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## Main Question or Discussion Point

What is d 'Alembert's solution? (in simple terms)

What does it mean physically

What does it mean physically

- Thread starter mcmzie
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What is d 'Alembert's solution? (in simple terms)

What does it mean physically

What does it mean physically

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You have two waves moving at eachother and passing by.

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For the wave equation, I think the intuition is basically like what Cyrus said. D'Alembert's solution says that you get two disturbances which propagate outwards in space and time, but in opposite directions. Imagine taking a jump rope and having two people hold it on either side. If another person comes and kicks the jump rope in the center, you will see that the rope moves outwards towards both people, in two different directions. That's kind of the intuition I think. However, I'd like to hear more people speak about the intuition as far as the general energy/momentum transfer goes.

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It can be shown that energy is conserved starting from the wave equation, but that has nothing to do with D'Alemberts solution.

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http://en.wikipedia.org/wiki/Wave_equation#Solution_of_the_initial_value_problemWhat is d 'Alembert's solution? (in simple terms)

it's a clean and simple solution to a partial differential equation, called the

[tex] \frac{\partial^2 f(x,t) }{ \partial t^2 } = c^2 \frac{ \partial^2 f(x,t) }{ \partial x^2 } [/tex]

is the physical description of what the displacement of a string,

the d 'Alembert's solution or " d 'Alembert's formula" is

[tex] f(x,t) = f_{+}(x,t) + f_{-}(x,t) [/tex]

where

[tex] f_{+}(x,t) = f_1(x-ct) [/tex]

and

[tex] f_{-}(x,t) = f_2(x+ct) [/tex]

or more simply

[tex] f(x,t) = f_1(x-ct) + f_2(x+ct) [/tex]

and where

now, if there are boundary conditions on

[tex] f(0,t) = f_{+}(0,t) + f_{+}(0,t) = 0 = f_1(-ct) + f_2(ct) [/tex]

or [tex] f_1(-x) = -f_2(x) [/tex]

and

[tex] f(L,t) = f_{+}(L,t) + f_{+}(L,t) = 0 = f_1(L-ct) + f_2(L+ct) [/tex]

or [tex] f_1(L-x) = -f_2(L+x) [/tex]

That gives you some symmetry properties. If it's a guitar string, and it's a slow, careful pluck (at t=0) where the string is deflected by the pick but is at rest at t=0 when the pick is released, the string's initial shape is assumed known

[tex] f(x,0) = f_1(x) + f_2(x) [/tex]

it turns out that if the string's velocity at time 0 is zero,

[tex] \frac{\partial f(x,t) }{\partial t}\Big|_{t=0} = 0 [/tex]

then

[tex] f(x,t) = ( f(x-ct,0) + f(x+ct,0) )/2 [/tex]

it's not the simple answer. but reasonably complete.

it means that the wave equation can be solved into a form of adding two wave functions together each representing waves going in opposite directions on the string and having equal wave speed.What does it mean physically

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Thanks for your clarification

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