Will a standing wave form when two waves travel at different directions in the same medium?

Question

Is the standing wave just the superposition of two waves travelling in opposite direction in same medium or does it need specific conditions to be formed like frequency and amplitude ?

Answer 1

Well, let's start more general, and then become more specific if needed. Let's just consider two waves traveling in opposite directions (I will use complex exponential functions so as to not have to deal with trig identities, but we can take just the real or imaginary part of any of these expressions to move back to the "real world"): $$y_1=A_1e^{i(k_1x-\omega_1t)}$$ $$y_2=A_2e^{i(k_2x+\omega_2t)}$$ Note that I am assuming the amplitudes, frequencies, and wave numbers are constant in space and time.

Adding these together we have: $$y_1+y_2=A_1e^{i(k_1x-\omega_1t)}+A_2e^{i(k_2x+\omega_2t)}$$

There is not much we can do here now, so let's assume that $A_1=A_2=A$, then we have $$y_1+y_2=Ae^{i(k_1x-\omega_1t)}+Ae^{i(k_2x+\omega_2t)}=A[e^{i(k_1x-\omega_1t)}+e^{i(k_2x+\omega_2t)}]$$

Ok, it is getting a bit better. Let's assume $k_1=k_2=k$. $$y_1+y_2=A[e^{i(kx-\omega_1t)}+e^{i(kx+\omega_2t)}]=Ae^{ikx}[e^{-i\omega_1x}+e^{i\omega_2t}]$$

Almost there. Let's finally assume $\omega_1=\omega_2=\omega$ $$y_1+y_2=Ae^{ikx}[e^{-i\omega t}+e^{i\omega t}]=2Ae^{ikx}\cos(\omega t)$$ Or, taking the imaginary part of our expression: $$Im[y_1+y_2]=2A\cos(\omega t)\sin(kx)$$

This is the form of a standing wave (If you want, you can think of it as a sine wave in space whose amplitude varies cyclically as $2A\cos(\omega t)$, i.e. a standing wave).

Of course if we are looking at something like waves on a string, we would need to confine the region of the x-axis we look at to be such that there are nodes at the ends of each interval. But it looks like you need to have two waves with the same amplitude, frequency, and wave number to have a standing wave (of course I have just shown these conditions are sufficient, not necessary, but I think it should still hold based on my own tests).

Answer 2

All waves in below example have the same wavelength.

Interference is just the effect of the resultant force applied by two different waves(string particles). Nodes are created when the force on a particle is constantly zero, i.e. the two waves create equal and opposite displacement on the same point.

For two waves with equal frequency and amplitude travelling in the opposite direction, the nodes will be formed as on some points where the displacement produced by each wave is the equal and opposite.

But, in the case of two waves with different amplitude, there cannot be a point with constant zero displacement as the change is curvature (amplitude) for the two waves with time will be different.

For waves having different frequencies and same amplitude $A$, suppose initially the displacement is zero at a point $P$(produced by $\frac{A}{2}$ of one wave and $-\frac{A}{2}$ for other) then the displacement for $P$ will change after some time as the shape of one wave changes faster than the other.

Answer 3

If you want a standing wave and nothing else then you need them to be traveling in different directions with the same wavelength, speed, and amplitude.

If they are traveling in opposite directions in a closed tube (or electrons in a wire etc) then maybe they can maintain about the same amplitude for a reasonably long distance. In 2D or 3D it will be hard to arrange that they have the same amplitude across much of the surface or volume, so it kind of makes sense to say they should travel in opposite directions.

If you don't mind having some nonstationary waves happening along with your stationary waves, then you don't need them to have the same wavelength. All you need is that after fourier analysis they each have a component that has the same frequency. That will give you a standing wave, and everything else will get superimposed on top of it.