We learned in class that a wave will be a function of position $x$ and time $t$. The wave function that describes this wave tells you the displacement of the medium at any position and time, and is given by: $$y(x,t) = A\sin(kx-\omega t)\label{wf}$$ with the definitions $$k\equiv \frac{2\pi}\lambda\nonumber$$ $$\omega\equiv \frac{2\pi}{T}\nonumber$$ where $\lambda$ is the wavelength, defined as the distance in space for 1 cycle, and $T$ is the period, defined as the time for 1 cycle to repeat. We also define the frequency $f$ as the number of cycles per second, and since 1 cycle repeats in $T$ seconds, we have $$f = \frac{1}{T}\nonumber$$ The wave also has a velocity of propagation, $v$, which measures the rate that the disturbance travels along the direction of propagation, here labeled by the $x$ coordinate. Since the wave travels a distance $\lambda$ in 1 cycle and it takes $T$ seconds to do so, then the velocity is given by $$v = \frac{\lambda}{T} = \lambda f\label{ev}$$ This wave function is the solution of the wave equation $$\frac{\partial^2y}{\partial x^2} - \frac{1}{v^2}\frac{\partial^2y}{\partial y^2}\nonumber$$ The wave function from equation $\ref{wf}$ is telling you how the position of the medium (not the wave) at any point $x$ and any time $t$.
In the simulation below, a blue wave pulse travels horizontally, left to right. The velocity of the pulse is
the same velocity as in equation $\ref{ev}$ above. Every so often, one of the points is marked by a yellow dot.
As the pulse propagates, you can see the direction of each yellow dot going vertical. This shows the difference
between the pulse velocity $v=\lambda f$ and the upward velocity of a point in the medium, given by
$$v_y = \frac{\partial y}{\partial t} = -A\omega\cos(kx-\omega t)\nonumber$$
Waves can be more than just pulses. Any periodic disturbance of a medium will generate periodic waves in the
medium. For instance, you can take a string, attach one end to a wall, and grab the other end and shake it up
and down sinusoidally. This will generate a sinusoidal wave on the string.
In the simulation below, we start with 2 traveling waves: a red and a blue. They start out moving in the
same direction with the same wavelength and frequency, and thus the same velocity. In the controls below,
you can modify the velocity and wavelength of either wave, and you can set the relative phase of the two
waves in the last column in units of $\pi$ (so if you type a 1.0 in that box, it will add a phase
of $1.0*\pi$ to the red wave). The black wave is the sum of the red and blue, using the principle of
superposition. To start the simulation, press "Start". Note that the frequencies of the two waves are the
same, and do not change when you change the wavelength or velocity because frequency is determined by the
source, not the medium.
Each wave travels along according to the wave function $\ref{wf}$ above. Before you press "Start"
(equivalent to setting $t=0$),
the simulation draws the 3 waves at $t=0$, with the same wavelength (and therefore velocity).
Note that after you start the simulation, if the waves are
traveling in the same direction, then the sum will also travel in the same direction, and at the same
speed since both waves are also traveling at the same speed. So if you have 2 waves with the same
frequency and wavelength, then when you add them together the resulting sum will be the same at all
time, moving along with the other two. So we can play with the relative phase at $t=0$, before the
simulation starts, and you can see constructive ($\phi = 0, 2\pi, 4\pi,$ etc) and destructive
($\phi = \pi, 3\pi, 5\pi,$ etc) or any value in between.
Next, change the velocity or wavelength of one of the waves and hit the "Reset" and then "Start" the
simulation. You should see the two waves moving with different velocities. and
you can see what happens to the resulting sum from the principle of superposition: the sum is no longer
constant in it's own reference frame, but has a complicated pattern.
What's also interesting is to see what happens when you have the two waves moving in opposite
directions. Start with the same wavelength and velocity, only opposite. For instance, you can
set the velocity of the red wave to $-1$ and the blue wave to $+1$ and see what is known as a "standing wave".
All of this from the principle of superposition!
Wave
Velocity ($v$)
Wavelength ($\lambda$)
Frequency ($f=1/T$)
Amplitude(%)
Relative Phase ($\phi/\pi$)
Red
1
200
.005
100
Blue
1
200
.005
100
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