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Solving the PDE in physics

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Consider a flexible string held stationary at both ends and free to vibrate transversely subject only to the restoring forces due to tension in the string. Deduce the PDE for such systems and define all parameters that distinguish the systems. Do this with a string of normal length l and using Cartesian coordinates.

Using a separation of variables and reasonable BC's solve the PDE generated above.

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This response provides instructions for solving the PDE using Cartesian coordinates and a separation of variables.

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Let's derive the wave equation for a uniform string with uniform linear mass distribution  under tension T

The mass of segment of length is

In the horizontal direction, since there is no motion, Newton's second law states that:

(1.1)
In the vertical direction, there is motion, so Newton's second law requires that:
(1.2)

We divide equation (1.2) by equation (1.1) to obtain:

(1.3)
But by definition the tangent of the angles is the value o the derivative at that point with respect to x.
Thus, equation (1.3) now becomes:
(1.4)

And as the left hand side of (1.4) becomes by definition the second order partial derivative of y with respect to x:
(1.5)
Now, the units of are mass per length while the tension has the units of force (mass times length per second squared)
Therefore the units of are:

So is the inverse of the square of the wave's speed c in the string,
So the wave's equation for a wave with propagation speed c is:
(1.6)
Now we have a string of length L that is clamped on both ends. We pluck the string - give it some initial shape f(x) and release it from rest. We need to show how the string behaves at any point x at any time t.

So first we have to state the system of this boundary/initial conditions system.
This is a second order partial differential equations. There is a second order derivative with respect to x and a second order derivative with respect to t. this means that to solve this ...

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