Waves in an arbitrary direction.

Let us now discuss how to describe a sinusoidal plane wave in an arbitrary direction denoted by the unit vector $\vec{n}$. A wave propagating along the $\hat{i}$ direction can be written as

$$\tilde{a}(\vec{r},t)=\tilde{A}e^{i (\omega t - \vec{k}\cdot \vec{r})}$$ (6.13)

where $\vec{k}= k \hat{i}$ is called the wave vector. Note that $\vec{k}$ is different from $\hat{k}$ which is the unit vector along the $z$ direction. It is now obvious that a wave along an arbitrary direction $\hat{n}$ can also be represented by eq. (6.13) if we change the wave vector to $\vec{k}=k \hat{n}$. The wave vector $\vec{k}$ carries information about both the wavelength $\lambda$ and the direction of propagation $\hat{n}$.

For such a wave, at a fixed instant of time, the phase $\phi(\vec{r},t)=\omega t - \vec{k}\cdot \vec{r}$ changes only along . The wave fronts are surfaces perpendicular to $\hat{n}$ as shown in Figure 6.6.

Problem: Show the above fact, that is the surface swapped by a constant phase at a fixed instant is a two dimensional plane and the wave vector $\vec{k}$ is normal to that plane.

The phase difference between two point (shown in Figure 6.6) separated by $\Delta \vec{r}$ is $\Delta \phi=-\vec{k}\cdot \Delta \vec{r}$.

Figure 6.6:

Problems

1. What are the wave number and angular frequency of the wave $a(x,t)=A \cos^2 (2 x - 3 t)$ where $x$ and $t$ are in ${\rm m}$ and ${\rm s}$ respectively? ( $4 \, {\rm m}^{-1}$, $6 \, {\rm s}^{-1}$)
2. What is the wavelength correspnding to the wave vector $\vec{k}=3 \hat{i} + 4 \hat{j} \, {\rm m}^{-1}$ ? ( $0.4 \, {\rm m}$)
3. A wave with $\omega=10 \, {\rm s}^{-1}$ and $\vec{k}=7 \hat{i} + 6 \hat{j} - 3 \hat{k} \, {\rm m}^{-1}$ has phase at the point $(0,0,0)$ at $t=0$. [a.] At what time will this value of phase reach the point $(1,1,1) \, {\rm m}$? [b.] What is the phase at the point $(1,0,0) \, {\rm m}$ at $t=1 \, {\rm s}$? [c.] What is the phase velocity of the wave? ([a.] $10 \, {\rm s}$ [b.] $24 \, {\rm rad}$ [c.] $1.03 \, {\rm m}\, {\rm s}^{-1}$
4. For a wave with $\vec{k}=(4\hat{i}+ 5 \hat{j}) {\rm m}^{-1}$ and $\omega = 10^8 \, {\rm m}^{-1}$, what are the values of the following? [a.] wavelength, [b.] frequency [c.] phase velocity, [d.] phase difference between the two points $(x,y,z)=(3,4,7) \,{\rm m}$ and $(4,2,8) \,{\rm m}$.
5. The phase of a plane wave is the same at the points $(2,7,5)$, $(3,10,6)$ and $(4,12,5)$. and the phase is $\pi/2$ ahead at $(3,7,5)$ . Determine the wave vector for the wave.[All coordinates are in ${\rm m}$.]
6. Two waves of the same frequency have wave vectors $\vec{k}_1= 3 \hat{i} + 4 \hat{j} \, {\rm m}^{-1}$ and $\vec{k}_1= 4 \hat{i} + 3 \hat{j} \, {\rm m}^{-1}$ respectively. The two waves have the same phase at the point $(2,7,8) \, {\rm m}$, what is the phase difference between the waves at the point $(3,5,8) \, {\rm m}$? ( $3 \, {\rm rad}$)