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.

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 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.

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'.

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.

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 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 $10\ \text{dB}$?

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 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 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?