Harmonic Wave Equation

reedc15

Dear Guys,

Does f(x,t)=exp[-i(ax+bt)^2] qualify as a harmonic wave? Please help!

Manish
Germany

mathman

Yes. Separate the real (cosine) and imaginary parts (sine).

reedc15

Ok, but what about the quadratic exponent? Would my wave equation still be harmonic?

TheDragon

i actually think not, cos(x^2) or cos(2x*t) is not an harmonic wave.
in general, an harmonic function f is a function that gives f''=A*f when A is a constant. the function you gave do not fulfil this requirement.

reedc15

Yes, cos(x^2) is not a harmonic wave, but cos[(kx+wt)^2] is, I think. "f''=A*f when A is a constant" this requirement is also fulfilled, as f comes from w, and it will take integer multiple (given by constant A)

TheDragon

I didn't understand what you mean,
d^2 f/dx^2= -f*(2xk^2+2kwt)-2k^2*sin((kx+wt)^2)
and nothing here suggest that there exist a constant A that for every t and every x
d^2 f/dx^2=Af.