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For the 1 dimensional wave equation,

$$\frac{\partial^2 u}{\partial x ^2} - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$

##u## is of the form ##u(x \pm ct)##

For the 3 dimensional wave equation however,

$$\nabla ^2 u - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$It appears that solutions need not be of the form ## u(\vec{k} \cdot \vec{r} - \nu t) ##, for instance spherical waves ## u = A/r \; \mathrm{exp}[-i(kr - vt)] ##

Am I right? Why is it so?

$$\frac{\partial^2 u}{\partial x ^2} - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$

##u## is of the form ##u(x \pm ct)##

For the 3 dimensional wave equation however,

$$\nabla ^2 u - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$It appears that solutions need not be of the form ## u(\vec{k} \cdot \vec{r} - \nu t) ##, for instance spherical waves ## u = A/r \; \mathrm{exp}[-i(kr - vt)] ##

Am I right? Why is it so?

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