Sound, Pipes and the Ear

Sound is a longitudinal wave in which regions of high and low pressure travel through a medium. The speed of sound in air is about 340 m/s, and the speed, frequency f and wavelength λ of any wave are related by v = fλ. When a source of sound and an observer move relative to each other, the observed frequency differs from the source frequency. This is the Doppler effect. For a source moving toward a stationary observer at speed vₛ, the observed frequency is f′ = f · v/(v − vₛ), and for a source moving away from the observer, f′ = f · v/(v + vₛ). The Doppler effect changes the frequency that reaches the observer. It does not change the speed of the wave in the air.

The loudness of a sound is described by the sound intensity level β, measured in decibels (dB): β = 10 log₁₀(I/I₀), where I is the intensity and I₀ = 10⁻¹² W/m² is close to the quietest sound a person can hear. A tenfold increase in intensity raises the level by 10 dB. For a small source in open air, the intensity falls in proportion to the inverse square of the distance from the source.

Part 1. A student studies resonance in two tubes of the same length, 0.85 m. Tube A is open at both ends, and Tube B is closed at one end and open at the other. She holds a tuning fork of variable frequency at the open end and records the frequencies at which the sound inside becomes much louder. Table 1 shows the lowest two resonant frequencies for each tube.

Table 1. Resonant frequencies of two tubes, each 0.85 m long.

Part 2. A student stands beside a straight road and records the frequency of an ambulance siren that emits a steady tone of 400 Hz as measured by the driver. The ambulance drives at constant speed past the student. Table 2 shows the frequency the student hears as the ambulance approaches and as it moves away.

Table 2. Frequency of a siren heard by a stationary student.

Part 3. The student then measures the sound intensity level from a small loudspeaker in an open field at several distances. Table 3 shows the results.

Table 3. Sound level at different distances from a loudspeaker.

The ear detects sound through a series of structures. The eardrum vibrates in response to pressure variations, and three small bones transmit the vibrations to the oval window, an opening in the fluid-filled cochlea. Inside the cochlea, the basilar membrane is narrow and stiff near the oval window (the base) and wide and flexible at the far end (the apex). Different frequencies cause the largest vibrations at different positions along the membrane, and hair cells at those positions send signals to the brain.