Jean le Rond d’Alembert was a famous French Mathematician, Physicist, and Philosopher in the 18^th century. He derived and established d’Alembert’s Formula which is used when researching wave patterns to solve for the general solution to the one-dimensional wave equation.

The equation:

- = 0, u( x, 0) = g(x), ( x, 0) = h(x), for -∞<x<∞ and t>0,

shows u as two waves each with constant velocities traveling in two opposite directions, with as the second derivative of u.

The first step in the set up of the equation is to demonstrate u as a function of both x and t.

u( x, t ) = F( x + ct) + G( x-ct), when u is if F and G are .

Now, since u( x, 0) = g(x)

then F(x) + G(x) = g(x),

And if ( x, 0) = h(x)

then cF’(x) – cG’(x) = h(x).

So after the integration of the previous equation, to get F’(x) and G’(x) as the original function:

cF(x) – cG(x) = y)dy +

Now to solve the new system of equations involving the newly integrated and first equation use:

F(x) + G(x) = g(x)

+ ( cF(x) – cG(x)) = ((y)dy + )

To get:

F(x) + G(x) = g(x)

- 1/c ( F(x) + G(x)) = 1/c ((y)dy + )

So:

2F(x) + 0 = g(x) + 1/c ((y)dy + )

Finally:

F(x) = 1/2g(x) + 1/c ((y)dy + ) and therefore,

G(x) = 1/2g(x) – 1/c ((y)dy + )

Now using the original equation that showed u as a function of x and t, u( x, t ) = F( x + ct) + G( x-ct), substitute (x) for (x - ct) in the F(x) equation and also substitute (x) for (x + ct) in the G(x) to show:

u( x, t) = + 1/2c (y)dy +

This result is d’Alembert’s Formula.

As mentioned before d’Alembert’s Formula is regarded as the general solution to the one dimensional scalar wave equation. The equation is useful in predicting a certain point on the wave form by using c as the velocity, F as the forward movement and G and any backwards movement of the wave.

In this clip http://www.youtube.com/watch?v=hEziafbLvAE from Discovery Channels’ new hit show Time warp, Mike Mangini (a famous solo drummer who now teaches at the Berkley College of Music) shows off his speed and precision on the drums. At 4:15 into the clip there is a demonstration the snare drum and the cymbals as they are struck in extreme slow motion. Both the snare drum and the cymbal demonstrate almost perfect examples of d’Alembert’s formula for wave equations. This monumental realization of motion is merely impossible to see with the naked eye. However, this phenomenon becomes visible with the use of super slow motion cameras, as used in this clip.

The simulation above is a computer’s prediction of the same event that occurs as a cymbal or drum head is struck. The simulation and the clip display how flexibility of an object used in making sound. This flexibility is what allows the sound waves to develop and travel through the air and into our ears. Thanks to mathematics and physics, us humans are able to enjoy the most beautiful sounds. So next time you hear your favorite song or tune don’t take it for grated because there is much more behind that sound than you ever could have possibly imagined.

Sound waves and d’Alembert’s Formula

By Dane Andrews