Waves – How to Add Two Plane Waves Propagating in Different Directions

Question

In the undergraduate course about the wave, there stated for two harmonic waves propagating in opposite direction, then the resulting wave will be a standing wave. In math, it is like

$$y_1 = A\sin(kx + \omega t), y_2 = A\sin(kx – \omega t)$$

so
$$
y = y_1+y_2 = A\sin(kx + \omega t) + A\sin(kx – \omega t) = 2A \sin(kx)\cos(\omega t)
$$

I am thinking what happen if we have the two plane waves propagating along two different direction (says making angle 60 degree, i.e. the two wave are making ). I know that if that's the case, we cannot write $kx$ but we need to consider the $k$ is a vector such that

$$y_1 = A\sin(\vec{k}\cdot\vec{r} + \omega t), y_2 = A\sin(\vec{k}\cdot\vec{r} – \omega t)$$

But if we look at the horizontal direction (i.e. x) and vertical direction (i.e. y), what can we tell about the resulting wave along x and along y? I am thinking from physical point of view, if we look at the horizontal direction, should the waves still added up to a standing wave because the x components of waves are propagating in opposite direction. But along the vertical direction, the y components of waves are propagating in the same direction so there is no standing wave. Is that correct? If so, how to prove that in math? The term $\vec{k}\cdot\vec{r}$ is very confusing!

Answer 1

The other answer are good, but this might help you visualise the result. It is easy to produce visualizations if you have access to a package like Mathematica (you could also do this with python+matplotlib, gnuplot or Matlab or just about anything really). I've generated plots of two waves in 2D, one going in the positive $x$ direction and the other going at an angle $\theta$ relative to the $x$ axis. The amplitudes, wavelengths and frequencies are the same. Here is the code:

wave1[x_, y_, t_] := Sin[x - t];
wave2[x_, y_, t_, \[Theta]_] := Sin[Cos[\[Theta]] x + Sin[\[Theta]] y - t];
frames[\[Theta]_] := frames[\[Theta]] = Table[Plot3D[wave1[x, y, t] + wave2[x, y, t, \[Theta]],
    {x, -10, 10}, {y, -10, 10}, PlotLabel -> "\[Theta] = " <> ToString[\[Theta]]], {t, 0, 10}];
Table[Export["twowaves_\[Theta]_" <> ToString[\[Theta]] <> ".gif", frames[\[Theta]]], {\[Theta], 0, 2 \[Pi], 0.5}]

Selected plots shown below. Note that the sum of waves simplifies to

$$ \sin(x-t)+\sin(\cos(\theta)x+\sin(\theta)y-t)=2 \cos\left(\frac{1}{2} x \left(\cos\theta-1\right)+\frac{1}{2} y \sin\theta\right) \sin\left(\frac{1}{2} x \left(\cos\theta+1\right)+\frac{1}{2} y \sin\theta-t\right). $$

You only get a standing wave if the space and time dependence seperates. So you need the $x$ and $y$ terms in the $\sin$ to vanish. This requires $\cos\theta=-1$ and $\sin\theta=0$, which has the unique (up to $2\pi$) solution $\theta=\pi$. So you only get standing waves if the two waves are counter propagating. Every other case gives you a travelling wave (the $\sin$ term) modulated by a space-dependent amplitude (the $\cos$ term).

Both waves in the positive $x$ direction:

Wave 2 going slightly up and to the right:

Wave 2 going nearly 90 degrees to wave 1:

Wave 2 nearly opposite wave 1: