Originally Posted by **DesertFox** The wave equation: **d²l/dx² = (1/k²)(d²l/dt²)**
Can you give me, please, a clear and intuitive explanation of the equation. For dummies. **t** is for time. **x** is probably for distance?
But what does **l** mean? **k** is constant and it represents the propagation... but i can't grasp the whole idea.
Please, explain me step by step and piece by piece... |

OK so maths seems to be working again here so I will try my explanation.

I don't know how much maths you know, but this is University/College level stuff.

But I am going to start with a simple agebraic equation

If they have any solutions at all, the solutions to algebraic equations are just numbers.

So the solution to ax - b = 0 is one number x= b/a (if a is not zero, but then you don't have an equation either!).

If you have a quadratic, say

$\displaystyle {x^2} - 4 = 0$

Then there are two solutions, x = -2 and x = +2.

So you need an extra piece of information to distinguish which one you want.

This is known as a boundary condition, say x < 0 which picks out one solution form the others.

Boundary conditions are very important in differential equations.

This is because once you have integrated (solved) the differential equation you have not finished as you have an unknown arbitrary term.

For ordinary differential equations this arbitrary term is an arbitrary constant.

It arises thus.

Take the simplest possible ordinary differential equation

$\displaystyle \frac{{dy}}{{dx}} = 1$

$\displaystyle dy = 1dx$

$\displaystyle \int {dy = \int {1dx} } $

$\displaystyle y = x + C$

Where C is an arbitrary constant which is found by applying a boundary condition, say y = 0 when x = 0, which leads to C = 0.

So the solution of a differential eqaution is not a number but a function y = f(x), where the variable y is a function of the variable x.

y is known as the dependent variable because it is set by the independent variable, x.

Note also the minimum number of variables for a differential equation is 2, but our example algebraic equation had only one variable.

Algebraic equations with 2 or more variables cannot be solved with one equation (except by a lucky guess).

Now your equation is a second order differential equation.

Second order ordinary differential equations have two arbitrary constants, one for each integration.

This requires two boundary conditions to find both constants.

Note I said ordinary differential equations, which have two variables.

But your equation has three variables, not two.

A differential equation with 3 or more variables is a partial differential equation, not an ordinary differential equation.

So your wave equation should be written

$\displaystyle \frac{{{\partial ^2}{\rm I}}}{{\partial {x^2}}} = \frac{1}{{{k^2}}}\frac{{{\partial ^2}{\rm I}}}{{\partial {t^2}}}$

Where the dependent variable $\displaystyle {\rm I}$ is a function of two variables, both of which are independent (of each other). The two independent variables are distance, x and time, t.

So the solution of this partial differential equation will be

$\displaystyle {\rm I} = f\left( {x,t} \right)$

If you think about the implications of this, when you partially differentiate with respect to one variable, you hold the other variable constant.

So the partial derivative of any function g(t)

**with respect to x**is the derivative of a constant which is therefore zero.

$\displaystyle \frac{\partial }{{\partial x}}\left[ {g\left( t \right)} \right] = 0$

Similarly the partial derivative of any function h(x)

**with respect to t**is the derivative of a constant which is therefore zero.

$\displaystyle \frac{\partial }{{\partial t}}\left[ {h\left( x \right)} \right] = 0$

So when we integrate a partial differential equation we have to include arbitrary functions of the other variable in each integration.

Again we must fall back on boundary conditions to reach a full solution.

This can be difficult to impossible for some partial differential equatiosn.

Fortunately the wave eqaution is an (relatively) easy one and as it is also important it is much studied.

From a physics point of view

The arbitrary functions determine the shape of the wave and permit the very useful process of modulation for sound and radio waves.