In teams of about 4, solve the following problem: Find all the vectors p= (px , py) that satisfy
for a given vector k = (kx , ky) . Take for instance k = (4,8). k, p, q are the norms of the vectors k, p, q, i.e. k=sqrt(kx2 + ky2) and similar definitions for p, q .
Background
The "grown-up" elementary representation of a wave in a two-dimensional
space is
where A is the amplitude of the wave, k is the wavevector, x =(x,y) is the position vector, W is the oscillation frequency (function of k) and c.c. denotes "complex conjugate". The function u(x,y,t) could represent, for instance, the elevation of the surface of lake Michigan as a function of position and time. The last two equations above indicate that the 3 vectors k, p, q form a triangle. This is necessary for wave interaction when the nonlinearity is quadratic. The first equation requires that the frequencies of the three waves add up to zero. This is the resonance condition which implies strong interaction between the waves. The specific problem above comes up when studying the nonlinear interaction of waves in rotating and/or stratified fluids (e.g. atmosphere or oceans) where W(k) = ky/k.