# Question about spherical means method

I am learning the spherical means method to solve three-dimensional wave equations, but I am confused about it. Here is my understanding of the method, I will mark where I found confusing.

First, we define some spherical means function, i.e., for any given point $$M(x_0,y_0,z_0)$$, any given radius $$r$$, $$\bar{\mu}(r,t)=\frac{1}{4\pi r^2}\iint_{S_{r}^{M}} u(x,y,z)dS$$, where $$S_{r}^{M}$$ means the surface of a ball with center $$M$$ and radius $$r$$.

Second, we need to dig some properties of $$\bar{\mu}$$, so we integrate the wave equation on a given ball with center $$M$$ and radius $$r$$ $$\iiint_{B_{r}^{M}}\mu_{tt}dxdydz=a^2\iiint_{B_{r}^{M}}(\mu_{xx}+\mu_{yy}+\mu_{zz})dxdydz$$, by Gauss, we have

$$a^2\iiint_{B_{r}^{M}}(\mu_{xx}+\mu_{yy}+\mu_{zz})dxdydz=a^2\iint_{S_{r}^{M}}(\frac {\partial{u}}{\vec{n}})dS=a^2\iint_{S_{r}^{M}}(\mu_xcos\alpha+\mu_y cos \beta+\mu_z cos \gamma)dS$$

where $$\vec{n}$$ is the normal vector. Then it states that $$a^2\iint_{S_{r}^{M}}(\frac {\partial{u}}{\vec{n}})dS=a^2\iint_{S_{r}^{M}}(\frac {\partial{u}}{r})dS$$. I know we can use the spherical coordinate($$\vec{n}=(sin\phi cos\theta,sin\phi cos\theta,cos\phi)$$), and by the chain rule, it seems right($$\mu_r=\mu_x x_r+\mu_y y_r+\mu_z z_r$$). $$\begin{cases}x=x_0+rsin\phi cos\theta \\ y=y_0+rsin\phi cos\theta\\ z=z_0+rcos\phi \end{cases}$$, $$r$$ is given, $$\phi \in [0,2\pi), \theta \in [0,\pi)$$

But my question is, since we integrate on a given ball ($$r$$ is given), $$\frac {\partial{u}}{r}$$ is differentiating with respect to some constant. It does not make sense to me.

• It's a normal derivative, not differentiation w.r.t. a constant: $\frac{\partial u}{\partial n}=\nabla u\cdot n$ Commented Nov 8, 2022 at 6:08

I can already answer my own question. If we use $$k$$ instead of $$r$$ when integrating. It will be clearer and things just work out.