Question

Derive an equation for moving string.​

Answer 1

Let's apply Newton's second law in the vertical y direction:

Fy  =  may.

The sum of forces in the y direction is

Fy  =  T sin θ2 − T sin θ1.

Using the small angle approximation, sin θ ≅ tan θ = ∂y/∂x. So we may write:

F = T(dy/dx)_2 - T (dy/dx)_1

So the total force depends on the difference in slope between the two ends: if the string were straight, no matter what its slope, the two forces would add up to zero. Now let's get quantitative. The mass per unit length is μ, so its mass dm = μdx. The acceleration in the y direction is the rate of change in the y velocity, so ay  =  ∂vy/∂t  =  ∂y2/∂t2. So we can write Newton’s second law in the y direction as

F = T((dy/dx)_2 - (dy/dx)_1) = mu dx d^2y/dt^2

Rearranging this gives

d^2y/dt^2 = (T/mu)((dy/dx)_2 - (dy/dx)_1)/ dx  

Now we have been using the subscript 1 to identify the position x, and 2 to identify the position (x+dx). So the numerator in the last term on the right is difference between the (first) derivatives at these two points. When we divide it by dx, we get the rate of change of the first derivative with respect to x, which is, by definition, the second derivative, so we have derived the wave equation:

d^2y/dt^2 = (T/mu)d^2y/dx^2)