According to Weber-Fechner's law, "loudness (L) is proportional to the logarithm of the actual intensity (1) measured with an accurate nonhuman instrument". This means that
L ∝ 1n I|
L = k  1n I
where k is a constant, which depends on the unit of measurement. The difference between two loudnesses, L₁ and L_o measures the relative loudness between two precisely measured intensities and is called as sound intensity level. Mathematically, sound intensity level is
ΔL = L₁-L_o = k In I₁- k In I₀ = k In \(\left[I_1\over I_0\right]\)
If k = 1, then sound intensity level is measured in bel, Therefore,
\(ΔL=In\)\(\left[I_1\over I_0\right]\)bel
However, this to express smaller unit, decibel. Thus, [decibel = \(1\over10\)bel] by multiplying and dividing by 10
\(ΔL=10\left(In\left[I_1\over I_0\right]\right){1\over 10}bel\)
\(ΔL=10In\left[I_1\over I_0\right]\)decibel with k = 10
For practical purposes,
\(ΔL=10log_{10}\left[I_1\over I_0\right]\)decibel.

A jerk is given on a stretched string at time t = 0 s. Assume that the wave pulse created during this disturbance moves along positive x direction with constant speed v as shown in Figure (a).

Represent the shape of the wave pulse, mathematically as y = y(x, 0) = j(x) at time t = 0s. Assume that the shape of the wave pulse remains the same during the propagation. After some time t, the pulse moving towards the right and any point on it can be represented by x' (read it as x prime) as shown in Figure (b). Then
y(x, t) =j(x') =j(x - vt)
Similarly, if the wave pulse moves towards left with constant speed v, then y =j(x + vt). Both waves y =j(x + v!) and y =j(x - vt) will satisfy the following one dimensional differential equation known as the wave equation
\({∂^2y\over ∂x^2}={1\over4 v^2}{∂^2y\over ∂t^2}\)
where the symbol p represent partial derivative (read \({∂y\over ∂x}\)  as partial y by partial x). Not all the solutions satisfying this differential equation can represent waves, because any physical acceptable wave must take finite values for all values of x and t. But if the function represents a wave then it must satisfy the differential equation. Since, in one dimension (one independent variable), the partial derivative with respect to x is the same as total derivative in coordinate x, we write So it can be written as
\({d^2y\over dx^2}={1\over v^2}{d^2y\over dt^2}\)

Sound waves can be classified in three groups according to their range of frequencies:
(1) Infrasonic waves : Sound waves having frequencies below 20. Hz are called infrasonic waves. These waves are produced during earthquakes. Human beings cannot hear these frequencies. Snakes can hear these frequencies.
(2) Audible waves : Sound waves having frequencies between 20 Hz to 20,000 Hz (20kHz) are called audible waves. Human beings can hear these frequencies.
(3) Ultrasonic waves : Sound waves having frequencies greater than 20 kHz are known as ultrasonic waves. Human beings cannot hear these frequencies. Bats can produce and hear these frequencies.

It works on the principle of multiple reflections.
It consists of three main parts:
(i) Chest piece
(ii) Ear piece
(iii) Rubber tube
(i) Chest piece: It consists of a small disc-shaped resonator (diaphragm) which is very sensitive to sound and amplifies the sound it detects.
(ii) Ear piece: It is made up of metal tubes which are used to hear sounds detected by the chest piece.
(iii) Rubber tube: This tube connects both chest piece and ear piece. It is used to transmit the sound signal detected by the diaphragm, to the ear piece. The sound of heart beats (or lungs) or any sound produced by internal organs can be detected, and it reaches the ear piece through this tube by multiple reflections.

The sound produced in a big hall or auditorium or theatre is absorbed by the walls, ceilings, floor, seats, etc. To avoid such losses, a curved sound board (concave board) is kept in front of the speaker, so that the board reflects the sound waves of the speaker towards the audience. This method will minimize the spreading of sound waves in all possible direction in that hall and also enhances the uniform distribution of sound throughout the hall. That is why a person sitting at any position in that hall can hear the sound without any disturbance.

Let us consider an ideal gas whose equation of state is
PV= n R T  ...(1)
where, P is pressure, V is volume, T is temperature, n is number of mole and R is universal gas constant. For a given mass of a molecule, equation (1) can be written as
\({PV\over T}=Constent\)   ........(2)
For a fixed mass m, density of the gas inversely varies with volume. i.e.,
\(ρ∝{1\over V}, V={m\over ρ}\)
Substituting equation (3) in equation (2), we get
\({P\over \rho}=T\)  ..........(4)
where c is constant
The speed of sound in air given in equation
\(v_A=\sqrt{B_A\over \rho}=\sqrt{\gamma P\over \rho}=\sqrt{\gamma v_r}\) can be written as
\(v=\sqrt{\gamma P\over \rho}=\sqrt{\gamma cT}\)
Since v∝√T  the speed of sound varies directly to the square root of temperature in kelvin.
Let v_o be the speed of sound at temperature at 0° C or 273 K and v be the speed of sound at any arbitrary temperature T (in kelvin), then
\({v\over v_0}=\sqrt{T\over 273}=\sqrt{273+t\over 273}\)
\(v=v_0\sqrt{1+{t\over 273}}≅v_0\left(1+{t\over 546}\right)\)
(using binomial expansion)
Since v_o = 331m s^-1 at 00C, vat any temperature in t₀C is
v = (331 + 0.60t) m s^-1
Thus the speed of sound in air increases by 0.61 ms^-1 per degree celcius rise in temperature. Note that when the temperature is increased, the molecules will vibrate faster due to gain in thermal energy and hence, speed of sound Increases.

(i) The number of cycles (or revolutions) per unit time is called angular frequency.
Angular frequency, \(\omega=\frac{2\pi}{T}=2\pi f\) (unit of radians/ second)
(ii) The number of cycles per unit distance or number of waves per unit distance is called wave number.
wave number, \(k=\frac{2\pi}{\lambda}\) (unit is radians/ meter)
(iii) In two, three or higher dimensional case, the wave number is the magnitude of a vector called wave vector. The points in space of wave vectors are called reciprocal vectors, \(\vec k\). Dimensions of \(\vec k\) is L^-1.

(i) For the propagation of the waves, the medium must possess both inertia and elasticity, which decide the velocity of the wave in that medium.
(ii) In a given medium, the velocity of a wave is a constant whereas the constituent particles in that medium move with different velocities at different positions. Velocity is maximum at their mean position and zero at extreme positions.
(iii) Waves undergo reflections, refraction, interference, diffraction and polarization.

(i) A tuning fork is struck on a rubber pad, the prongs of the tuning fork vibrate about their mean positions.
(ii) The prong vibrating about a mean position means moving outward and inward.
(iii) When a prong moves outward, it pushes the layer of air in its neighbourhood which means there is more accumulation of air molecules in this region.
(iv) Hence, the density and also the pressure increase. These regions are known as compressed regions or compressions.
(v) This compressed air layer moves forward and compresses the next neighbouring layer in a similar manner. Thus a wave of compression advances or passes through air.
(vi) When the prong moves inwards, the particles of the medium are moved to the right. In this region both density and pressure are low. It is known as a rarefaction or elongation.