Could someone please assist? Plane Waves

Bob

I'm currently reading a paper where the author discusses the propagation of errors in terms of plane waves. The paper aims to reduce the reflection of such errors by including appropriate boundary conditions at the domain boundaries.

The equation we're trying to solve is:

2*u_xt = K*u_xx + u_yy (where K is a constant and I have borrowed the notation "_x" to mean differentiation w.r.t x and so on)

The author states that the solution to such an equation is the set of plane waves f( ct - x*w1 - y*w2 ) which satisfies this equation if

c= -( K*w1^2 + w2^2 )/( 2*w1 ) so far fairly easy )

The author then states that this value 'c' is the "local wave speed" in the direction of the unit vector (w1,w2). Could someone please advise me if this is correct? my knowledge of wave mechanics is somewhat rudimentary and I am a tad confused.

My understanding is that the phase speeds are c/w1 & c/w2 in the x and y directions respectively. For the particular case I'm dealing with w1 is far smaller than w2 which means the phase speed should be much larger in the x-direction than in the y-direction. If the authors statement is correct, however, then I'd expect the wave to travel with local speed 'c' in the direction (w1,w2) which would be, more or less, in the y-direction.

I fear I've missed something important. If someone could please point out my error then I'd be very grateful.

Thanks for reading Bob