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In the larynx, sound is produced by the vibration of the vocal cords. The diagram in Figure 12.44 is a cross section of the vocal tract at one instant in time. Air flows upward (in the \(+z\) direction) through the vocal tract, causing a transverse wave to propagate vertically upward along the surface of the vocal cords. In a typical adult male, the thickness of the vocal cords in the direction of airflow is \(d=2.0 \mathrm{~mm} .\) High-speed photography shows that for a frequency of vibration of \(f=125 \mathrm{~Hz}\), the wave along the surface of the vocal cords travels upward at a speed of \(v=375 \mathrm{~cm} / \mathrm{s}\). Take \(t\) to be time, \(z\) to be displacement in the \(+z\) direction, and \(\lambda\) to be wavelength. What is the wavelength of the wave that travels on the surface of the vocal cords when they are vibrating at frequency \(f ?\) A. \(2.0 \mathrm{~mm}\) B. \(3.3 \mathrm{~mm}\) C. \(0.50 \mathrm{~cm}\) D. \(3.0 \mathrm{~cm}\)

Step by step solution

01

Understand the Relationship

The relationship between speed \( v \), frequency \( f \), and wavelength \( \lambda \) for wave motion is given by the equation \( v = f \cdot \lambda \). We are given the speed of the wave and the frequency of the wave and need to solve for the wavelength, \( \lambda \).

02

Identify Given Values

From the problem statement, we have:- Frequency \( f = 125 \text{ Hz} \)- Speed \( v = 375 \text{ cm/s} \)

03

Solve for Wavelength

Using the formula \( v = f \cdot \lambda \), rearrange to solve for \( \lambda \):\[ \lambda = \frac{v}{f} \]Substitute the given values into the formula:\[ \lambda = \frac{375 \text{ cm/s}}{125 \text{ Hz}} = 3 \text{ cm} \]

04

Match the Solution to the Given Options

The calculated wavelength is \( 3 \text{ cm} \), which matches option D. Therefore, the answer to the question, based on the calculated wavelength, is option D: \( 3.0 \text{ cm} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Motion

Wave motion is a fundamental concept in physics that describes how waves travel through a medium. In the case of sound, waves carry energy as they pass through air, water, or any other medium. Waves are characterized by their amplitude, wavelength, frequency, and speed.

A simple way to visualize wave motion is by imagining a pebble dropped into a pond. The ripples that form on the water surface are similar to sound waves moving through air. These ripples propagate outwards, transferring energy without moving the water itself much, just like air molecules oscillate back and forth rather than travel with the sound waves.

Sound waves, like the ones produced by vocal cords, are longitudinal waves. This means the oscillation of air particles is parallel to the direction the wave travels. Understanding wave motion helps in grasping how sound is produced and reaches our ears.

Vocal Cord Vibration

Vocal cord vibration is a critical aspect of how humans produce sound. The vocal cords, also known as vocal folds, are two bands of muscle located in the larynx. When air passes through the larynx, it causes these vocal cords to vibrate, creating sound waves.

The frequency of these vibrations determines the pitch of the sound produced. In a typical adult male, the frequency might be around 125 Hz, which corresponds to a low pitch. Women's vocal cords are thinner and shorter, generally leading to higher pitch sounds.

These vibrations are regulated by the tension and length of the vocal cords, which are altered by small muscles in the larynx. This control allows us to change the pitch and intensity of our voice, an essential feature for speech and singing.

Frequency and Wavelength Relationship

The frequency and wavelength relationship is a core principle in understanding sound waves. Wavelength (\(\lambda\)) is the distance between consecutive points in phase on a wave, such as from crest to crest. Frequency (\(f\)) is the number of waves that pass a given point per second and is measured in Hertz (Hz).

The speed (\(v\)) of the wave can be determined using the relationship:\(v = f \cdot \lambda\). This equation shows that for a given wave speed, increasing frequency would result in a shorter wavelength, and vice versa.

In the context of vocal cords, if the frequency is 125 Hz and the wave speed is 375 cm/s, the wavelength, using the equation, is \(\frac{375 \text{ cm/s}}{125 \text{ Hz}} = 3 \text{ cm}\). Understanding this relationship aids in comprehending how different sounds are produced.

Vocal Tract Acoustics

The vocal tract acoustics provides the framework for shaping the sound produced by vocal cord vibrations into recognizable speech and sound patterns. The vocal tract includes the throat, mouth, and nasal passages, playing a critical role in sound modulation.

The shape and size of the vocal tract change to adjust resonant frequencies, also known as formants, to produce different vowels and consonant sounds. Alterations in vocal tract shape are achieved by movements of the tongue, lips, and other articulators.

By varying the vocal tract's configuration, we can produce a wide variety of sounds and phonemes, which form the basic building blocks of speech. This ability highlights the intricacy of human speech creation and how dynamic the acoustics of the vocal tract can be.