= Solution
Take constant complex <amplitude> $A$ and real <wavenumber> $k$ and <angular frequency> $\omega$. Substitution of the <complex exponential> gives
$$
y_t=-i\omega y,\qquad y_{xx}=-k^2y,\qquad
im(-i\omega)y=k^2y.
$$
Thus every nonzero <plane wave> of the given form solves the equation precisely when
$$
\boxed{m\omega=k^2.}
$$
The <wavenumber> specifies spatial <wave phase> change per unit length, with <wavelength> $\lambda=2\pi/|k|$. The <angular frequency> specifies temporal <wave phase> change per unit time, with period $2\pi/|\omega|$ and ordinary <frequency> $|\omega|/(2\pi)$. The constant $m$ in this equation has dimensions time divided by length squared; without an additional dimensional coefficient, it is not an ordinary physical <mass>.
For propagating oscillatory solutions assume real $m\ne0$. If $m=0$, the equation becomes $y_{xx}=0$ and only $k=0$ among these spatial <plane waves> is possible; the usual <dispersion relation> branch and velocity discussion would degenerate.
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