Why if I start from the equation of a wave not with cosine but with sine (Asin(kx+wt)-Asin)(kx+wt)) I get an antinode in the origin. I cant make physiscal sense of it. I should still be able to satisfy the boundary conditions
This is a really good question, and you're probably exposing some "smoke and mirrors" sloppy math in my derivation. I'll check it out and get back to you. -- Zak
Here's my answer for now: there is definitely a lack of generality in the assumption of cosine wave functions or sine wave functions. At the upper div. level, you'd write down a differential equation, find the general solution (probably in terms of complex exponentials), then nail down the coefficients of the different pieces by applying both boundary conditions for the ends of the string. This allows you to match any boundary conditions (but I haven't looked at those details in a long time). You're right that plugging in Asin(kx-wt) and -Asin(kx+wt) results in a problem where there cannot be a node at x=0, and this is a symptom of our lack of generality. So . . . the reasoning here pedagogically is to avoid dealing with the heavy math in an introductory course by pretending that a special case is actually totally general! In a sense, this derivation is "half-general" because after fortunately choosing a special case with a node at x=0, we were able to show how to nail down the wave function by enforcing the second boundary condition at the right side of the string, but the systematic application of boundary conditions to a general solution was glossed over. Thanks for the great question -- I'll dig into my old books to refresh my memory and at least mention to my students that there is more going on here than the standard introductory treatment pretends. -- Zak
@@ZaksLab Yeah! I was also searching for the statement of the problem in terms of differential equations. Huge thanks for your detailed respose. Very very good videos
Thankyou SO much! All your waves videos are life savers with the easy to understand explanations and cool animations. :)
glad I could help! z
Thank you so much! I really need this to understand the particle-wave section in my modern physics class!
good to hear! z
Why if I start from the equation of a wave not with cosine but with sine (Asin(kx+wt)-Asin)(kx+wt)) I get an antinode in the origin. I cant make physiscal sense of it. I should still be able to satisfy the boundary conditions
This is a really good question, and you're probably exposing some "smoke and mirrors" sloppy math in my derivation. I'll check it out and get back to you. -- Zak
Here's my answer for now: there is definitely a lack of generality in the assumption of cosine wave functions or sine wave functions. At the upper div. level, you'd write down a differential equation, find the general solution (probably in terms of complex exponentials), then nail down the coefficients of the different pieces by applying both boundary conditions for the ends of the string. This allows you to match any boundary conditions (but I haven't looked at those details in a long time).
You're right that plugging in Asin(kx-wt) and -Asin(kx+wt) results in a problem where there cannot be a node at x=0, and this is a symptom of our lack of generality. So . . . the reasoning here pedagogically is to avoid dealing with the heavy math in an introductory course by pretending that a special case is actually totally general! In a sense, this derivation is "half-general" because after fortunately choosing a special case with a node at x=0, we were able to show how to nail down the wave function by enforcing the second boundary condition at the right side of the string, but the systematic application of boundary conditions to a general solution was glossed over.
Thanks for the great question -- I'll dig into my old books to refresh my memory and at least mention to my students that there is more going on here than the standard introductory treatment pretends. -- Zak
@@ZaksLab Yeah! I was also searching for the statement of the problem in terms of differential equations. Huge thanks for your detailed respose. Very very good videos