Some on the list prefer to define a wave as any function of space and time
which satisfies the wave equation. They raise counterexamples to
definitions based on the characteristics a wave function must possess. Here
is an example of a function which satisfies the wave equation in a
non-trivial way, but doesn't appear to be a wave to me.
Take the one dimensional wave equation to be:
d^2 f(x,t)/dx^2 = (1/v^2)*d^2 f(x,t)/dt^2 (1)
Where "d ()" is the partial differentiation operator.
Let f(x,t) =a*( x^2 + v^2*t^2) where a is a constant which provides the
correct dimensions for the disturbance of interest.
f(x,t) satisfies (1). Is this a counterexample to using the wave equation
to define waves?