A crude model of the human vocal tract treats it as a pipe closed at one end. Find the effective length of the vocal tract in a person whose fundamental tone is \(620 \mathrm{Hz}\). Sound speed in air at body temperature is \(354 \mathrm{m} / \mathrm{s}\)

Acoustic resonance is a phenomenon that occurs when an object vibrates at particular frequencies, producing a loud, clear sound. The process is similar to pushing a child on a swing at the right moment; when timed perfectly, the child swings higher with each push. Similarly, with sound waves, when the frequency of the vibrations matches the natural frequency of an object, resonance occurs, amplifying the sound.

In the human vocal tract, this resonance is vital. It enhances certain harmonics based on the shape and length of the vocal tract, which, in turn, helps to form the unique sounds of our voices. It's not just the vocal cords themselves creating the sound you hear when someone talks; it's also due to this resonance within the air-filled cavity that starts from the vocal cords and extends through the mouth and nose.

The concept of standing waves in pipes is used to explain how certain musical instruments, such as flutes and organs, as well as the human vocal tract, produce sound. When air is blown across or through these instruments, the air inside vibrates back and forth. Under certain conditions, these vibrations form standing waves - patterns of oscillation that remain in a constant position.

How Waves Behave in Closed Pipes

In a pipe that is closed at one end and open at the other (like the human vocal tract), standing waves occur at frequencies where a node exists at the closed end and an antinode at the open end. This means the air at the closed end cannot move (node), while the air at the open end moves freely (antinode). These constrained conditions create the standing wave pattern and determine which frequencies will resonate in the pipe.

Wavelength is a fundamental concept in understanding sound waves. It is the distance between two corresponding points of consecutive cycles of a wave, typically measured from peak to peak or trough to trough—depending on the method and type of wave. The wavelength of sound plays a crucial role in determining its pitch; the longer the wavelength, the lower the sound pitch, and vice versa.

When it comes to the human vocal tract, the wavelength can be related to the pipe model using the formula for the fundamental frequency or first harmonic of a pipe closed at one end: \( f = \frac{v}{4L} \). Here, \( L \) represents the length of the tract, while \( v \) is the speed of sound in air. Knowing the speed of sound and the fundamental frequency, as provided in the exercise, allows us to calculate the effective length of the vocal tract, which is directly related to the wavelength of the sound resonating in it.