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 foetus.
2. b) A bat locating insects in the dark.
3. c) A seismograph recording a distant earthquake.
4. d) A fishing sonar to locate shoals of fish.

The sound intensity level is defined as $\beta = 10 \times \log \frac{I}{I_0}$, measured in decibels ($\text{dB}$), where $I_0 = 10^{-12}\ \text{W/m}^2$ is the threshold intensity of hearing.

Calculate the intensity level of a sound of $I = 10^{-5}\ \text{W/m}^2$.

A point sound source emits with power $P = 0.5\ \text{W}$ uniformly in all directions. The intensity at a distance $r$ is $I = \frac{P}{4 \times \pi \times r^2}$ (take $\pi = 3.14$). The diagram represents isotropic emission.

Calculate the intensity at $5\ \text{m}$ from the source.

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.

A loud conversation has an intensity of $I = 5 \times 10^{-6}\ \text{W/m}^2$ (data: $I_0 = 10^{-12}\ \text{W/m}^2$; $\log 5 \approx 0.7$).

Calculate its intensity level in decibels using $\beta = 10 \times \log \frac{I}{I_0}$.

A passing lorry produces a level of $80\ \text{dB}$ and a quiet library is at $40\ \text{dB}$.

1. a) Without using a calculator, use the logarithmic definition of intensity level to reason how many times more intense (in $\text{W/m}^2$) the sound of the lorry is compared to that of the library.
2. b) Generalise the result: by what factor is the intensity multiplied for every additional $10\ \text{dB}$?

A public address loudspeaker emits sound with a power of $P = 3\ \text{W}$ (data: $\pi = 3.14$; $I_0 = 10^{-12}\ \text{W/m}^2$; $\log 2.4 \approx 0.38$).

1. a) Calculate the intensity at $10\ \text{m}$ from the loudspeaker using $I = \frac{P}{4 \times \pi \times r^2}$.
2. b) Calculate the intensity level at that point, in decibels.

At $2\ \text{m}$ from a drill, the intensity level is $90\ \text{dB}$ (given: $\log 2 \approx 0.3$).

1. a) Moving away to $4\ \text{m}$, the intensity is divided by $4$: calculate by how many decibels the level decreases and its new value.
2. b) Repeat the calculation for $8\ \text{m}$.
3. c) State the rule of thumb that can be deduced: how many decibels are lost each time the distance to the source is doubled?

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 fan produces an intensity level of $70\ \text{dB}$ at a point (given: $\log 2 \approx 0.3$).

1. a) When a second identical fan is turned on, the total intensity at that point doubles: calculate the new intensity level.
2. b) Reason why the result is not $140\ \text{dB}$ and explain how decibels are 'added'.

The intensity level of a normal conversation is $60\ \text{dB}$ (given: $I_0 = 10^{-12}\ \text{W/m}^2$).

1. a) Rearrange $\beta = 10 \times \log \frac{I}{I_0}$ for $I$ and calculate the intensity of that sound.
2. b) How many times greater than the threshold of hearing is this intensity?

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 millimetres.
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: reason why the bat uses such high frequencies to hunt small insects.

A pneumatic drill produces an intensity level of $100\ \text{dB}$ at a distance of $1\ \text{m}$ (given: $I_0 = 10^{-12}\ \text{W/m}^2$).

1. a) Calculate the intensity at $1\ \text{m}$ from the machine.
2. b) Assuming isotropic propagation ($I \propto \frac{1}{r^2}$), calculate the intensity and the intensity level at $10\ \text{m}$ and at $100\ \text{m}$.
3. c) Formulate the general rule relating the decrease in level to multiplying the distance from the source by ten.

At $15\ \text{m}$ from a concert loudspeaker, the measured intensity level is $90\ \text{dB}$ (data: $I_0 = 10^{-12}\ \text{W/m}^2$; $\pi = 3.14$).

1. a) Calculate the intensity at that point.
2. b) Assuming isotropic radiation, calculate the power of the loudspeaker using $P = I \times 4 \times \pi \times r^2$.
3. c) Comment on two reasons why the result is only an estimate of the real power (think about absorption by air, directivity of the loudspeaker or reflections from the surroundings).

Occupational health and safety regulations set $85\ \text{dB}$ as the recommended maximum level for $8\ \text{h}$ working days. A workshop machine produces $95\ \text{dB}$ at $2\ \text{m}$ (data: $I_0 = 10^{-12}\ \text{W/m}^2$; $10^{0.5} \approx 3.16$).

1. a) Calculate the intensities corresponding to $95\ \text{dB}$ and $85\ \text{dB}$.
2. b) Using $I \propto \frac{1}{r^2}$, calculate the minimum distance from the machine at which a worker could work without exceeding $85\ \text{dB}$.
3. c) Propose two additional protective measures and justify them physically.

An industrial machine produces a level of $75\ \text{dB}$ in the machine room.

1. a) Reason, using the logarithmic definition of intensity level, how many identical machines operating at the same time would be needed to reach $85\ \text{dB}$ at the same point.
2. b) And to reach $95\ \text{dB}$?
3. c) Explain why doubling the number of machines does not double the decibel level.

In an open field without obstacles, a human voice emits sound with a power of $10^{-5}\ \text{W}$, approximately isotropic (data: $I_0 = 10^{-12}\ \text{W/m}^2$; $\pi = 3.14$; $\sqrt{7.96 \times 10^{5}} \approx 892$).

1. a) State the condition for the threshold of hearing $I = I_0$ and rearrange for the maximum distance at which the voice could still be heard.
2. b) Calculate that distance.
3. c) The actual range of the voice is much smaller: cite two physical reasons that explain this.

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 kilometres 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 millimetres.
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.