Longitudinal Wave meets Transverse

Question

Okay, so the title may seem like a YouTube viral, but yes, my friend asked me this question today:

What if a longitudinal wave superposes a transverse wave? Is it possible? What would happen then?

I felt it was an absurd thing to ask, but the thought of it drew a considerable bit of my attention. I couldn't explain why something like that couldn't happen, or even if it did, what would be the result. To demonstrate what I mean -

Imagine you have a slingy. You clip its two ends horizontally (its length parallel to the floor) and you pull a part of the slingy horizontally (parallel to its springy length).

It is now showing longitudinal compression and rarefaction. There, you already have longitudinal waves.

Now you pull it downwards and release (as if a bowstring! Imagine launching an arrow from your slingy upwards...).

What happens now? The transverse wave occurs in your slingy, making it move up and down. But at the same time it is continuously moving back and forth within itself. So -

1. Will both kinds of waves show up?
2. Will they look just as usual, or affect each other, "distorting" their individual appearances?
3. Will this be a chaotic system (I honestly do not know this term well enough to use it, but I'm assuming it means that the system will be predictable but just too hard to predict - "sensitive dependence on initial conditions"...) ?

Just curious.

Answer 1

First, let's mention that purely traverse/longitudinal modes only exist for plane waves, and these only exist in infinite media. In any finite media you will have some combination of these two.

Now, let's consider a medium where we have both transverse and longitudinal waves, for example, an elastic solid. In such a medium you could have 2 waves of the form:

$$\mathbf{u}_\text{long} = \begin{pmatrix}A e^{i\omega(t - \alpha x)}\\0\\0\end{pmatrix}\, ,$$

for longitudinal waves, the motion of the particles looks like the following animation.

For the transverse wave we have

$$\mathbf{u}_\text{trans} = \begin{pmatrix}0\\ B e^{i\omega(t - \beta x)}\\0\end{pmatrix}\, ,$$

In this case, we chose a wave propagation in the $x$-direction and the transverse polarization in the $y$-direction. $\alpha$ and $\beta$ are the phase speeds for P- and S-waves.

So, since we are talking about a linear medium, we can superimpose those two displacements to obtain

$$\mathbf{u}_\text{total} = \begin{pmatrix}A e^{i\omega(t - \alpha x)}\\B e^{i\omega(t - \beta x)}\\0\end{pmatrix}\, ,$$

where each mode makes material points ("particles") oscillate in a different direction. But, since $\alpha > \beta$ the wavelengths won't be the same.

Answer 2

Here is an example. You pluck a harp string. You can see the string vibrating sideways. That's a transverse wave.

But the string is connected to the sounding board, and what the string does to the sounding board is pull in it. The string pulls more when its center is moving sideways than when it is shortest, so for the sounding board this is strictly a compression wave. Or I guess a stretching wave.

But then what the sounding board does in response is to vibrate back and forth, though unlike the string it doesn't do it enough to see. It has to be an invisible transverse wave.

And it makes sound in the air -- another compressive wave.

They get all mixed together. Don't worry too much about it.

But when it comes to light, the big deal is that the way you make a longitudinal electromagnetic wave, is to take an electric charge and move it alternately toward and away from the target you're aiming the wave at. And then the force goes down by the square of distance. But if you move it back and forth perpendicular to the target, then the force is only reduced linearly with distance. So after a little distance, nothing matters but the transverse waves. Everything else dissipates fast.

Answer 3

Suspend a small weight from a spring. If you stretch the spring by pulling the weight down a little, you can get a longitudinal wave.

If you move the weight sideways without stretching the spring, you can get a transverse wave when it moves like a pendulum.

Make the weight just heavy enough that both motions have the same period.

Then the transverse wave will inevitably have a longitudinal component. There is more stretching force when the weight is at the bottom. (Just like when you swing in a playground swing.)

Now adjust the weight differently, so they have different periods. Start it swinging. Each swing you start a new longitudinal wave at a different time. The longitudinal wave affects the transverse wave -- it's like a pendulum where the length of the string keeps changing.

I expect you could get some weird effects. Maybe you could develop the math in closed form, but I bet it would be easier to approximate the result with simulation.