Physics, asked by tariqwaheedtari2003, 5 months ago

when does the principle superposition not hold for wave equation?​

Answers

Answered by Aaravnasascientist
1

Answer:

light waves

The superposition principle states that when two or more waves overlap in space, the resultant disturbance is equal to the algebraic sum of the individual disturbances.

Answered by Anonymous
0

Answer:

If a wave f(x,t) is something that satisfies the wave equation Lf=0 where L is the differential operator ∂2t−c2∇2 then, because L is linear, any linear combination λf+μg of solutions f and g is again a solution: L(λf+μg)=λLf+μLg=0.

In general, there might be things that propagate (not exactly waves, but since the question is for waves of any kind) determined by other differential equations. If the equation is of the form Lf=0 with L a linear operator, the same argument applies and the superposition principle holds.As coconut wrote, the superposition principle comes from the linearity of the operator involved. This is the case for electromagnetic radiation in vacuum. Approximations to water waves are also linear (since it is an approximation) but probably will have small non-linear parts. Free quantum field theory is also linear, therefore you have a superposition principle there. With interactions and renormalization, I think it is not linear any more.

As coconut wrote, the superposition principle comes from the linearity of the operator involved. This is the case for electromagnetic radiation in vacuum. Approximations to water waves are also linear (since it is an approximation) but probably will have small non-linear parts. Free quantum field theory is also linear, therefore you have a superposition principle there. With interactions and renormalization, I think it is not linear any more.Gravity as described by general relativity is highly non-linear. Therefore it does not have any superposition principle. Gravitational waves do not have a superposition principle. However, at very large distances these waves can be approximated. And then this operator might be linear and you can reasonable speak of superpositions again.

As coconut wrote, the superposition principle comes from the linearity of the operator involved. This is the case for electromagnetic radiation in vacuum. Approximations to water waves are also linear (since it is an approximation) but probably will have small non-linear parts. Free quantum field theory is also linear, therefore you have a superposition principle there. With interactions and renormalization, I think it is not linear any more.Gravity as described by general relativity is highly non-linear. Therefore it does not have any superposition principle. Gravitational waves do not have a superposition principle. However, at very large distances these waves can be approximated. And then this operator might be linear and you can reasonable speak of superpositions again.The usual approximation to a wave,

As coconut wrote, the superposition principle comes from the linearity of the operator involved. This is the case for electromagnetic radiation in vacuum. Approximations to water waves are also linear (since it is an approximation) but probably will have small non-linear parts. Free quantum field theory is also linear, therefore you have a superposition principle there. With interactions and renormalization, I think it is not linear any more.Gravity as described by general relativity is highly non-linear. Therefore it does not have any superposition principle. Gravitational waves do not have a superposition principle. However, at very large distances these waves can be approximated. And then this operator might be linear and you can reasonable speak of superpositions again.The usual approximation to a wave,(1c2d2dt2−∇2)ϕ(x,t)=0

As coconut wrote, the superposition principle comes from the linearity of the operator involved. This is the case for electromagnetic radiation in vacuum. Approximations to water waves are also linear (since it is an approximation) but probably will have small non-linear parts. Free quantum field theory is also linear, therefore you have a superposition principle there. With interactions and renormalization, I think it is not linear any more.Gravity as described by general relativity is highly non-linear. Therefore it does not have any superposition principle. Gravitational waves do not have a superposition principle. However, at very large distances these waves can be approximated. And then this operator might be linear and you can reasonable speak of superpositions again.The usual approximation to a wave,(1c2d2dt2−∇2)ϕ(x,t)=0is linear by definition. A lot of waves can be described well as linear waves with non-linear perturbations (water waves, EM waves in medium). Strictly speaking, they are non-linear from the start once there is the smallest non-linear perturbation to them.

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