Relationship between Simple Harmonic Motion Equation and Wave Equation

Question

I am very familiar with the equation:

$$f(t)=A\sin(\omega t+\phi)$$

Used to describe the instantaneous value $f(t)$ of a wave with amplitude $A$, frequency $\omega$, and phase shift $\phi$ at time $t$. This equation is very intuitive to understand: As $t$ increases the value within the $\sin$ operator will increase from $\phi$ upwards at a rate proportional to $\omega$, so the $\sin$ function will then oscillate between $-1$ and $1$, and the function $f(t)$ will oscillate between $-A$ and $A$.

However, in one of my modules the equation:

$$f(x,t)=A\cos(kx-\omega t)$$

Is now being used instead, with no explanation to the equivalence between this and the previous equation or what it really means. I would really like to understand this equation as intuitively as I do the first. I think $k$ is the wavenumber (number of waves per unit length), and $x$ is the distance along the wave.

Can someone please provide a written explanation (In words as opposed to math) for the second equation?

Also what is the relationship or difference between the two equations, and why is the second equation used instead of the first? Also why are there two arguments for the second equation? What does this actually mean, and could I just as easily say $f(\phi,t)=\sin(\omega t +\phi)$ ?

Thanks!

Answer 1

The first equation is just simple harmonic motion, the particle moves in $1$-D up and down and up and down.

The second equation is a wave equation. At any particular position $x=a$, it is SHM in time, and if you move along $x$ keeping $t$ time constant, you will sine wave.

Here is the example of second wave equation here http://www.animations.physics.unsw.edu.au/jw/travelling_sine_wave.htm and here is the graph on SHM that varies with time http://commons.wikimedia.org/wiki/File:Simple_harmonic_oscillator.gif

Note that wave equation is also function of (independent) space coordinate $x$ and the y-axis is dependent coordinate (displacement from mean position).

Okay here one thing to note what confuses most student who encounteres it for firs time is.

1. In SHM you plot $y$ vs $t$ graph and that looks like sine curve.

2. In Wave equation you plot $y$ vs $x$ graph and use static image and again get since curve. You don't get static sine curve. You have to visualize that the static image changes when you see your watch. If you plot $y$ vs $t$ in SHM then it's doesn't change with time. If you don't want to plot $y$ vs $t$ graph then you will get one dimensional oscillating spring.