Difference between the equation of SHM and a travelling wave

Question

The displacement of an oscillator is written as $y ( t ) = a \sin (ω t ± φ)$ whereas the equation of a wave is written as $y ( x , t ) = a \sin (ω t − kx )$.

The differences I can enumerate are-

1. The former, SHM is an equation of two dimensions. SHM is the function of time. And the former, travelling wave equation is a three dimensional equation- a function of displacement and time.

2. SHM depends upon the initial phase $φ$, whereas the wave equation depend upon $k$= wave number, and $x$= displacement.

3. The wave equation is the partial differentiated solution of the SHM.

I cannot find any other differences. Are there any other differences?

Answer 1

In general they are simply solutions to different differential equations. The simple harmonic oscillator obeys the following differential equation:

$$\frac{d^2 y}{d t^2} = -ky,$$

where $k$ is a constant. The solution to this equation can indeed be written as a sinusoid with a given frequency and phase.

The wave equation is a differential equation given by:

$$\frac{\partial^2 y}{\partial x^2} - \frac{1}{c^2} \frac{\partial^2 y}{\partial t^2} = 0,$$

where the solution is arbitrary function of the form $y(x,t) = F(x-ct) + G(x+ct)$ for any two functions $G$ and $F$. This can be satisfied by a sinusoid, and hence the solution to the SHO differential equation is also a solution to the wave equation. However, not all solutions to the wave equation are simple harmonic oscillators. Fortunately, one can see via Fourier analysis that any function can be decomposed into a sum of simple harmonic oscillators.