Small Angle Approximation with a Stiff String

[rwilliams184]

Hi all,

Could someone please help me understand a small but significant step in the derivation of the wave equation for a string with stiffness.

I am trying to follow the notes here:

http://courses.physics.illinois.edu/phys406/Lecture_Notes/Waves/PDF_FIles/Waves_2.pdf

The statement I don't understand is near the bottom of page 2 where ∂ϕ(x,t)/∂x ≅ −(∂2y(x,t)/∂x2)

I know sin(phi) ~ tan(phi) ~ opp/adj but this doesn't seem to help.

Any help greatly appreciated. :-)

 

[jvicens]

Hello,

First of all, I'm glad to see that you are actively trying to understand the derivation of the wave equation for a string with stiffness. It can be a complex topic, so it's great that you are seeking clarification.

To understand the statement near the bottom of page 2, it is important to first understand the concept of a small angle approximation. This approximation is commonly used in physics and engineering when dealing with trigonometric functions. It states that for small angles, the sine and tangent of the angle can be approximated by the angle itself. In other words, sin(phi) ~ phi and tan(phi) ~ phi. This approximation is valid as long as the angle is small, typically less than 10 degrees.

Now, let's apply this concept to the statement in question. The derivative of a function is essentially the slope of the function at a given point. In this case, we are looking at the derivative of the displacement of the string (y) with respect to position (x). The equation is stating that this derivative (∂y/∂x) can be approximated by the negative second derivative (∂2y/∂x2). This is similar to saying that the slope of a curve at a point can be approximated by the curvature of the curve at that point.

To better understand this, let's look at an example. Imagine a string with a slight bend in it. The displacement of the string (y) at a certain point (x) can be described by a sine function. The derivative of this function (∂y/∂x) would be a cosine function, which is essentially the slope of the string at that point. However, for small angles, the cosine of the angle is approximately equal to the angle itself. Thus, we can approximate the derivative (∂y/∂x) with the angle (x) itself. This is where the small angle approximation comes in. And since the angle is small, the tangent of the angle is also approximately equal to the angle itself. Therefore, we can further approximate the derivative (∂y/∂x) with the negative second derivative (∂2y/∂x2), which is essentially the curvature of the string at that point.

I hope this explanation helps you understand the statement better. If you have any further questions, please don't hesitate to ask. Keep up the good work in understanding the derivation!