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 

$$ \omega = 2 \pi f $$

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

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

• Mechanical traveling waves require a medium, but the particles of that medium do not travel with the wave: they propogate the wave by oscillating about some position.
• Electromagnetic waves are self-propogating; they do not require a medium and can travel through vacuum.
• The wave speed $v = \lambda \, f$ is not the speed of the oscillators, it is the speed of the feature of the wave itself.  For example, a traveling wave on a string the oscillators are the tiny pieces of string that move up and down; the wave speed describes the speed of the crest that travels along the string, not the up and down motion.
• Not understanding the SHM nature of each particle in the wave.
• Speed of wave vs. speed of the particles in the wave.
• Use of degrees vs. radians for the function arguments.
• Direction of wave motion not discerned from equation; not connect to sinosoidal graph.
• Water waves are neither transverse or longitudinal waves, they are a combination of the two.
• The linear mass density of a string is the total mass divided by the total length.  Experimentally a commom mistake is to measure the total mass and divide by the length between two fixed ends of the string when the string is clamped and ready to oscillate back and forth to observe traveling waves. 

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