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Hi All,

First off, thanks to all the old hands at physicsforums, you guys are truly an amazing resource.

I was thinking about a system today that at first glance, appears to violate local conservation of energy for two mechanical wave pulses interfering with each other.

Consider a ripple tank, in which we have two identical plane wave pulses travelling perpendicularly toward each other. The energy of each wave is equal to the kinetic energy of the wave motion, and the potential energy of the wave height due to gravity.

At some point, they overlap, and interfere. The amplitude of the wave at this point is doubled. The potential energy of this region of constructive interference, however, is four times that of each individual wave. How is this possible?

I did a little research, and found these interesting papers relating to the subject:

What happens to energy and momentum when two oppositely-moving wave pulses overlap? - N. Gauthier, 2003

Superposition and energy conservation for small amplitude mechanical waves. W. N. Matthews Jr., 1985

The first paper derives how two 1D wave pulses propagating toward each other and interfering will conserve energy, due to kinetic energy being transformed into potential energy. The second paper makes the argument that any two waves must produce an equivalent amount of destructive interference for any constructive interference produced, therefore conserving energy.

I tried to consider if the kinetic energy of the waves had been transformed into potential energy, however I believe the kinetic energy of the two waves do not cancel as they do in the first paper, since the x and y velocities are orthogonal, and the z component of the water waves will interfere constructively in half of the interference region, and destructively in the other half. I also don't believe there can be any destructive interference, since this is a wave pulse.

First off, thanks to all the old hands at physicsforums, you guys are truly an amazing resource.

I was thinking about a system today that at first glance, appears to violate local conservation of energy for two mechanical wave pulses interfering with each other.

Consider a ripple tank, in which we have two identical plane wave pulses travelling perpendicularly toward each other. The energy of each wave is equal to the kinetic energy of the wave motion, and the potential energy of the wave height due to gravity.

At some point, they overlap, and interfere. The amplitude of the wave at this point is doubled. The potential energy of this region of constructive interference, however, is four times that of each individual wave. How is this possible?

I did a little research, and found these interesting papers relating to the subject:

What happens to energy and momentum when two oppositely-moving wave pulses overlap? - N. Gauthier, 2003

Superposition and energy conservation for small amplitude mechanical waves. W. N. Matthews Jr., 1985

The first paper derives how two 1D wave pulses propagating toward each other and interfering will conserve energy, due to kinetic energy being transformed into potential energy. The second paper makes the argument that any two waves must produce an equivalent amount of destructive interference for any constructive interference produced, therefore conserving energy.

I tried to consider if the kinetic energy of the waves had been transformed into potential energy, however I believe the kinetic energy of the two waves do not cancel as they do in the first paper, since the x and y velocities are orthogonal, and the z component of the water waves will interfere constructively in half of the interference region, and destructively in the other half. I also don't believe there can be any destructive interference, since this is a wave pulse.

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