A traveling wave, like a water wave or sound wave, is a collection of oscillators. Each part that makes up the wave feature simply oscillates back and forth in some manner, but the wave feature itself, propagates. This feature transfers energy and allows objects to interact at a distance, e.g. talking or seeing.
DoodleScience briefly explains the two types of traveling waves.ult body text.
Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.
A traveling wave is comprised of a bunch of individual oscillators moving in a seamingly cooperative way such that a new feature arises from the their collective motion. Below is an example of a logintudal wave, one where the motion of the particles that the wave is comprised of is parallel to the axis of the wave's direction of travel - sound travels much this way.
Notice the particles themselves (red dot) do not travel over appriciable distances but rather wiggle back and forth about a fixed point. Even though the particles don't themselves travel, the wave feature does, carrying with it energy. This means there are two velocities of interest, that of the individual particles, which follows a sinosdial change as a function of time, and the wave feature, which is a constant with respect to time.
Below is an example of a transverse wave, where the motion of the particles is perpendicular to velocity of the wave. This is how the particles comprising a rope move when a wave is generated in the rope.
A wave is not only an oscillation of as a function of time but also a function of space. To model them we now need a sinodial function that depends on both $x$ and $t$. You can see this traveling sinosodial wave in the below animation.
The function can be studied by taking snapshots of the displacement as a function of time. The wavelength is the distance between points in space where the wave is equivalently displaced.
The displacement as a function of time and space can be mathematically modeled.
In the above equation $D(x,t)$ represents the displacement from equilibrium as a function of position and time. The wave has some max amplitude, which for a wave on a rope, would be how far a piece of the the rope travels from equilibrium, measured in meters. Whether you choose a postive or negative, and whether the motion is modeled as a sine or cosine function, depends on the initial conditions (I.C.) of the motion. The quantity $\omega$, as we learned before studying single oscillations, is equal to $\frac{2 \pi}{T}$, where T is the period. It comes from the rate of the oscillation through time. Simliarly, a wave oscillates through space. The wavenumber of the wave ($k$) is what holds the information about the repetition through space.
$k=\frac{2 \pi}{\lambda}$, where $\lambda$ is the wavelength of the wave
The frequency, which is related to the period of the source $f=\frac{1}{T}$, depends entirely on the source generating the wave. When you talk at different pitches you are determing the rate at which you're vocal chords vibrate back and forth. The speed of the wave through a medium is dependent on the properties of the medium and not the source. There is a fundamental relationship between the speed ($v$) of a wave and its frequency and wavelength.
$v=f \lambda$
With the fact that the frequency is determined by the source and the speed by the medium, it follows that the wavelength simply must be what satisfies this relationship.
To see the connection between Uniform Circular Motion, Simple Harmonic Motion, and Traveling waves, check out the animation below.
UCM-SHM-waves.mp4
Now, take a look at the pre-lecture reading and videos below.
