1. The problem statement, all variables and given/known data The horizontal component of a wave is in the y-direction, and the vertical component is in the x-direction. If the horizontal and vertical components of a wave on a string have the same amplitude and are 90 degrees out of phase (say δv=0 and δh= 90°). At a fixed point z, show that the string moves in a circle around the z-axis. 2. Relevant equations ƒi(z,t)=A Cos(kz-ωt+δi) 3. The attempt at a solution First of all, don't tell me ƒv2+ƒh2=A2 therefore it is a circle. I know that, and I don't like it. If that is the ONLY way to solve the problem so be it, but I think I can do it a different way. I think I should be able to express ƒ in cylindrical coordinates, ƒ=s(t)s+φ(t)φ where s and φ are the s and φ unit vectors, and show that s(t) = constant and that φ(t)≠constant and φ(t) has range s.t. φ(t)max≥2π (i.e. will go around a full circle, not do some silly oscillation between 0 and π or something. That wouldn't really make any physical sense I don't think, but I think it is necessary for the proof.). So here we go with my attempt(forgive my notation, hopefully you can make sense of it, if not let me know and ill do everything proper) : f_v=A Cos(kz-wt) x_unitvect. f_h=A Cos(kz-wt+π/2) y_unitvect. = - A Sin(kz-wt) y_unitvect. f=f_v+f_h= A [Cos(kz-wt) x_unitvect. - Sin(kz-wt) y_unitvect.] Converting to cylindrical: f= A [Cos(kz-wt) Cos(φ) - Sin(kz-wt) Sin(φ)] s_unitvect. - A[Cos(kz-wt)Sin(φ)+Sin(kz-wt)Cost(φ)] φ_unitvect. Using Cos/Sin (A±B) Trig Identities: f= A Cos[kz-wt+φ] s_unitvect. - A Sin[kz-wt+φ] φ_unitvect. If φ= wt (?), Then reduces to f= A Cos[kz] s_unitvect. - A Sin[kz] φ_unitvect. which would make both time independent. Yikes.