Prove that sound waves propagate with a speed given by Equation 17.1. Proceed as follows. In Figure 17.3, consider a thin cylindrical layer of air in the cylinder with face area A and thickness Δx. Draw a free-body diagram of this thin layer.
Show that ∑Fx = max implies that ─ [∂(ΔP)/∂x] A Δx = pA Δx (∂2s/∂t2). By substituting ΔP = ─ b (∂s/∂t2). To a mathematical physicist, this equation demonstrates the existence of sound waves and determines their speed. As a physics student, you must take another step or two. Substitute into the wave equation the trial solution s(x, t) = smax cos (kx─wt). Show that this function satisfies the wave equation provided that w/t = √B/p. this result reveals that sound waves exist provided that they move with the speed v = fA = (2πf) (A /2π) = w/k = √B/p.