# Superposition principle in sinusoidal waves

In sinusoidal wave equations that produce interference we simply add their displacements by superposition principle, however superposition position principle can be applied to only linear equations. So what's the deal?

What allows superposition is that the wave equation

$$\frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2}$$

is linear in the “waving” variable $$u$$... for example, the string displacement for a string stretched along $$x$$. This linearity means that if $$u_1(x,t)$$ and $$u_2(x,t)$$ are solutions, then so is their sum. The fact that this linear differential equation has sinusoidal (and many other) solutions $$u(x, t)$$ that, for fixed $$t$$, are not linear functions of the coordinate $$x$$ is irrelevant.

• By u do you mean y displacement?? If yes then let me just ask this for confirmation linearity should be in the main variable whose change I am considering?? – naruto_022 Sep 28 '19 at 17:52
• If the string is vibrating in the $xy$ plane, then the displacement $u$ is $y$. Yes, linearity of the dynamical equation in the variable that is changing is what matters. – G. Smith Sep 28 '19 at 18:06

The wave equation is a linear operator, ie: $$L(x+y)=L(x)+L(y)$$.
Define the wave equation operator $$L(u)=u_{xx}-\frac{1}{c^2}u_{tt}$$ where u is the solution and c is the speed of propagation. Then the wave equation is $$L(u)=0$$ and since its linear$$L(u_1+u_2)=0$$ where $$u_1$$ and $$u_2$$ are any functions that have first and second derivatives.

• Google 'linear operator'. 'operator' is a function that operates on functions instead of numbers--like a normal function does. Linearity is one of the most important concepts you will run into. It applies to both functions and operators--and is basically defined by my equation for $L(x+y)=L(x)+L(y)$. – user45664 Sep 28 '19 at 19:03