Measurements are presented of spin waves observed in the nearly free-electron metals sodium and potassium, utilising a transmission microwave spectrometer operating at 80 GHz. With very pure material and careful sample preparation it has been possible to detect up to fifty distinct spin waves. The samples, studied at 1.4K, have values of omega tau of the order of 150 where omega is the observational frequency and tau is the momentum relaxation time determined from the linewidth and amplitude decay of the spin waves. Consequently the signals approximate very closely to the infinite omega tau limit. This allows a simple comparison of theory with experiment yielding values for B₀, B₁ and B₂, the first three parameters characterising the many-body spin interaction. For sodium B₀=-0.210+or-0.015, B₁=-0.010+or-0.015 and mod B₂ mod <or=0.02 while for potassium B₀=-0.293+or-0.01, B₁=-0.076+or-0.01 and mod B₂ mod <or=0.01.