The Back-Row Bound: How Far a Teacher's Voice Carries Intelligibly in Classroom Noise

Intelligibility at the back of a classroom is set by signal-to-noise ratio, not by how loud a teacher feels. This paper computes the distance at which a raised voice falls to a stated 12 dB signal-to-noise criterion against 55 dBA of occupied-room noise, compares normal and amplified voice, and converts the result into the fraction of a standard 8 metre deep room that lies beyond intelligible reach. The bound is a clean limit: in the stated noise, unaided speech cannot serve a deep room, however experienced the teacher. Computed results: d_max_raised_m = 0.79, d_max_normal_m = 0.45, d_max_amplified_m = 2.51, noise_dBA = 55.0, frac_depth_lost = 0.9. Every number in this paper is produced by the accompanying script `compute.py`. SIMULATED, NOT MEASURED.