Relationship between Simple Harmonic Motion Equation and Wave Equation

[CraigH]

I am very familiar with the equation:

$$f(t)=Asin(ωt+ϕ)$$

Used to describe the instantaneous value f(t) of a wave with amplitude A, frequency ω, and phase shift ϕ at time t. This equation is very intuitive to understand: As t increases the value within the sin operator will increase from ϕ upwards at a rate proportional to ω, 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:

$$f(x,t)=Acos(kx−ωt)$$

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(ϕ,t)=sin(ωt+ϕ) ?

Thanks!

 

[berkeman]

I am very familiar with the equation:

$$f(t)=Asin(ωt+ϕ)$$

Used to describe the instantaneous value f(t) of a wave with amplitude A, frequency ω, and phase shift ϕ at time t. This equation is very intuitive to understand: As t increases the value within the sin operator will increase from ϕ upwards at a rate proportional to ω, 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:

$$f(x,t)=Acos(kx−ωt)$$

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(ϕ,t)=sin(ωt+ϕ) ?

Thanks!

It is the equation for a traveling wave: http://en.wikipedia.org/wiki/Travelling_wave#Traveling_waves