How can I calculate the value in an equation?

gamma=1.4
V_M_2 = 20;


V_M_2 = sqrt((gamma+1)/(gamma-1)) * atan(sqrt((gamma-1)/(gamma+1)*(M_2^2-1)))-atan(sqrt(M_2^2-1))

How can I find the M_2 from this equation? Thank you.

Answer 1

[Sam Chak]

@Ali Deniz

Please verify if the nonlinear equation is correct.

Looks like there are no real solutions for this one.

[Ali Deniz]

x values will be X^2.

[Torsten]

x values will be X^2.

Won't make a difference - no roots.

[Ali Deniz]

That's right, I should find another way. Thank you.

[Sam Chak]

@Ali Deniz, Thanks for the correction.

By the way, the nonlinear function has a limit, .

So, if , then it has roots. Please check again.

gamma = 1.4;
V_M_2 = 2.0;

% function to be solved
f = @(x) sqrt((gamma + 1)/(gamma - 1))*atan(sqrt((gamma - 1)/(gamma + 1)*(x^2 - 1))) - atan(sqrt(x^2 - 1)) - V_M_2;

% initial guess
x0 = 20;                

% root-finding solver
xsol = fzero(f, x0)

xsol =

17.9584

Answer 2

[Torsten]

I don't see a root:

gamma=1.4
V_M_2 = 20;
fun = @(M_2)V_M_2 - (sqrt((gamma+1)/(gamma-1)) * atan(sqrt((gamma-1)/(gamma+1)*(M_2-1)))-atan(sqrt(M_2-1)));
M_2 = -20:0.1:200;
plot(M_2,fun(M_2))