I Wave motion and a stretched string

AI Thread Summary
The discussion centers on understanding Griffiths' treatment of wave motion in a stretched string, particularly regarding the small angle approximation and the derivation of second partial derivatives. The small angle assumption is crucial, as it ensures that the string's displacement from equilibrium is minor, allowing the sine function to be approximated by the tangent function. The second derivative arises from applying Taylor series expansion, which provides a method to relate the change in the function to its derivatives. Clarification is sought on the significance of the term O(Δz²) in the expansion, indicating higher-order terms that become negligible for small displacements. Overall, grasping these concepts is essential for accurate analysis in electromagnetic wave theory.
mondo
Messages
27
Reaction score
3
I continue my reading of Griffiths electrodynamics (chapter 9, electromagnetic waves) and I got stuck on this:
237fe636e4cbb4c33a445f778634b7a4.png

Author tries to prove a stretched string supports wave motion and I found it very difficult to grasp.
In the second equation, why can we replace sin function with a tangents really? What guarantees that the angles are small? Is he trying to match it to partial derivative formula?
Next, how a difference of partial derivatives $\frac{\partial_{f}}{\partial_{z}}$ became a second partial derivative in the second equation?

Thank you.
 
Physics news on Phys.org
It is the string's displacement from its stable equilibrium position (assumed horizontal) that is small (compared with the length of the string); a consequence of this is that the angle between the horizontal and the tangent to the string is everywhere small. If this assumption is not valid, then neither are the results of the analysis.

For your second question, you have from the Taylor series expansion of \frac{\partial f}{\partial z} wrt z that <br /> \left.\frac{\partial f}{\partial z}\right|_{z+ \Delta z} = \left.\frac{\partial f}{\partial z}\right|_{z} + \left.\frac{\partial^2 f}{\partial z^2}\right|_{z}\Delta z + O(\Delta z^2).
 
mondo said:
What guarantees that the angles are small?
It's an assumption.
 
pasmith said:
It is the string's displacement from its stable equilibrium position (assumed horizontal) that is small (compared with the length of the string); a consequence of this is that the angle between the horizontal and the tangent to the string is everywhere small. If this assumption is not valid, then neither are the results of the analysis.

For your second question, you have from the Taylor series expansion of \frac{\partial f}{\partial z} wrt z that <br /> \left.\frac{\partial f}{\partial z}\right|_{z+ \Delta z} = \left.\frac{\partial f}{\partial z}\right|_{z} + \left.\frac{\partial^2 f}{\partial z^2}\right|_{z}\Delta z + O(\Delta z^2).
Thanks for a repones.
As for the first part about the small angles - yes I think I got it. Thanks for mentioning the assumed position is horizontal - that helped me to visualize the movement.

As for the second derivative derivation, what is the last term in your formula -O(\Delta z^2). ?
I think in the book formula T (\frac{\partial_{f}}{\partial_{z}}|_{z+\delta z} - \frac{\partial_{f}}{\partial_{z}}|_{z}) is something (in the parenthesis) that at a glance looks very close to the derivative formula - we take value at time z+\delta z and subtract from a value at time z. The only missing thing (for it to be a derivative) is a division by this delta time. So, now, if we multiply it by this missing delta, and rewrite it as a derivative (which in this case is a second derivative) we get what's in the book. Am I right?
 
Thread 'Question about pressure of a liquid'
I am looking at pressure in liquids and I am testing my idea. The vertical tube is 100m, the contraption is filled with water. The vertical tube is very thin(maybe 1mm^2 cross section). The area of the base is ~100m^2. Will he top half be launched in the air if suddenly it cracked?- assuming its light enough. I want to test my idea that if I had a thin long ruber tube that I lifted up, then the pressure at "red lines" will be high and that the $force = pressure * area$ would be massive...
I feel it should be solvable we just need to find a perfect pattern, and there will be a general pattern since the forces acting are based on a single function, so..... you can't actually say it is unsolvable right? Cause imaging 3 bodies actually existed somwhere in this universe then nature isn't gonna wait till we predict it! And yea I have checked in many places that tiny changes cause large changes so it becomes chaos........ but still I just can't accept that it is impossible to solve...
Hello! I am generating electrons from a 3D gaussian source. The electrons all have the same energy, but the direction is isotropic. The electron source is in between 2 plates that act as a capacitor, and one of them acts as a time of flight (tof) detector. I know the voltage on the plates very well, and I want to extract the center of the gaussian distribution (in one direction only), by measuring the tof of many electrons. So the uncertainty on the position is given by the tof uncertainty...
Back
Top