In the equation regarding an array of masses connected by springs in wikipedia the step from(adsbygoogle = window.adsbygoogle || []).push({});

$$\frac {u(x+2h,t)-2u(x+h,t)+u(x,t)} { h^2}$$

To

$$\frac {\partial ^2 u(x,t)}{\partial x^2}$$

By making ##h \to 0## is making me wonder how is it rigorously demonstrated. I mean:

$$\frac {\partial ^2 u(x,t)}{\partial x^2} = \lim_{h\to 0} \frac {u_x(x+h,t)-u_x(x,t)} { h}$$

But we have

$$\lim_{h\to 0}\frac {u(x+2h,t)-2u(x+h,t)+u(x,t)} { h^2}=\lim_{h\to 0}\frac{\frac {u(x+2h)-u(x+h)}{h}-\frac {u(x+h)-u(x)}{h}}{h}$$

How do we demonstrate that these two expressions are equal?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Very simple: second order derivative in wave equation

**Physics Forums | Science Articles, Homework Help, Discussion**