1. D’Alembert’s Solution
There is an elegant approach to solve the wave equation by introducing new variables:

4. Example

6. It appears to have a wave moving to the right
It appears to have a wave moving to the left

7. Wave Propagation
Initial condition

8. When t>0, the disturbance splits into two parts, one propagating
To the right while the other propagating to the left, as shown below.
Propagate to the right
Propagate to the left

9. Characteristic Lines t x-ct=constant x+ct=constant f(x-ct)=const
Slope -c
x-ct=constant
f(x-ct)=const
Slope c x

10. Physical Interpretation
3Dx=3cODt x
Dx=cODt
2Dx=2cODt

11. Nonhomogeneous Wave Equation