`Thermal' ambience and fluctuations in classical field theory

A plane monochromatic wave will not appear monochromatic to a noninertial observer. We show that this feature leads to a `thermal' ambience in an accelerated frame {\it even in classical field theory}. When a real, monochromatic, mode of a scalar field is Fourier analyzed with respect to the proper time of a uniformly accelerating observer, the resulting power spectrum consists of three terms: (i)~a factor $(1/2)$ that is typical of the ground state energy of a quantum oscillator, (ii)~a Planckian distribution $N(\Omega)$ and---most importantly---(iii)~a term $\sqrt{N(N+1)}$, which is the root mean square fluctuations about the Planckian distribution. It is the appearance of the root mean square fluctuations that motivates us to attribute a `thermal' nature to the power spectrum. This result shows that some of the `purely' quantum mechanical results might have a classical analogue. The `thermal' ambience that we report here also proves to be a feature of observers stationed at a constant radius in the Schwarzschild and de-Sitter spacetimes.