Why does the wave equation change between sine and cosine forms?

[quietrain]

a wave equation is given as A = A cos (kx - ωt)

so why if someone describes the wave equation to be A = A sin (ωt - kx) , the argument of the sin function changes by a minus sign?

and is there a meaning to it?

also i still don't really understand why the minus sign in the first equation signifies wave moving forward whereas a + sign signifies wave moving backwards ...

thanks a lot for the help!

 

[Bob_for_short]

kx - ωt = kx' - ωt' if both x' > x and t' > t. => The same wave value A cos(kx - ωt) propagates with time in positive x direction.

 

[jimmy neutron]

You don't have a wave equation there, you have solutions to a wave equation.

As well, since cos(-z) = cos(z), i.e. cos is an even function you can write the cos solution as A cos (ωt - kx) if you wish.

The general solution to the wave equation is
Acos(wt - kx) + Bsin(wt -kx)
where A and B are determined by the initial or boundary conditions
If you wrote the general solution in terms of (kx - wt) then the sign of the factor multiplying the sin function would change to accoomdate this.

 

[quietrain]

oh...

so cos (kx - ωt) = cos (ωt- kx ) because it is an even function.

so what's the difference if we choose to write it in sin instead?

 

[Bob_for_short]

...so what's the difference if we choose to write it in sin instead?

You can write in either way. The initial/boundary conditions will determine the signs and values of coefficients A and B in the general solution Acos(wt - kx) + Bsin(wt -kx).

 

[quietrain]

oh i see thanks