As has been indicated in the comments, two different cases can be distinguished with equations of the form $\,u_{tt}=au_{xx}\,$
(apart from the case $a=0$). Without loss of generality (i.e. renaming of a variable and scaling in that dimension) we have:
$$
a \gt 0 \quad \Longrightarrow \quad \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0
\quad \mbox{with} \quad a=c^2
\\ a \lt 0 \quad \Longrightarrow \quad \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
\quad \mbox{with} \quad a=-1\;,\; t=y
$$
In my Indefinite double integral answer it is proved with Operator Calculus that the first equation, the "wave equation", has the following general solutions:
$$
u(x,t) = F(x-ct) + G(x+ct)
$$
Everything real-valued.
This can be interpreted physically as the superposition of a wave travelling forward and a wave travelling backward.
On the other hand, the second equation, an "elliptic" equation, has the following general solutions:
$$
u(x,y) = F(z) + G(\overline{z})
$$
These are related to holomorphic functions in the complex plane ($z=x+iy$) and they are of a completely different nature.
I think that nothing else is needed to answer the question as formulated in the header.
EDIT.
In the first place, one should get rid of the idea that a wave is something "oscillating".
In the present context it can as well be a bump that is propagating forward or backward. Or it can even be a standing wave,
as a result of the superposition of both.
Assuming that $a \gt 0$ the issues as described in the body of the question boils down to the folowing. Are the solutions as obtained
by separation of variables compatible with the general solutions as presented above: $\,u(x,t) = F(x-ct) + G(x+ct)\,$? Let $\,a=c^2\,$
in what follows.
First case for $\,\lambda \gt 0\,$:
$$
u(t,x)=\left(c_1e^{-\sqrt{\lambda}.t}+c_2e^{\sqrt{\lambda}.t}\right)\left(c_3e^{-\sqrt{\lambda/a}.x}+c_4e^{\sqrt{\lambda/a}.x}\right) \\
= c_1c_3e^{-\sqrt{\lambda}.t-\sqrt{\lambda/a}.x} + c_1c_4e^{-\sqrt{\lambda}.t+\sqrt{\lambda/a}.x}
+ c_2c_3e^{+\sqrt{\lambda}.t-\sqrt{\lambda/a}.x} + c_2c_4e^{+\sqrt{\lambda}.t+\sqrt{\lambda/a}.x} \\
= \left[c_2c_3e^{-\sqrt{\lambda/a}(x-ct)} + c_1c_4e^{+\sqrt{\lambda/a}(x-ct)}\right] \\
+ \left[c_1c_3e^{-\sqrt{\lambda/a}(x+ct)} + c_2c_4e^{+\sqrt{\lambda/a}(x+ct)}\right] \\ = F(x-ct) + G(x+ct)
$$
Second case for $\,\lambda \lt 0\,$, in very much the same way:
$$
u(t,x)= \left[c_2c_3e^{-i\sqrt{-\lambda/a}(x-ct)} + c_1c_4e^{+i\sqrt{-\lambda/a}(x-ct)}\right] \\
+ \left[c_1c_3e^{-i\sqrt{-\lambda/a}(x+ct)} + c_2c_4e^{+i\sqrt{-\lambda/a}(x+ct)}\right] \\ = F(x-ct) + G(x+ct)
$$
These solutions are oscillating and real-valued, for suitable choices of the constants $c_k$.
It should be noticed that the above comes even closer to the general solution by considering linear combinations,
resulting in expressions of the form
$$
u(x,t) = \sum_m C_me^{\pm\sqrt{\lambda_m/a}(x\pm ct)}
$$
where $\,\sqrt{\lambda_m/a}\,$ can be real or imaginary.
At last we have, for $\,\lambda = 0\,$, a solution that doesn't fit the bill:
$$
u(t,x)=(d_1+d_2t)(d_3+d_4x)\ne F(x-ct) + G(x+ct)
$$
No surprise, because the accompanying degenerate partial differential equations system can hardly be called a "wave equation":
$$
\frac{\partial^2 u}{\partial t^2}=0 \quad ; \quad \frac{\partial^2 u}{\partial x^2}=0
$$
In an answer belonging to Wave Equation by separation of variables it is said that $\lambda = 0$ is not a valid eigenvalue.