Confusion between superposition of SHMs I am a high school student and i am little confused between superposition of Simple harmonic motions{SHM's}, suppose a spring of spring constant $k_1$ has time period $T_1$ and another spring of spring constant $k_2$ and time period $T_2$ when connected with a block separately, if both are now connected in parallel or series, we say that the resultant motion will also be SHM, but why? we have studied that if angular frequencies of two SHM are different{which is the case here as time periods are different for both} and they get superimposed then their resultant motion will not be a SHM, then why do we say that, here it is SHM, is it different from SHM? if yes then why? and please also give some practical examples other than this where superposition of SHM can be seen?
 A: You seem to be confused with the superposition of two oscillators. The following discussion will be useful for you :
Waves are usually described by variations in some parameters through space and time—for example, height in a water wave, pressure in a sound wave, or the electromagnetic field in a light wave. The value of this parameter is called the amplitude of the wave, and the wave itself is a function specifying the amplitude at each point.
The point that should be noted is that for superposition you must have at least two waves (at least two degrees of freedom) otherwise you can't apply the superposition principle.
In an abstract way, If $\psi_1$ and $\psi_2$ satisfy the equation for linear differential equation so does $$\psi=c_1\psi_1+c_2\psi_2.$$
Take a simple example to digest this

If you go about solving this equation, you will find out that
$$M\frac{d^2\psi_a}{dt^2}=k(\psi_b-\psi_a)-k\psi_a$$
$$M\frac{d^2\psi_b}{dt^2}=-k(\psi_b-\psi_a)-k\psi_b$$
The general solution for such a system is the superposition of two-equation, like
$$\psi_a=A_1\cos(\omega_1 t+\phi_1)+A_2\cos(\omega_2 t+\phi_2)$$
That's what means to be a superposition of two oscillators, Note that you can decouple these oscillators into two independent systems.

On OP's problem, you have a differential equation which looks like
$$M\frac{d^2}{dt^2}(x_1+x_2)=-(k_1x_1+k_2x_2)$$
which seem to have two variable (two degrees) but this is not true you need to fulfill the condition on joint (Newton's third law) so that $x_1k_1=x_2k_2$. Substitute this to your differential equation so
$$M\frac{d^2x_1}{dt^2}=-\frac{k_1k_2}{k_1+k_2}x_1$$
You see actually there is only one independent degree of freedom in the system and so you don't have the superposition of two waves. You just need to tell $x_1$ to find out where the mass is (state of the system).

The answer is already too long but I'm adding one more example of a parallel spring system on OP's request.

The following is the system in my mind. It's obvious that displacement for both the spring will be the same, say $x$.The force equation for the mass
$$F=m\ddot{x}=-k_1x-k_2x=-(k_1+k_2)x=-Kx$$
The fact remains the same. Until you don't have two degrees of freedom, you can not have two different frequencies. I will recommend OP to look on JAlex answer
on this question, When you let there is mass, you see there was a superposition solution but as you let that one mass is zero, the second frequency vanishes.
I hope this will help you. Best Wishes!
