# What happens to the velocity distribution during constructive interference?

Two pulses(one inverted & having velocity in the opposite direction) moving towards each other with same wavelength & amplitude after undergoing destructive interference do re-appear. Why? Because though the displacements became zero, velocity got added as its distribution for both pulses were same.

But during constructive interference, velocity distribution of one pulse gets cancelled by other. Then how can after the interference, the pulses reappear as there was no velocity to preserve the motion?

Suppose one pulse is moving to the right and another same but inverted pulse is moving towards left. The left part of the first pulse has downward velocities while the right part has upward velocities. The other pulse has also downward velocities & the other part upward velocities. When both the pulses meet, the displacements get cancelled but the velocity distribution didn't as both pulses have same velocities. They preserve the memory of them at zero displacement and for that the two pulses get re-coverd. This is implied by stating that reversing the signs of both $\dfrac{\partial y}{\partial x}$ and $\dfrac{\partial x}{\partial t}$ in $$v_y = - \dfrac{\partial y}{\partial x} \cdot \dfrac{\partial x}{\partial t}$$ leaves $v_y$ unchanged. Thus the transverse displacements cancel, but transverse velocities add.

This is not the case in constructive interference:

Suppose now two same pulses are coming towards each other but no one is inverted. Then the left part of the first pulse has velocities downward while the left part of the second pulse has velocities upward; the velocities of the elements of the right part of the first pulse is upward while that of the second pulse is downward. So, when they interfere constructively, though the displacements get doubled, the velocities get cancelled due to their opposite distributions for either pulses. If the velocities get cancelled, then how can the original pulses again recover from the interference?

• If I am not 'earnest' enough, should I still answer your question ? Feb 11, 2015 at 13:16
• But during constructive interference, velocity distribution of one pulse gets cancelled by other?
– Paul
Feb 27, 2015 at 2:20
• @Paul: . . . During constructive interference of two pulses coming from opposite direction, though amplitude gets add up, but the velocities get cancelled(why? , I've precisely written it in the quo). If so, then how can the waves get revived? Or am I mistaking?? Thanks for the comment; no one but you atleast noticed it:-|
– user36790
Feb 27, 2015 at 2:34
• Hmmm..I am travelling on bus somewhere.so I cant do the calculation.But my intuition said velocities should get cancelled. Imagine two water waves,one wave tries to move the level of water up and one wave tries to move the level of water down from normal position.but as a combined effect water level remains same.thus water molecules did have zero velocity???
– Paul
Feb 27, 2015 at 4:08
• @Paul: Velocities do add up during destructive interference. This is exactly written in A.P.French's Vibrations & Waves.
– user36790
Feb 27, 2015 at 5:45

As you mentioned, and as was discussed in the answers to the question "Why do traveling waves continue after amplitude sum=0", when two waves interfere destructively, there is an instant where the amplitude is zero, but at that moment points on the wave have significant velocity: all the energy in the wave is kinetic energy at that moment.

The converse is true when two waves travel in opposite directions and interfere constructively. In that case, there is a moment when their velocities exactly cancel, but the amplitudes add up. The result is a wave with twice the amplitude, and zero velocity: the kinetic energy will be zero, and all the energy is elastic.

A moment later the waves will "reappear" from that stationary state. The energy for this is the stored (elastic, potential) energy in the stationary wave (with twice the amplitude, you have four times the elastic energy: that is the elastic energy for each of the two waves, plus their kinetic energy).

The situation is not unlike that for a string that is plucked: if you pull on it, and let go, waves will travel outwards to the support points, where they reflect and return. If this is an infinite string, and you remove the support points at the moment you release the string, the wave will just continue traveling outwards. Whatever the initial shape is, a wave with half that amplitude will travel to the left, while another wave will travel to the right.

A wave "wants to" move (there is no stationary solution to the wave equation). If it starts stationary it wants to move left and right at the same time so you end up with two pulses... And from the uniqueness theorem - if that is "a" solution it is "the" solution.

• +1; thanks for answering. While there was resultant transverse velocity in the destructive interference, there is resultant displacement in the constructive interference. But what makes it to resurrect the pulses? In destructive interference, it was done by the transverse velocity distribution as they made it possible to displace each element in the pulse transversely further. But what happens in constructive interference? The superposed pulse is stationary. How do the pulses get resurrected? There is no velocity now that would displace each element further, isn't it?
– user36790
Apr 5, 2016 at 1:24
• A wave "wants to" move (there is no stationary solution to the wave equation). If it starts stationary it wants to move left and right at the same time so you end up with two pulses... And from the uniqueness theorem - if that is "a" solution it is "the" solution. Apr 5, 2016 at 1:27