Pressure Amplitude and Musical Sound

The branch of physics that deals with the process of the production, transmission and reception of sound is called acoustics. Some of the important fields of acoustic are

  1. The design of acoustical instruments.
  2. Electroacoustic i.e microphones, amplifiers, loudspeakers etc.
  3. Architecture acoustics dealing with the designs and construction of buildings, music balls, recording rooms in radio and television stations.
  4. Musical acoustic dealing with the design of musical instruments.

Pressure Amplitude

A longitudinal wave can be represented by a sinusoidal wave with a definite frequency, wavelength and amplitude. Such waves, travelling in x-direction can be represented as

$$ y = a\sin (\omega t – kx) \dots (i) $$

Where a = amplitude, ω = angular frequency, and k = wave number. The displacement y in a longitudinal wave is along the direction of wave travel and so, x and y are parallel to each other. The amplitude is the maximum displacements of a particle in the medium from its equilibrium position which is also called displacement amplitude.

Let \(\Delta P\) be the instantaneous pressure fluctuation in a sound wave at a point x at time t. The absolute pressure at that point is \( P + \Delta P\) where P is the atmospheric pressure. Consider a cylinder of air of cross-section area A and axis along the direction of wave travel as shown in the figure. Volume of the cylinder having length \(\Delta x\) at normal condition is \( V = A\Delta x\) when there is waves. When a wave is present, left cross-section of the cylinder displaces through a distance y1 and the right cross-section through a distance y2. If y2 > y1, the volume of the cylinder increases and this cause a decrease of the pressure. If y2<y1, the volume of cylinder decreases causing an increase in pressure. The pressure fluctuation depends on the difference between the displacements at neighboring points in the medium. So, the change in volume,

\begin{align*} \Delta &= A(y_2 –y_1) = a\Delta y \\ \text {In the limit as} \: \Delta x \rightarrow 0, \: \\ \text {the fractional change in volume dV/V is} \\ \frac {dV}{V} &= \lim_{\Delta x \to 0} \frac {A\Delta y}{A\Delta x} = \frac {\delta y}{\delta x} \dots (ii) \\ \end{align*}The pressure variation in the cylinder due to fractional change in volume is given by \begin{align*} \Delta P &= -B \frac {\Delta V}{V} \dots (iii) \\\end{align*}

\begin{align*} \text {where B is the bulk of modulus of air. }\\ \text {From equations} \: (ii)\: \text {and} \: (iii), \: \text {we have} \\ \Delta P = -B\frac {\delta y}{\delta x} = Bak\cos (\omega t – kx) \dots (iv) \\ \text {Let} \Delta P_m = Bak, \end{align*} the pressure amplitude which is the maximum increase or decrease in pressure due to wave. The above equation can be written as \begin{align*}\Delta P = \Delta P_m \cos (\omega t – kx) \dots (v) \\ \end{align*}

The compressions where the points have lowest pressure and density are points of zero displacement and the rarefaction where the points have lowest pressure and density are points of zero displacement. Substituting the value B from equation\begin{align*} v = \sqrt {B/ \rho } , \text {the pressure amplitude is given by} \\ \Delta P_m = Bak = v^2 \rho ka \dots (vi) \\ \end{align*}

Thus pressure amplitude is directly proportional to the displacement amplitude and this amplitude is very small.

Musical Sound and Noise

Musical Sound

A desirable sound that produces a pleasing effect on the listeners is called musical sound. Such sound is produced by regular and periodic vibrations. Sounds produced by tuning a fork, flute, piano etc are music sounds.

Noise

The sound that produces an unpleasant effect on the listener is called noise. It is an unpleasant discontinuous sound produced by an irregular succession of disturbances. All sounds other than musical notes are noises.

Difference between Musical and Noise

S.N.

Musical Sound

Noise

1.

It consists of a quick, regular and periodic succession of compressions and rarefactions.

Compressions and rarefactions in noise are not quick, regular and periodic in succession.

2.

Frequency is generally high.

Frequency is usually low.

3.

Changes in amplitude are not sudden.

Changes in amplitude may be sudden.

4.

It is usually of long duration.

It may be of very short duration.

5.

It produces a pleasing effect on the listeners.

It produces a displeasing effect on the listeners.

6.

It is desirable sound.

It is undesirable sound.

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 branch of physics that deals with the process of the production, transmission and reception of sound is called acoustics. 

A longitudinal wave can be represented by a sinusoidal wave with a definite frequency, wavelength and amplitude. 

 The amplitude is the maximum displacements of a particle in the medium from its equilibrium position which is also called displacement amplitude.

A desirable sound that produces a pleasing effect on the listeners is called musical sound. 

The sound that produces an unpleasant effect on the listener is called noise.

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Rajeshwari Baniya

how to derive equation of pressure and amplitude ?