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Traveling Waves

INTRODUCTION TO TRAVELING WAVES
Traveling waves are waves that move through a medium as opposed to standing waves which exhibit a constant wave pattern in a domain. For example, open ocean waves are traveling waves, as their peaks move across the ocean’s surface.
Both traveling waves and standing waves are solutions to the wave equation, and therefore may show up in many structural dynamics problems such as the vibrating string problem.
MATHEMATICS
Consider the wave equation in one dimension.
(1) (∂2u(x, t))/(∂t2) = C2(∂2u(x, t))/(∂x2)
This equation can be factored into two parts: a left traveling wave and a right traveling wave.
(2) ((∂)/(∂t) − C(∂)/(∂x))((∂)/(∂t) + C(∂)/(∂x))u = 0
Solutions to the wave equation that can be written in the following form are examples of traveling wave solutions:
(3) u(x, t) = u0(x + Ct)   u(x, t) = u0(x − Ct)
Where u0 = u(0, t).