The Decibel Scale (AQA A Level Physics): Revision Note

Relative Intensity Levels of Sound

• When the sound waves enter the auditory canal, a stationary wave is set up

  • This can be modelled as a tube with an antinode forming at the open end and a node forming at the closed end

• As a result, the auditory canal acts as a closed tube resonator which enhances the sensitivity of sounds in specific ranges

  • In the case of human hearing, this is what enables us to hear sounds in the range of 20 Hz to 20,000 Hz

  • The highest sensitivity region is around 2 to 5 kHz, which corresponds to frequencies of speech

Resonance in the Auditory Canal

The auditory canal works as a closed tube resonator which is sensitive to specific ranges of frequencies

The Decibel Scale

• The ear’s ability to detect small changes in sound intensity is referred to as its sensitivity

  • This is measured in decibels (dB)

• The sensitivity of human hearing depends on the frequency of the sound

• The graph of sound intensity and frequency shows that the ear is most sensitive to sounds with a frequency of about 3000 Hz

The Relationship between Sound Intensity and Frequency 

The human ear is most sensitive to sounds with a frequency of around 3000 Hz 

• For frequencies much greater, or lower than 3000 Hz, the intensity of sound must be several orders of magnitude higher to be detected

  • This large range of intensities shows that the response of the ear is logarithmic

• The decibel (dB) scale is also logarithmic 

  • Hence, the decibel scale is a useful way of measuring the response of the human ear to changes in sound intensity

The Decibel Scale

The decibel scale is used to represent the large range of intensities that the ear can detect on a logarithmic scale

Loudness & Decibels

• Intensity of a sound wave is defined as:

  The amount of sound energy that passes a point per second per unit area

• Intensity can be described by the equation:

$I=\frac{P}{A}$

• Where

  • $I$ = intensity of sound wave (W m⁻²)

  • $P$ = power, or energy transferred per second (W)

  • $A$ = area through which sound waves pass (m²)

• For a sound wave that spreads out uniformly in all directions, it can be assumed to propagate as a spherical shell

  • The surface area of a sphere is $4\mathrm{\pi }{r}^2$

  • The radius $r$ of this sphere is equal to the distance between the source of the sound and the receiver

• This means the intensity of the sound spreads out according to an inverse square law

$I=\frac{P}{4\mathrm{\pi }{r}^2}$

Inverse Square Law of Sound Intensity

When the source of the sound is twice as far away, it has spread over four times the area, hence the intensity is four times smaller

• The human ear can detect and respond to a very large range of intensities

• The response of the ear to changes in sound intensity, or the perceived change in loudness, is directly proportional to the logarithm of the percentage change in intensity

perceived change in loudness $\propto \log \left(\frac{I_2}{I_1}\right)$

• Where

  • $I_1$ = initial intensity of sound wave (W m⁻²)

  • $I_2$ = final intensity of sound wave (W m⁻²)

• This is known as the logarithmic response of sound intensity

  • This means when the intensity of a sound doubles, and doubles again the loudness increases in fixed steps (for a given frequency)

  • For example, an increase in sound intensity from 1 × 10⁻¹² W m⁻² to 2 × 10⁻¹² W m⁻² is perceived to be the same change in loudness as 1 W m⁻² to 2 W m⁻²

• The threshold of hearing $I_0$ is defined as:

  The lowest intensity that a normal human ear can detect at a frequency of 1 kHz

• This is equal to about 1 × 10⁻¹² W m⁻²

• The intensity level of a sound can be defined as

intensity level = 10 log $\left(\frac{I}{I_0}\right)$

• Where:

  • Intensity level is measured in decibels (dB)

  • $I$ = intensity of the sound wave (W m⁻²)

  • $I_0$ = threshold of hearing intensity = 1 × 10⁻¹² W m⁻²

• The threshold of hearing is assigned a sound level of 0 decibels, or 0 dB

  • An intensity level of 10 dB is 10 times louder than $I_0$, or 1 × 10⁻¹¹ W m⁻²

  • An intensity level of 20 dB is 100 times louder than $I_0$, or 1 × 10⁻¹⁰ W m⁻²

  • An intensity level of 30 dB is 1000 times louder than $I_0$, or 1 × 10⁻⁹ W m⁻²