Can two standing waves combine to form a traveling wave?

Two traveling waves with equal and opposite wave-vectors form a standing wave, which is easy to prove.

But what about combining two standing waves?

Can we derive the velocity of the wave/disturbance created by combining two standing waves?

For example take two standing waves:

$$y_1(x,t)=A_1sin(k_1x)cos(\omega_1t)$$ $$y_2(x,t)=A_2sin(k_2x)cos(\omega_2t)$$

And combine them:

$$y(x,t)=y_1+y_2$$

Does this result in a traveling wave? If so, what is its wave-vector and velocity?

• Let $k_1=k_2,\omega_1=\omega_2, A_2=iA_1$ Commented Nov 20, 2021 at 17:40
• How do you “prove” two equal and opposite waves superposed form a standing wave? To me that is a definition
– Rol
Commented Nov 22, 2021 at 19:06

By this trigonometric formula $$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$$ the two standing waves \begin{align} y_1(x,t)&=\sin(kx)\cos(\omega t)\,,\quad y_2(x,t)=\cos(kx)\sin(\omega t) \end{align} form the travelling wave $$y(x,t)=\sin(kx+\omega t)\,.$$

In general adding two standing waves can't create a traveling wave.

To see this we could look at the amplitude of the resultant wave. That is obtained by adding the amplitudes of the standing waves at each point.

A traveling wave has equal amplitude of oscillation for each part of the wave, but at $$A$$ the amplitude is the same as that of the green standing wave. At $$B$$ the amplitude is the same (or lower) than that of the red standing wave.

That would mean that the amplitude at $$A$$ was greater than that at $$B$$, so unless there were some very particular conditions met (perhaps as in RoderickLee's comment), then the traveling wave wouldn't occur.

Let $$k_1 = k_2 = k_3$$ $$\omega_1 = \omega_2 = \omega_3$$ $$A_1 = A_2 = A_3$$ $$\phi_1 = 0, \space \phi_2 = 2\pi/3, \space \phi_3 = 4 \pi/3$$

• Do you mean it can be done by adding three standing waves? Could you explain what you mean by this please? Commented Nov 21, 2021 at 9:12