It is well known that the solutions to a homogeneous linear differential equation form a vector space. This means that any linear combination of solutions is also a solution. The Law of Superposition is simply this statement.
But why call it “superposition”? The meaning becomes clear once we consider linear combinations of solutions. If we know two solutions \( \phi(x,t) \) and \( \psi(x,t) \), then any linear combination
is also a solution. On its own this doesn’t tell us much, so we usually impose initial conditions. The initial condition fixes which combination of solutions gives the particular solution we want.
In other words, the actual solution lies in a superposition of all possible linear combinations. Only after imposing the initial condition do we “collapse” into a specific solution.
Analogy to Quantum Mechanics
This can be compared to Schrödinger’s wave function: a quantum system is in a superposition of states until observed, at which point it collapses into a single state. Similarly, our PDE solution exists in all possible states (linear combinations), but once constrained by an initial condition, it collapses into the unique solution of the problem.
This could be why it’s called superposition: the general solution is literally a superposition of all possible particular solutions. However, this was my interpretation of the convention, not necessarily the truth.