Question about the wave equation

[Sunfire]

Hello,

is it possible for one to assume a straight-line propagation of an e.m. wave and constant velocity c? If so, is it possible to simplify the wave equation

u_tt=c²u_xx

by expressing the spatial variable x through the time variable t?
x must be a function of t, since the motion is rectilinear with constant c.

Then, x = x(t) and x is no longer independent variable. Then the above PDE should be rewritten in terms of t only, since u(x,t)=u(x(t),t)=u(t)

Does this make sense?

 

[sankalpmittal]

Hello,

is it possible for one to assume a straight-line propagation of an e.m. wave and constant velocity c? If so, is it possible to simplify the wave equation

u_tt=c²u_xx

by expressing the spatial variable x through the time variable t?
x must be a function of t, since the motion is rectilinear with constant c.

Then, x = x(t) and x is no longer independent variable. Then the above PDE should be rewritten in terms of t only, since u(x,t)=u(x(t),t)=u(t)

Does this make sense?

Yes , you can convert it into differential forms :

http://en.wikipedia.org/wiki/Wave_equation
http://people.ccmr.cornell.edu/~muchomas/P214/Notes/OtherWaves/node18.html
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/waveq.html

 

[Jano L.]

is it possible for one to assume a straight-line propagation of an e.m. wave and constant velocity c?

Yes, of course, the light of a pocket laser is a good example of such wave.

is it possible to simplify the wave equation...by expressing the spatial variable x through the time variable t?

The quantity u in the wave equation is some physical quantity ascribed to point in space x at some time t (e.g. electric field), so it is usually thought of as a function of both x and t.

If you have some function x1(t), you can define new function by
$$u1(t) = u(x1(t),t).$$

For example, if x1 is function giving the position of electron, u1 gives the electric field acting on the electron at time t.

However, the function x1 has to be inferred from other sources; there is nothing in the wave equation that would give such a function.

True, there is the motion of the maxima of the wave crests and one could ascribe such function x1(t) to one of them, but there is no good reason for doing so - there is no particle there - so it would seem to be just an empty exercise.

 

[sophiecentaur]

Yes, of course, the light of a pocket laser is a good example of such wave.

Or the signal at a few wavelengths distant from any radio transmitter. That behaves pretty well like a plane wave.

 

[pmacias]

Yes, it does make sense. The wave equation describes the propagation of an electromagnetic wave in a medium, and assuming a straight-line propagation with constant velocity c is a valid simplification. In this case, x is no longer an independent variable but a function of time, and the wave equation can be rewritten in terms of t only. This simplification can be useful in certain situations, but it is important to note that it may not accurately represent the behavior of the wave in all cases. It is always important to consider the assumptions and limitations of any mathematical model.