# Why is the displacement dependent function stated as the amplitude of a standing wave?

The equation for stationary wave is :

$$y(x,t)=A\sin(kx) \cos(\omega t)$$. From this equation the amplitude of the stationary wave is said to be $$A\sin(kx)$$, in other words the displacement dependent function. So why can't it be the other way around. Like why can't the amplitude be $$A\cos(\omega t)$$ or in other words the time dependent function?

You could have $$A\cos(\omega t) \sin(kx)$$ and talk about the ‘amplitude’ of the time dependent function $$\cos$$ if you wanted to. It’s a matter of interpretation, the result is the same.

To see this, take out the time dependence (scale it to 1, the same as considering the wave at time $$t=0,2\pi,4\pi,...$$). Now you just have a wave in space.

As you change $$A$$ bigger or smaller, you make the wave in space taller or shorter. If you set $$A$$ negative, then the wave ‘inverts’ and what was positive becomes negative and vice versa.

Now imagine that you move $$A$$ back and forth, growing and shrinking your wave and inverting it, with a regular motion that can be described in time by $$\cos(wt)$$. This function is $$A\cos(\omega t)$$. So then we have added time dependence back in.

What the function $$\cos(\omega t)$$ does is it scales each point in space by a factor (between -1 and 1) that depends on time in a certain way. So each point in space ‘waves’ up and down between its maximum value and its minimum value.

If you thought of $$A$$ attached to $$\cos$$ rather than $$\sin$$, then the time dependent function scales each point in space between $$-A$$ and $$A$$. So the scaling with time becomes stronger if $$A$$ is large, and weaker if $$A$$ is small.

The time-dependent cosine makes oscillations in time, it is "a wave". What is its amplitude? It is a factor standing at the time dependent cosine. For each point in space the amplitude is different. "Hand waving" implies constant $$x$$ (you) and time-dependent hand motion (a wave).

• I liked your example. However you gave an example of space dependent amplitude. So can you explain what will it be for a time dependent amplitude? – Aslan Jan 25 '19 at 15:01