Exercises: Sound waves and their properties

The three physiological properties of sound are pitch, intensity, and timbre.

1. a) Associate each property with the physical quantity of the wave that determines it: frequency, amplitude (energy transported), or wave shape.
2. b) Give a musical example of each: two sounds that differ only in pitch, only in intensity, and only in timbre.

The human ear perceives frequencies roughly between $20\ \text{Hz}$ and $20\ \text{kHz}$. The diagram shows the three regions of the acoustic spectrum:

Classify the following sound waves into infrasound, audible sound, or ultrasound, and justify the classification:

1. a) $15\ \text{Hz}$ (communication between elephants).
2. b) $440\ \text{Hz}$ (the note "A" of a tuning fork).
3. c) $45\ \text{kHz}$ (echolocation of a bat).
4. d) $3\ \text{MHz}$ (medical ultrasound scan).

A tuning fork vibrates at $440\ \text{Hz}$ (speed of sound in air: $340\ \text{m/s}$).

1. a) Calculate the period of the sound wave generated.
2. b) Calculate its wavelength in air.

Match each application with the type of sound wave it uses (infrasound, audible sound, or ultrasound) and justify the choice:

1. a) An ultrasound scan to observe a fetus.
2. b) A bat locating insects in the dark.
3. c) A seismograph recording a distant earthquake.
4. d) A fishing sonar to locate schools of fish.

A violin and a flute play the same note ($440\ \text{Hz}$) with the same intensity, and yet they can be distinguished perfectly.

1. a) Which property of sound allows this, and with which wave quantity is it related?
2. b) Explain the role of harmonics (frequency components that are multiples of the fundamental frequency) in this difference.
3. c) Draw in your notebook, qualitatively, the expected waveform for each instrument and compare it with that of a pure tone.

You can do it on paper and send a photo with your answer.

We cannot hear the ultrasound emitted by bats ($f > 20\ \text{kHz}$) nor the infrasound with which elephants communicate ($f < 20\ \text{Hz}$).

1. a) Explain why, using the concept of hearing range.
2. b) The wavelengths of infrasound are very large: reason what advantage this provides for communicating over long distances, considering the phenomenon of diffraction around obstacles in the terrain.

An ultrasound scanner emits ultrasound waves at $3\ \text{MHz}$ that propagate through soft tissues at $1540\ \text{m/s}$. The diagram shows the path of the pulse.

1. a) Calculate the wavelength of the ultrasound in the tissue.
2. b) The echo from the back wall of an organ returns $1.3 \times 10^{-4}\ \text{s}$ after emission: calculate the depth at which it is located, bearing in mind the round trip of the pulse.

A bat emits ultrasound waves at $80\ \text{kHz}$ (speed of sound in air: $340\ \text{m/s}$).

1. a) Calculate the wavelength and express it in millimeters.
2. b) The echo produced by a moth returns $0.05\ \text{s}$ after emission: calculate the distance to the prey.
3. c) Objects much smaller than the wavelength barely reflect the wave: explain why the bat uses such high frequencies to hunt small insects.

In an abdominal ultrasound scan, a single pulse generates two successive echoes: that from the front wall of an organ returns at $6 \times 10^{-5}\ \text{s}$ and that from the back wall at $9 \times 10^{-5}\ \text{s}$ (speed of sound in tissue: $1540\ \text{m/s}$). The diagram shows the two echoes.

1. a) Calculate the depth of each wall, keeping in mind the round trip.
2. b) Deduce the thickness of the organ.
3. c) Explain how the ultrasound scanner reconstructs a two-dimensional image from many such pulses and echoes, scanning in different directions.

Whales communicate with sounds of around $20\ \text{Hz}$.

1. a) Calculate the wavelength of that sound in air ($340\ \text{m/s}$) and in seawater ($1500\ \text{m/s}$).
2. b) Reason, from the phenomenon of diffraction, why these low frequencies allow communication over hundreds of kilometers despite obstacles on the seabed.
3. c) The absorption of water increases with frequency (given): justify why whales do not use the high frequencies that dolphins use for short distances for long-distance communication.

In ultrasound, the minimum structure size that can be resolved is of the order of one wavelength (given: speed of sound in tissue $1540\ \text{m/s}$).

1. a) Calculate the wavelength and the corresponding resolution for a $3\ \text{MHz}$ probe and for a $10\ \text{MHz}$ probe, in millimeters.
2. b) Tissue absorption increases with frequency (given): discuss the trade-off between resolution and depth of exploration.
3. c) Justify which probe you would choose to examine a deep organ such as the liver and which for a superficial structure such as the thyroid gland.