Can the energy in two waves cancel out? If so, why?

[asada]

TL;DR Summary

2 waves that are same amplitude, same frequency and 180 degree out of phase. Can they cancel each other out?

I know the answer would be yes, but why?

In class, I learned that energy is scalar and cannot be negative (at least in undergraduate class). Thus adding two sources of energy should result in a higher level of energy in general. But here for wave, if we have 2 waves that do destructive interference with each other, we won't observe any movement of the medium, thus no energy?

Take the standing wave for example. At the node, since it's stationary, it doesn't have any potential or kinetic energy, but the waves that combine and construct that node obviously have energy in it. We know that using vectors we can explain the "lost" of movement, but how about using the energy theory?

Thank you in advance. I haven't taken any higher Physics class so this is my understanding so far.

 

[Dale]

Destructive interference doesn’t destroy energy, it just moves it. If you have destructive interference in one place then you have constructive energy in another place. Energy moves away from the region of destructive interference and towards the region of constructive interference. The total energy is conserved.

 

[vanhees71]

Finally one should always be aware that plane waves are an idealization, not describing accurately real-world phenomena, because their total energy is infinity. One should rather take them as "modes", i.e., a complete set of orthonormal systems of function defined in an appropriate Hilbert space you can use to "Fourier decompose" any real-world wave field.

 

[kuruman]

In a traveling wave, energy flows in the direction of travel. When two harmonic waves travel in opposite directions and form a standing wave, say in a plucked string, energy does not flow and does not "cancel out" either but stays confined between the fixed ends of the string.

The points on the string execute harmonic motion, as if they were masses at the ends of spring. Some have the largest possible amplitude (at the antinodes) and some have zero amplitude (at the nodes). The sum total of the energies of all the oscillators is the energy contained in the standing wave.

 

[Mister T]

Summary:: 2 waves that are same amplitude, same frequency and 180 degree out of phase. Can they cancel each other out?

What do you mean by "cancel each other out"? What it means in this context is that if there was only one of the two waves passing through a location, the displacement would be some positive number. Likewise, if we had only the other wave the displacement would be the opposite, it would be the negative of the other wave's displacement. No one is saying those displacements are real. They are just the displacements we'd have if there was just one wave. To get the actual displacement we add the two displacements we would have if there was only one wave.

In class, I learned that energy is scalar and cannot be negative (at least in undergraduate class). Thus adding two sources of energy should result in a higher level of energy in general. But here for wave, if we have 2 waves that do destructive interference with each other, we won't observe any movement of the medium, thus no energy?

Same argument applies here. The energies are the energies that each wave would have if it existed separately. In this case, though, adding the two energies doesn't give us the energy of the wave at that location.

 

[sophiecentaur]

Summary:: 2 waves that are same amplitude, same frequency and 180 degree out of phase. Can they cancel each other out?

We know that using vectors we can explain the "lost" of movement, but how about using the energy theory?

The best answer to that is that there isn't an "energy theory" that's relevant. Imagine you had a string with a vibrator at one end. If the vibrator is 'powerful enough' to ensure that the amplitude was constant and if the fixing at the other end were ideal then (if the string was exactly the correct length (say one wavelength) then there would be a node at the middle because the phase of the returning wave would be exactly opposite to the phase of the incident wave (at that point). The two wave displacements would be equal and opposite at all times. Easy so far.
But what about the Energy flow? It would go to the right and then be reflected towards the left and then, when it reached the driving oscillator?? The wave would be the same amplitude and phase as the oscillator so the oscillator could supply no energy to maintain the up down motion of the string. The left moving wave would effectively be reflected and go to the right and so on and so on. The energy 'in' the string would be unchanging - just distributed along the string as sinusoidal vibrations with amplitudes depending on position (i.e. a stationary wave) with no energy in the middle.
In a real system, the string will be 'lossy' and the end termination may not reflect perfectly so the neat standing wave pattern will change. The peaks will be less and the node will be somewhat filled in and there will be some energy at the node.