Waves in three dimensions.

We have till now considered waves which depend on only one position coordinate $x$ and time $t$. This is quite adequate when considering waves on a string as the position along a string can be described by a single coordinate. It is necessary to bring three spatial coordinates into the picture when considering a wave propagating in three dimensional space. A sound wave propagating in air is an example.

We use the vector $\vec{r}=x \hat{i}+ y \hat{j}+ z \hat{k}$ to denote a point in three dimensional space. The solution which we have been discussing

$$\tilde{a}(\vec{r},t)=A e^{i(\omega t - k x)}$$ (6.12)

can be interpreted in the context of a three dimensional space. Note that $\tilde{a}(\vec{r},t)$ varies only along the direction and not along $y$ and $z$. Considering the phase $\phi(\vec{r},t)=\omega t - k x$ we see that at any particular instant of time $t$, there are surfaces on which the phase is constant. The constant phase surfaces of a wave are called wave fronts. In this case the wave fronts are parallel to the $y-z$ plane as shown in Figure 6.5. The wave fronts move along the $+x$ direction with speed $v_p$ as time evolves. You can check this by following the motion of the $\phi=0$ surface shown in Figure 6.5.

Figure 6.5: