In this paper we report an analogue for the vacuum state in classical field theory and its Planckian nature with respect to uniformly accelerated observers. 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 none of which have a simple classical meaning. Specifically, the three terms are (i) a factor (1/2) that is typical of the ground state energy of a quantum oscillator, (ii) a Planckian distribution N(Ω) and - most importantly - (iii) a term √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. Such a power spectrum also arises when we Fourier analyze a real, monochromatic, plane electromagnetic wave in the frame of a uniformly accelerating observer. We also present a model of a detector whose response is the Fourier spectrum of the field with respect to its proper time, which illustrates that it should, in principle, be possible to physically measure the power spectrum we have obtained. These results show that some of the "purely" quantum mechanical results might have a classical analogue.