  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.
