Thepurposeofthepresentpaperistocallforattentiontothefollowingquestion:Whichoftheinitialdata(nonsmall)admitglobalsmoothsolutionstotheCauchyproblemfornonlinearwaveequations.Afewcasesandexamplesaresketched,showingthatthegeneralanswerofthisquestionmaybequitecomplicated.