BIO Deep-sea divers often breathe a mixture of helium and oxygen to avoid getting the 'bends' from breathing high-pressure nitrogen. The helium has the side effect of making the divers' voices sound odd. Although your vocal tract can be roughly described as an open-closed tube, the way you hold your mouth and position your lips greatly affects the standing-wave frequencies of the vocal tract. This is what allows different vowels to sound different. The 'ee' sound is made by shaping your vocal tract to have standing-wave frequencies at, normally, 270 Hz and 2300 Hz. What will these frequencies be for a helium-oxygen mixture in which the speed of sound at body temperature is 750m/s ? The speed of sound in air at body temperature is 350m/s .

Verified step by step guidance

1. Step 1: Understand the relationship between the speed of sound, frequency, and wavelength. The fundamental equation is \( v = f \lambda \), where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength. For standing waves in a tube, the wavelength is determined by the geometry of the tube.
2. Step 2: Recognize that the vocal tract can be modeled as an open-closed tube. For such a tube, the fundamental frequency corresponds to a wavelength \( \lambda \) that is four times the length of the tube \( L \), i.e., \( \lambda = 4L \). Higher harmonics occur at odd multiples of the fundamental frequency.
3. Step 3: Use the ratio of the speeds of sound in helium-oxygen mixture and air to determine the new frequencies. Since the wavelength \( \lambda \) is fixed by the geometry of the vocal tract, the frequency \( f \) scales directly with the speed of sound \( v \). The relationship is \( f_{\text{new}} = f_{\text{old}} \times \frac{v_{\text{new}}}{v_{\text{old}}} \).
4. Step 4: Substitute the given values into the scaling equation. The speed of sound in air at body temperature is \( v_{\text{old}} = 350 \, \text{m/s} \), and the speed of sound in the helium-oxygen mixture is \( v_{\text{new}} = 750 \, \text{m/s} \). The original frequencies are \( f_{\text{old,1}} = 270 \, \text{Hz} \) and \( f_{\text{old,2}} = 2300 \, \text{Hz} \).
5. Step 5: Calculate the scaled frequencies using \( f_{\text{new}} = f_{\text{old}} \times \frac{750}{350} \). Perform this calculation for both \( f_{\text{old,1}} \) and \( f_{\text{old,2}} \) to find the new frequencies in the helium-oxygen mixture.

Key Concepts

Speed of Sound

The speed of sound is the rate at which sound waves propagate through a medium. It varies depending on the medium's properties, such as density and temperature. In gases, the speed of sound increases with lower molecular weight; hence, helium, being lighter than nitrogen, allows sound to travel faster, affecting the frequencies produced by the vocal tract.

Standing Waves

Standing waves occur when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other, creating nodes and antinodes. In the context of the vocal tract, the shape and length of the tract determine the specific frequencies at which these standing waves form, which correspond to different vowel sounds.

Resonant Frequencies

Resonant frequencies are specific frequencies at which a system naturally oscillates with greater amplitude. For the human vocal tract, these frequencies depend on its physical dimensions and the medium through which sound travels. When divers breathe helium, the change in the speed of sound alters these resonant frequencies, resulting in a higher pitch of the voice.