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.