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 Jul 9th 2017, 10:32 AM #1 Junior Member   Join Date: Jul 2017 Posts: 27 the wave equation 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... Last edited by DesertFox; Jul 9th 2017 at 10:34 AM.
 Jul 9th 2017, 04:46 PM #2 Senior Member   Join Date: Aug 2010 Posts: 308 Yes, x is the distance along the wave, t is the time the wave has been moving. l may have many meaning depending upon what kind of wave you are talking about. If we have a rope or string pulled taut, l would be the height above (or below if negative) the original line. If it is a light wave, then I would be the "intensity" of the electro-magnetic wave. k is the wave speed.
Jul 10th 2017, 04:15 PM   #3
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 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.

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 xis 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 tis 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.

Jul 22nd 2017, 04:56 PM   #4
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 Originally Posted by DesertFox Hello buddies!
Hello buddy!

 Originally Posted by DesertFox The wave equation: d²l/dx² = (1/k²)(d²l/dt²)
I've seen you put equations in bold before where they don't belong in bold. That's a very bad idea. The last time you did it I thought you were talking about vectors since its only vector quantities which are supposed to appear in bold. A and A mean two different things.

 Originally Posted by DesertFox 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...
We need an example to drive this home. Consider a vibrating string. Let I be he height of the string from its equilibrium position (i.e. where it is when its not vibrating). Then

t = time
x = is the position along the length of the string at which you wish to find the height of the string.

For a great description of this phenomena, where the wave equation comes from etc see: https://en.wikipedia.org/wiki/String_vibration

For everything you wanted to know about the wave equation but was afraid to ask, see: https://en.wikipedia.org/wiki/Wave_equation

 Originally Posted by studiot Note I said ordinary differential equations, which have two variables.
ODE's can have more than two. See: Ordinary Differential Equation -- from Wolfram MathWorld

 Originally Posted by studiot A differential equation with 3 or more variables is a partial differential equation, not an ordinary differential equation.
A diff eq might be a system of equations and can have more than 3 variables. In fact there is no limit to the number of variable they can have. This is true when working with systems of equations. Consider the following example (from page 162 of Ordinary Differential Equations - 3rd Ed by Finizio and Ladas)

dx/dt = y + t

dy/dt = -2x + 3y + 1

The term ordinary differential equations (ODE) refers to any equations in which only ordinary derivatives appear. Equations involving partial derivatives are called partial differential equations. An ODE which is linear in the derivatives is called a linear differential equation (LED).

The wave equation he's talking about is a LDE.

 Originally Posted by studiot So your wave equation should be written $\displaystyle \frac{{{\partial ^2}{\rm I}}}{{\partial {x^2}}} = \frac{1}{{{k^2}}}\frac{{{\partial ^2}{\rm I}}}{{\partial {t^2}}}$
Yes siree Bob!

 Originally Posted by studiot The two independent variables are distance, x and time, t.
Note: Its better to refer to x is as a position coordinate. This is critical terminology when one gets to the wave equation in higher dimensions such as E(x, y, z, t) where x, y, z denote the coordinates where E is determined at time t. There are other ways to describe a wave in which x could be an angle and not the distance from a reference point. After all we don't know how his "x" is defined. A wave equation is a wave equation regardless of what letters are used. Some freaky authors use Greek letters, the jerks!

 Originally Posted by studiot So the solution of this partial differential equation will be $\displaystyle {\rm I} = f\left( {x,t} \right)$
Note on notation: Typically one would write I = I(x, t)

Last edited by Pmb; Jul 22nd 2017 at 05:09 PM.

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