When solving a traveling wave problem you will most likely be using the wave equation $D(x,t) = Asin(kx - \omega t)$. Remember that $k = \frac{2 \pi}{\lambda}$ and that $\omega = 2 \pi f$ where $\lambda$ is the wavelegnth and $f$ is the frequency of the wave.

Graphical problems:

Draw a physical representation of your system and label any quantities given in the problem statement. It is a good idea to either draw a sketch of $x$ versus $t$ or an overhead snapshot of the situation, depending which is appropriate - for example, for the motion of a particle in a vibrating string, $x$ versus $t$ is more appropriate; for analyzing ripples on a pond radiating from a point, the overhead view will be more appropriate. Label wavelengths and/or periods.

Graphical problems will typically ask you about how many cycles will pass a point in a given amount of time, or how many wavelengths will fit in some area. Remember that a wavelength is a distance in meters that tells you how far a wave must travel to be at the same point in its cycle; the period is a time that tells you how long a wave must propogate for before it returns to the same point in their cycle. 

Other problems:

Identify the type of wave - is it mechanical or electromagnetic? If it is an electromagnetic wave then you immediately know that its traveling with speed $ c = 3 * 10 ^ 8 $ m/s, and you can quickly find the wavelength if given the frequency, and vice-versa. Recall that $v_{wave} = f \lambda$. A common question when considering traveling waves is how long it takes a wave to travel from point A to point B. Understanding the wavespeed makes this problem easy - the distance is just $d = v t$.

 All of the mathematical operations inside of the cosine/sine function can make the wave equation look intimidating, but are simple to handle case-by-case. For example, say you have the following equation - $D(x,t) = A sin( 30x - 50t) $ - and you are asked what are the wavelength and period of the wave. A good way to remember which terms are associated with wavelength and frequency is to inspect the equation and notice that there is an $x$, which we typically measure in meters, and $t$, which we measure for time. Since we want the argument inside of a trig function to be dimensionless, we know that $x$ is associated with wavelength and $t$ with the frequency or period.

Since the wave equation is $D(x,t) = Asin(kx - \omega t)$, we know that in the previous example

$$k = \frac{2 \pi}{\lambda}$$

$$⇒ \lambda = \frac{2 \pi}{30} = 0.21 m $$

and 

$$ \omega = 2 \pi f $$

$$⇒ f = \frac{\omega}{2 \pi} = 8 s^{-1} $$

$$ T = \frac{1}{f} = 0.125 s $$

There are two basic traveling waves: Logitudinal, Transverse. Traveling waves may be a combination of both types. Longitudinal waves travel in a line, whereas, transverse waves propogatae up and down.

$\nu = f \lambda$

$\kappa = \frac{2 \pi}{\lambda},\; \omega = \frac{2 \pi}{T}$

$\nu_{string} = \sqrt{\frac{F^{T}}{\mu}}$ 

$\nu_{sound} \propto \sqrt{Temperature}$

$x(t) = \pm x_{max} \frac{\sin \text{or}}{\cos} (\kappa x \pm \omega t)$

Velocity ($\nu$), Frequency ($f$), Wavelength ($\lambda$), Wave number ($\kappa$), Angular frequency ($\omega$),

Linear mass density ($\mu$), Force tenstion ($F^{T}$), Period ($T$)

This is a snapshot graph that depicts a a wave at different points in time.

A history graph describes the history of a specific position with respect to time

Traveling waves are comprised of longitudinal and transverse waves. Longitudinal waves propogate in a line, whereas, transverse waves propogate up and down. Traveling waves may also be comprised of both types of of basic waves. For example, electromagnetic waves are a combination of both transverse and longitudinal traveling waves.

Longitudinal and transverse waves experiments are easily performed using a slinkly for longitudinal waves or a rope for trasnverse waves. Check out the videos below to see the experiments. 

Longitudinal 

Transverse