Solve $\sin(ax) / \sin(x) = a/2$ for $x$

Question

I am currently trying to solve $\sin(ax)/\sin(x) = a/2$ for $x$, where $x$ is between $0$ and $\pi$ and $a$ is a constant.

As I have limited skills in math, I cannot seem to solve this problem without a brute force solution using Matlab to calculate a numerical value for this equation, I was wondering if there is a mathematical solution to this.

$a$ is non-zero, and around $500,000$ for my application.

Any help would be appreciated, thank you.

Answer 1

You always have the trivial solution $x=0$.

The next root is a small value (on the order of $1/a$), so that the sine is very well approximated by $x$ (the next term being $-x^3/6$). The equation is rewritten

$$\frac{\sin ax}{a\sin x}\approx\frac{\sin ax}{ax}=\text{sinc }ax=\frac12,$$ hence $$x\approx\frac1a\text{sinc}^{-1}\frac12=\frac{1.895494267034\cdots}a$$

For large $a$ this initial approximation is excellent, if not just sufficient.

Then no other root is found until the end of the range, very close to $\pi$. Making the change of variable $\dfrac ta=\pi-x$, we can approximate the sine as $\dfrac ta$. The equation becomes

$$\sin(a\pi-t)\approx\frac t2.$$

By the periodicity of the sine, we can replace $a\pi$ by $b=a\pi\bmod 2\pi$, a more manageable coefficient.

Anyway, uness $a$ is integer, there is no further simplification here and the equation has the general form of the intersection of a straight line with a sinusoid, $\sin p=\alpha p+\beta$, that can have up to three solutions and needs to be solved numerically.

To get started, the two arches of the sinusoid can be approximated by $4p(\pi-p)/\pi^2$ and $-4(p-\pi)(2\pi-p)/\pi^2$, leading to quadratic equations.

UPDATE: numerical issue

Actually, finding the root(s) close to $\pi$ is an ill-posed problem. Indeed, its location depends on the value of $a\pi\bmod2\pi$ so that $a$ needs to be known with, say $2$ exact decimals ($8$ significant digits). If this is not the case, all you can say is that $|\dfrac t2|<1$ and $x$ is in the range $(\pi-2/a,\pi)$.

Answer 2

Let $y = ax$, so $\sin(ax)/\sin(x) = a/2$ becomes $\sin(y)/\sin(y/a) = a/2$.

Since $a$ is large, under the assumption that $y/a$ is small, I will approximate $\sin(y/a)$ by $y/a$, so this becomes $a \sin(y)/y = a/2$ or $\sin(y)/y = 1/2$.

The solution to this, according to alphy, is $y \approx ±1.89549$, so that $x = y/a \approx ±1.89549/a $.