# Superposition Waves Ver. - Interference Basics
Remember [[Newton's Law of Gravitation - The Intro#Gravitation Superposition|superposition]]? Well, if you haven't, it's when a bunch of different forces converge onto an object to all act on it at once. Now, if that happens to our dear wave here, how are we going to model it?
When two or more waves collide, we call it 'Interference'. There are two types of wave interference:
1. **Constructive Interference** - when the crests and troughs of the waves align
2. **Destructive Interference** - when the waves are *not in phase*, meaning their crests and troughs do not line up for one reason or another.
%%Diagram of destructive / constructive interference%%
With everything we've done, this'll be a jog. Let's begin!
We can algebraically express this as a combination of two [[Waves Foundations#More Transverse Waves|transverse wave equations]] - the wave after we've interfered with it is given prime notation:
$y'(x,t) = y_{1}(x,t) + y_{2}(x,t)$
In string form, this gives us the total overlap displacement of the wave - just a simple addition! For now, the waves will have the same amplitudes and frequencies, though the second wave will be shifted a little - giving us a phase angle to worry about:
$y_{1}(x,t) = y_{m}\sin(kx - \omega t)$
$y_{2}(x,t) = y_{m}\sin(kx - \omega t + \phi)$
Now we add these waves up. [[Trig Identities - Derivations DLC Ver. (Maths)#Angle Addition Identities|Additional trig identities]] mandate that we can get a generalised form for the added equation:
$y'(x,t) = y_{m}\sin(kx - \omega t) + y_{m}\sin(kx - \omega t + \phi)$
Now we apply the addition identity $\sin A + \sin B = 2\sin \frac{1}{2} (A+B)\cos \frac{1}{2}(A-B)$:
$y'(x,t) = \left[ 2y_{m}\cos \frac{1}{2}\phi \right]\sin\left( kx-\omega t+\frac{1}{2}\phi \right)$
This gives us an amplitude of ...:
$y'_{m} = 2y_{m}\cos \frac{1}{2}\phi$
If the phase angle is $\phi$, interference is perfectly constructive and we won't have to deal with this anymore! We get a reduced equation:
$y'(x,t) = \left[ 2y_{m} \right]\sin\left( kx-\omega t \right)$
# Interference for Acoustic Waves
*This links back to* [[All about Acoustics (Sound Waves)]].