# Solution to 2D Laplacian in polar coordinates yields complex solution

I was studying a textbook: Physical Foundations of Technical Acoustics (referenced by a research paper) and I tried to solve the Laplacian of a wave equation by following the steps given. It was equation (2.35) from page 29:

$$\nabla^2 \Phi = {\frac 1r^2} {\frac\partial {\partial r}} \left(r^2 {\frac{\partial \Phi} {\partial r}}\right)+ {\frac 1{r^2\sin \theta}} {\frac\partial {\partial \theta}} \left(\sin \theta {\frac{\partial \Phi} {\partial \theta}}\right)$$

I did follow the separation of variables method, but I hit the roadblock when I was putting the solutions back together, following equation (2.42) from page 32. The book uses the spherical wave of zero order and substitutes $j_0 ,p_0,$ and $h_0^{(2)} (kr)$.

Using equation (2.42a), $$h_m^{(2)} (kr) = j_m(kr)+i n_m(kr),$$

I obtained: $${\frac {A_0}{kr}(\sin(kr)-i\cos (kr))e^{i\omega t}}$$

Which, in complex notation: $${\frac {A_0}{kr}(ie^{-ikr})e^{i\omega t}}$$

(Why is $i$ attached to the exponent $e$? According to the results below, it shouldn't be there.)

The results disagrees with the solution given by equation (2.43) on page 34: $${\frac {a\Phi_\textrm{max}}{r}e^{i(\omega t-kr-\phi)}}$$

Now the question: Why is the solution different and my answer has an imaginary unit, which makes the solution imaginary? Did I make a fatal mistake?

Side Question: How did $\frac {A_0}{kr}$ become $\frac {a\Phi_\textrm{max}}{r}$ (where did the $k$ go?) and how does the $\phi$ show up in the solution?

It seems that if I put the value of the Bessel function $j_0$ into the Neumann function in equation (2.42a) and vice versa, I will get the solution similar to the book's solution except with the constants. This also goes for another solution in the book with first order waves.

Glossary: $j_m$ Bessel function, $n_m$ Neumann function, $h_m^{(2)}$ Hankel function.

Note: I have given a books.google link for the book. The pages I referenced are accessible.

• Are you sure it's not a typo where they have -kr instead of -ikr? The key difference is, ikr would oscillate in space like a wave should, whereas -kr just drops off with distance and all oscillation is in phase with time. That sounds like what you get when a wave is evanescent, such as in a reflecting boundary layer where the phase at the boundary is picked out. So aside from the mathematics, you should be able to tell from the physical situation. Commented Oct 9, 2016 at 16:20
• I obtained: A0kr(sin(kr)−icos(kr))eiωt is it the possible functions $\Phi$ obtained by using variable seperable? Commented Oct 9, 2016 at 16:26
• What you have written is not the wave equation, but an expression for the laplacian. Can you clarify your question? Commented Oct 9, 2016 at 19:29
• @KenG Are you referring to equation 2.43? I missed the parenthesis on the power of e, which I have fixed. Commented Oct 10, 2016 at 8:01
• @ConfusinglyCuriousTheThird It is only the Laplacian part of the wave equation that is being discussed in that part of the book. I'm going to edit the question to clarify. Commented Oct 10, 2016 at 8:04