Overview
The one dimensional wave equation is given here
{% \displaystyle \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} %}
Trigonometric Solution
As a guess, one could try a solution of the form
{% y(x, t) = sin(x+ct) %}
Taking derivatives, we obtain
{% \displaystyle \frac{\partial ^2 y}{\partial x^2} = -sin(x+ct) %}
and
{% \displaystyle \frac{\partial ^2 y}{\partial t^2} = -c^2sin(x+ct) %}
This can be plugged into the wave equation and verified as a solution.
General Solution
In general, any equation of the form
{% y = f(x + ct) %}
is also a solution to the wave equation.
{% \displaystyle \frac{\partial ^2 y}{\partial x^2} = f''(x+ct) %}
and
{% \displaystyle \frac{\partial ^2 y}{\partial t^2} = c^2f''(x+ct) %}
where
{% f'(z) = \frac{df}{dz} %}