech, which fall in the range of 2 to 5 kHz.

• Occurs when sound waves match natural frequency
• Important for amplifying speech frequencies
• Makes certain sounds louder, aiding hearing

Hence, resonance is essential for enhancing the sensitivity of the ear to specific key frequencies.

Speed of Sound

The speed of sound is crucial when discussing standing wave frequencies. It's defined as the rate at which sound waves propagate through a medium and, in air, it's approximately 343 meters per second. The speed of sound is the factor that, alongside the length of the auditory canal, determines the fundamental frequency of standing waves.
For pipes that are open at one end, like the auditory canal, the fundamental frequency is calculated with the equation: \[ f_1 = \frac{v}{4L} \] Where \(f_1\) is the fundamental frequency, \(v\) is the speed of sound, and \(L\) is the length of the pipe.

• Approximately 343 m/s in air
• Influences frequency calculations in acoustic systems
• Key in determining the fundamental frequency of sound waves

The knowledge of the speed of sound helps us accurately identify which sound frequencies will be naturally amplified in the auditory canal.

Speech Recognition

Speech recognition is significantly impacted by how frequencies are naturally amplified in the ear. Frequencies crucial for understanding speech typically range from 2 to 5 kHz. Due to resonance in the auditory canal, these frequencies can be naturally boosted, which could aid in better perception and recognition of speech.
This process enhances the listener's ability to recognize and differentiate sounds essential for effective communication. Understanding how resonance and harmonics interact with the speed of sound further enhances the clarity of speech sounds.

• Involves recognizing frequencies critical to speech
• Enhanced by naturally amplified frequencies in the ear
• Affected by resonance which boosts specific speech frequencies

In this way, the structure of the auditory canal heavily influences our ability to comprehend speech by highlighting critical frequencies necessary for effective speech recognition.