- Note
- Things to remember

It is a sensation experienced by a listener. It is not a subjective term which cannot be measured quantitatively. The pitch of a sound depends upon the frequency of vibration of air. If the frequency of a sound is high, its pitch is high and if the frequency of sound is low. The high pitch sound is shrill and the low pitch is grave. The voice produced by ladies and children is high pitched.

Loudness in a subjective sensation depends on the intensity and sensitiveness of the listener’s ear. The intensity of sound is sound energy flowing per second per unit perpendicular to the direction of travel of the sound. Loudness or intensity depends upon the following factors:

- The amplitude of the vibration of source: Greater is the amplitude of vibration of the source, larger is the intensity and sound.
- The motion of the medium: If the wind is blowing in the direction of propagation of sound, the loudness of the sound increases.
- The presence of other bodies: The loudness of sound is increased due to the presence of other bodies near the source of a sound.
- The surface area of the vibrating body: The greater the surface area of a vibrating body, the larger is loudness of sound.
- The frequency of sound: The loudness of a vibrating body is directly proportional to the square of the frequency of the vibrating body.

It is a measure of the complexity of sound which enables us to distinguish between two sounds of the same pitch and loudness produced by two different sources. The note produced by an instrument does not contain a single frequency but consists of a fundamental and overtones with smaller intensity.

**Difference between Intensity of Sound and Loudness**

S.N. | Intensity of Sound | Loudness |

1. | It is a physical quantity which is measured in terms of rate of flow of energy per unit area. | It cannot be measured but compared like intensity. |

2. | It is independent of the sensitivity of the ears. | It depends on the sensitivity of ears. |

3. | It has an objective existence i.e. the personal physiology of the listener has no effect on its magnitude. | It has a subjective existence i.e. its magnitude depends on the personal physiology of the listener. |

The lowest intensity of sound that can be heard by an ear is called the threshold of hearing. The magnitude of hearing sensation depends upon both the intensity and frequency of the sound. The threshold of hearing is 10^{-12} Wm^{-2} for a pure tone of frequency 1 kHz.

Intensity of sound is defined as the average rate of transfer of energy per unit area cross a surface perpendicular to the direction of propagation of the sound. So, we can write intensity of sound,

\begin{align*} I = \frac PA \\ \end{align*} where P is the times rate of energy transfer of power of the sound wave and A, the surface area intercepting the sound. Consider a sound wave traveling in x-direction. Displacement of a particle when the wave is passing through the medium is \begin{align*} y = a\sin (\omega t – kx) \dots (ii) \\ \end{align*}

The speed of an oscillating particle of the medium is \begin{align*} \\ u = \frac {dy}{dt} = a \omega \cos (\omega t – kx ) \dots (iii) \\ \end{align*}

The power is equal to the product of force and velocity. So, the power per unit area is equal to the product of pressure and velocity.

\begin{align*} \\ \text { So,} \\ \Delta P \times u = [Bak\cos (\omega t – kx)] \times [a\omega \cos (\omega t – kx)] \\ = Bk\omega a^2 \cos ^2 (\omega t – kx) \\ \end{align*}

Since the intensity is the time average rate, the average value of the function

\begin{align*}\: \cos ^2 (\omega t – kx) \: \text {over a period T} = 2\pi /\omega: \text {is}\: \frac 12 \: \text {and so} \\ I = \frac 12 Bk\omega a^2 \dots (iv) \\ \text {As} \: \omega =2\pi f = 2\pi v/\lambda , \: \text {and} \: v^2 = B/\rho, \: \text {we have} \\ I = \frac 12 \sqrt {B\rho } \omega ^2 a^2 \dots (v) \\ \text {This equation can be modified as} \\ I = \frac 12 \rho v^2 k \omega a^2 = \frac 12 \rho v\omega ^2 a^2 \dots (vi) \\ \end{align*}

The intensity of sound can be expressed in terms of pressure amplitude as

\begin{align*} \\ I = \frac 12 Bk\omega a^2 \\ \text {or,} \: I = \frac {\omega \Delta P_m^2 }{2Bk} = \frac {v\Delta p_m^2}{2B} \dots (vii) \\ \text {As} \: v^2 = B/\rho , \: \text {then} \\ I = \frac {\Delta P_m^2}{2\rho v} = \frac {\Delta P_m^2}{2\sqrt {\rho B}} \dots (viii) \\ \end{align*}

Comparing equations (v) and} (vii),we see that the sound wave of the same intensity but different frequency have different amplitudes a but the same pressure amplitude, \(\Delta P_m. \)

The intensity of sound which the human ear can hear from 10^{-12} W/m^{2} to 1 W/m^{2}. Experiment shows that to produce an apparent doubling in loudness, the intensity of sound must be increased by a factor of 10. Intensity level \(\beta \) is defined as

$$ \beta = \log _{10} \frac {I}{I_0} $$

Where I is the measured intensity and I_{0} is a reference intensity chosen as 10^{-12} W/m^{2}, threshold of hearing at 1000 Hz. The sound intensity levels are expressed in bels in SI-units, a unit named to honour Alexander Graham Bell. However, it is a larger unit and a smaller unit, called the decibel is used.

$$ \beta = (10\: dB) \log _{10} \frac {I}{I_0} \: [\because 1 bel =10\: dB ]$$

References

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. *Principle of Physics*. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. *A text Book of Physics*. Kathmandu: Surya Publication, 2003.

The pitch is a sensation experienced by a listener.

Loudness in a subjective sensation depends on the intensity and sensitiveness of the listener’s ear.

The intensity of sound is sound energy flowing per second per unit perpendicular to the direction of travel of the sound.

The quality of timber is a measure of the complexity of sound which enables us to distinguish between two sounds of the same pitch and loudness produced by two different sources.

The lowest intensity of sound that can be heard by an ear is called the threshold of hearing.

The intensity of sound is defined as the average rate of transfer of energy per unit area across a surface perpendicular to the direction of propagation of the sound.

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May 20, 2019

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