Question

how the theory of black-body radiation was formulated by Rayleigh and Jeans

Answer 1

Rayleigh and Jeans considered that the radiation inside the cavity form standing wave at absolute temperature. As the cavity is 3D hence they generalize 1D standing waves in the string in 3D. At the end of the wall, nodes will form. The number of standing waves in the frequency interval between per unit volume can be given as

This formulation is independent of the shape of the cavity. Using the classical theory, the average energy per degree of freedom is where is Boltzmann constant.

For harmonic oscillator, the no. of degree of freedom is two hence average energy will be

Therefore the total energy perunit volume in the cavity for frequency interval between will be

This was  black-body radiation formula derived by Rayleigh and Jeans.