How Does the Plane Wave Equation Define Wavefronts?

[iScience]

Plane wave equation:

$$\psi(t) = \psi_0e^{i(\vec{k}\cdot\vec{r}-\omega t)}$$

The part that makes the domain of \psi(t_i) a plane is the k dot r part.

I'm reading a book that takes this term and imposes the following condition:

$$\vec{k}\cdot\vec{r}=Const.$$

which, i understand its necessity, but if we just plug in the LHS of the equation, the information on the RHS is lost no? i mean, we didn't use it; we just got rid of it. Can someone clarify this part for me please.

 

[Simon Bridge]

Plane wave equation:

$$\psi(t) = \psi_0e^{i(\vec{k}\cdot\vec{r}-\omega t)}$$

The part that makes the domain of \psi(t_i) a plane is the k dot r part.

I'm reading a book that takes this term and imposes the following condition:

$$\vec{k}\cdot\vec{r}=Const.$$

which, i understand its necessity, but if we just plug in the LHS of the equation, the information on the RHS is lost no? i mean, we didn't use it; we just got rid of it. Can someone clarify this part for me please.

$\vec k\cdot\vec r = \text{const.}$ would mean that $\psi$ is a function of time alone.
Isn't a plane wave also a function of space?
http://en.wikipedia.org/wiki/Plane_wave#Arbitrary_direction

You don't "plug in" the LHS of that equation - the equation is a definition of what the LHS means. If you already know what $\psi(t)$ is, then what extra information could the RHS possibly supply?

 

[iScience]

aha! i get it! thanks

 

[vanhees71]

Hm, I don't get it. What's the book intending to derive/demonstrate? Could you quote more details?

 

[ehild]

Plane wave equation:

$$\psi(t) = \psi_0e^{i(\vec{k}\cdot\vec{r}-\omega t)}$$

The part that makes the domain of \psi(t_i) a plane is the k dot r part.

I'm reading a book that takes this term and imposes the following condition:

$$\vec{k}\cdot\vec{r}=Const.$$

A wave is a traveling disturbance, and the disturbance described by the function ψ depends both on place and time.
$$\vec{k}\cdot\vec{r}=Const.$$ is the equation of a wavefront, a plane, where the phase of the wave is the same at each point. Consider a wavefront where C=0 at t=0, that is, $\vec{k}\cdot\vec{r}=0$ . The equation represents a plane at the origin that is perpendicular to the wave vector $\vec k$. At a later time t, the points where the phase is zero are on the plane
$$\vec{k}\cdot\vec{r}-ωt=0$$ In case $\vec k$ is parallel with the x axis, $\vec k =k\hat e_x$, the plane is perpendicular to the x-axis and its position is determined by $k x -ωt=0$, that is, at $x=ω/k t$: the wavefront travels in the positive x dirction, with speed ω/k. ω/k is the propagation velocity or phase velocity of the wave.

ehild

 

[nrqed]

A wave is a traveling disturbance, and the disturbance described by the function ψ depends both on place and time.
$$\vec{k}\cdot\vec{r}=Const.$$ is the equation of a wavefront, a plane, where the phase of the wave is the same at each point. Consider a wavefront where C=0 at t=0, that is, $\vec{k}\cdot\vec{r}=0$ . The equation represents a plane at the origin that is perpendicular to the wave vector $\vec k$. At a later time t, the points where the phase is zero are on the plane
$$\vec{k}\cdot\vec{r}-ωt=0$$ In case $\vec k$ is parallel with the x axis, $\vec k =k\hat e_x$, the plane is perpendicular to the x-axis and its position is determined by $k x -ωt=0$, that is, at $x=ω/k t$: the wavefront travels in the positive x dirction, with speed ω/k. ω/k is the propagation velocity or phase velocity of the wave.

ehild

What they are saying is NOT that $\vec{k} \cdot \vec{r} = constant$ everywhere. What they are saying is this: the vector $\vec{k}$ is a constant, and the vector $\vec{r}$ can be anything. Now, you pick constant C. Then all the points satisfying the condition $\vec{k} \cdot \vec{r} = C$ lie on a plane, right? (and not that that plane will be perpendicular to $\vec{k}$) What we know is that everywhere on that plane the wave function has the same phase at any given instant (fixed t). So all the points on that plane correspond a fixed phase. This is the definition of a plane wave.