# Do all waves of any kind satisfy the principle of superposition?

Is it an inherent portion of defining something as a wave?

Say if I had something that was modeled as a wave. When this thing encounters something else, will it obey the principle of superposition. Will they pass through each other?

• Well, I would first ask what you mean by wave. The best answer I can think of is whipping out some "wave equation". "If it satisfying this/one of these equations, it is a wave." A lot (not sure if all) the stuff we call waves are linear. But do realize that our classification of waves is arbitrary. – Ben S Jan 20 '17 at 6:41
• Related question: How can one tell if a PDE describes wave behaviour?. There are non-linear PDEs that have wave solutions (see e.g. solitons) that don't satisfy the superposition principle. – Winther Jan 20 '17 at 10:11
• @Winther Actually some nonlinear equation show a sort of a superposition principle and this is indeed one of the special feature presented by solitons. In these cases, you add two solitons and you end up with a new soliton. – Diracology Jan 20 '17 at 13:12
• My initial thought is that a "handwave" doesn't, but having thought about it some more perhaps it does.. Hands colliding cancel each other out/interfere, and a crowd of waving hands is amplified in the sense that it can be seen from a further distance than a single handwave might... Hmm... – kwah Jan 21 '17 at 11:48
• The answer you accepted is not... that correct; for example according to that answer high intensity laser light is "not exactly waves". This might be a better choice. Sometimes the first answer that pops up and looks correct gets most upvotes. Doesn't mean it's the best answer. – user Jan 29 '17 at 7:54

If a wave $f(x,t)$ is something that satisfies the wave equation $Lf=0$ where $L$ is the differential operator $\partial_t^2-c^2\nabla^2$ then, because $L$ is linear, any linear combination $\lambda f+\mu g$ of solutions $f$ and $g$ is again a solution: $L(\lambda f + \mu g)=\lambda Lf+\mu Lg=0$.

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.

• In other words: "Yes" (given a reasonable assumption of what a "wave" is). – Todd Wilcox Jan 21 '17 at 7:17
• (The point being: plenty of waves are nonlinear, and do not obey the principle of superposition. Linearizing may or may not make sense depending on the situation, and there are plenty of cases where it doesn't. Solitons and breaking waves are easy examples of things you do want to include under the term "wave", but which don't follow the principle of superposition.) – Emilio Pisanty Jan 21 '17 at 16:44
• This answer is mathematically correct but ignores physics. For example two waves in a solid may both be linear, but the combination of them may exceed the elastic limit of the material and hence be nonlinear. But as Feynman said, "Physicists always have a habit of taking the simplest example of any phenomenon and calling it 'physics,' leaving the more complicated examples to become the concern of other fields." – alephzero Jan 21 '17 at 17:13
• @coconut The point is that the question literally asks "do all waves of any kind satisfy the superposition principle", and there are tons of examples that don't; see my answer for more. Your answer just hides its head in the sand and arbitrarily decrees that a bunch of wave phenomena, from optical solitons to sonic booms to waves at the beach, are not "waves" because your answer doesn't like the mathematics they follow. That's an untenable position, I should say. – Emilio Pisanty Jan 21 '17 at 18:50
• I am surprised by the number of up votes and the green tick for this answer which does not really answer the question. One of the interesting features of this site is that the first (few) answer(s) are often not the best but often get the most credit. I side very much with @EmilioPisanty in terms with his comments and with all the other informative answers. – Farcher Jan 23 '17 at 7:08

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, $$\left(\frac{1}{c^2} \frac{\mathrm d^2}{\mathrm dt^2} - \nabla^2 \right) \phi(x, t) = 0 \,$$ is 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.

• Surprising and counter-intuitive. Are you sure? Classical static gravity surely follows the superposition (I can simply add the gravitation of two masses. Anything else would be impossible.) Is it really so that gravitational waves do not superimpose? What else do they do? – Peter - Reinstate Monica Jan 20 '17 at 11:50
• The Einstein field equations are non-linear second order differential equations. If you have two black holes, one cannot just take the Schwarzschild solution twice, one needs to find a new solution. – Martin Ueding Jan 20 '17 at 12:09
• @PeterA.Schneider Classical static gravity doesn't have waves, and yes, it is linear, though it also only works with point objects, not fields. It also doesn't agree with GR - it just approximates well enough for external observers of relatively low-energy fields. I'm sure you'll find lots of examples of "the old theory is simpler but more or less wrong" in science :) After all, SR is another nice example - the old kinetic theory assumed velocities added linearly, Lorentz and co. showed that isn't really true. – Luaan Jan 20 '17 at 12:45
• @Luaan Valid points ;-). – Peter - Reinstate Monica Jan 20 '17 at 12:50
• Are water waves and electromagnetic waves (aside from those in vacuo) really linear? What about solitons? – steeldriver Jan 21 '17 at 4:20

# No.

Despite what several answers on this thread will tell you, there are plenty of phenomena which are perfectly deserving of the term "wave" which do not satisfy the superposition principle. In technical language, the superposition principle is obeyed whenever the underlying dynamics are linear. However, there are plenty of situations that do not obey this assumption.

Some examples:

• Breaking waves on a beach: the underlying dynamics of water surface waves is linear when the amplitude is small, but this assumption breaks down when the amplitude is comparable to the depth of the water.

Everyday experience should tell you that a taller wave will break further from the shore, while a wave with a smaller amplitude will break closer to the beach. This is patently incompatible with the superposition principle.

• Solitons, which rely on nonlinear effects to maintain their shape even in the presence of dispersion, and which show up as water surface waves and in fiber optics, as well as more esoteric domains.

• Light propagating in a material at sufficiently high intensities, at which point the Kerr effect (i.e. a nonlinear modulation of the index of refraction $$n=n_0+n_2I$$ depending on the intensity $$I$$) will kick in, resulting in useful effects (like Kerr-lens modelocking) as well as harmful ones (like catastrophic runaway self-focusing).

• More broadly, optics is only linear in vacuum (and even then, at some point you start to run into pair-production and light-light scattering). In the presence of media, there are plenty of useful phenomena that use the nonlinear response of materials, falling into what's known as nonlinear optics.

This goes from perturbative phenomena like Kerr lensing and frequency-mixing processes like second-harmonic generation (such as employed in green laser pointers) all the way up to highly nonperturbative processes like high-order harmonic generation, where doubling the intensity can dramatically change the spectrum of the emitted harmonics (i.e. almost double the cutoff of the harmonic orders that you can produce).

• Sound waves that are strong enough to enter the nonlinear acoustics regime, including sonic booms, acoustic levitation and medical ultrasound imaging.

• Hydraulic jumps, which form everywhere from dams to tidal bores to your kitchen sink.

• The nonlinear wave dynamics of the quantum mechanics of Bose-Einstein condensates which obey the Gross-Pitaevskii and nonlinear Schrödinger equations, and related models.

• ... the latter of which, by the way, is also useful for modelling nonlinear behaviour in fiber optics and in water waves.

• Come to think of it, from a ground-up perspective, all fluid dynamics is inherently nonlinear. The first approximation is indeed nonlinear, but many phenomena are well described by the next step up, i.e. including a weak nonlinearity, giving you something called cnoidal waves.

I could go on, but you get the point. You can, if you want to, restrict the term "wave" to only phenomena that obey linear dynamics. However, if you do so, you are explicitly leaving all the above phenomena out, and I would argue that that's not really what we mean by the term.

Simply calling something a "wave" isn't enough for a superposition of solutions to satisfy the governing wave equation. When deriving wave equations linearity is achieved by requiring "small amplitude" oscillations, so in nature when large amplitudes are involved the superposition principle does not hold true in general.

• I'm using the term "linearity" as described by the answers above. – I.E.P. Jan 20 '17 at 7:40
• Could you explain the distinction between large and small amplitudes? I never understood what distinction there might exist. Can't you make any small amplitude large just by changing your units to something minuscule? How can that affect the physics? – user541686 Jan 20 '17 at 8:16
• As an elementary example consider the classical derivation of the linear wave equation for a 1d string. The only way we actually yield the linear PDE is by approximating that terms of order $(df(x,t)/dx)<< 1$ (where f(x,t) is the displacement of the string at position x, time t). This is what constitutes "small oscillations." Another way of thinking of this requirement is that in this regime we are analyzing the "long wavelength" behavior of our system. – I.E.P. Jan 20 '17 at 9:06
• (cont'd) Say $f(x,t)= A\text{sin}(kx-\omega t)$, thus $df(x,t)/dx= Ak\text{cos}(kx-\omega t)$. The term $Ak$ is what is small and notice that the spatial dimensions cancel--thus rescaling space will have no effect on the governing physics of our system. – I.E.P. Jan 20 '17 at 9:08
• D'oh!! Of course, that makes sense... I totally didn't realize that's what you're talking about. Somehow I completely missed what you meant by your second sentence, even though it's crystal clear now in hindsight. Sorry about that and thanks for the (re-)explanation! – user541686 Jan 20 '17 at 9:17

Actually, none of them satisfies the superposition completely. First, superposition requires linearity, and linearity isn't perfect in most cases. Even in the case of linear theories, the theory is only a model and it has its borders.

For example, the Maxwell-equations are linear, and thus light waves superpose. If you cross two laser beams, they totally pass through each other without any change. But:

• If the beams are enough strong to induce pair production, it is not true any more.
• If the beams have enough significant mass-energy tensor to induce General Relativistic effects, they will affect each other gravitationally (which is quite interesting, for example it can be even repulsive).

Of course none of these effects are strong enough to be induced by a laser pointer.

• Note that the Maxwell equations are in principle only linear in vacuum; in the presence of a medium there will typically be nonlinear components of the electric and magnetic susceptibilities which are easily accessible to in-the-lab intensities. – Emilio Pisanty Jan 22 '17 at 14:17

Linear waves are mostly just an approximation -- as soon as some nonlinearity is present, linearity breaks and superposition isn't true anymore. In fact, you usually get production of higher harmonics. Most cases that involve matter have at least some nonlinearity that becomes more pronounced at larger amplitudes.

Maxwell equations give rise to a perfectly linear wave equation in vacuum, but in matter, you have nonlinear effects, such as Kerr effect. Nonlinear optics makes use of that -- for self-focusing beams, generation of higher harmonics (frequency doubling for lasers is used in some laser pointers to produce green from infrared).

Water waves are very well known examples which are nonlinear (just look at the wave shape changing and tumbling over itself when it gets to the shore).

For sound waves in gases, nonlinearity becomes apparent when the sound pressure becomes comparable to the ambient pressure (meaning that the low density parts of the sound wave are close to vacuum), and even before that, as the ideal gas law no longer holds well. Nonlinearity can lead to formation to shockwaves.

In general: any nonlinear response of the medium on the displacement will have as consequence:

• superposition no longer holds
• dependence of behaviour (frequency, propagation speed) on the amplitude
• harmonic sinusoidal waves will not hold their shape with time
• higher harmonics will be produced
• wave interferes with itself through nonlinearity and as such, alters its direction/shape/frequency