So I have been taking a wave course, and one thing that I don't understand is how does reflection even allows normal modes of vibration. I'll try to explain my confusion:
Suppose we have a string which is fixed at one of its ends, and I set up a travelling wave of some frequency on the string. This wave travels along the string in the positive x direction and then it reaches the fixed end. Upon reflection a reflected wave with phase shifted by π travels in the negative x direction and so this means the two destructively interfere and give no wave; in all the books I have read (French, Crawford, Howard...) they say that setting up a travelling wave with a certain specific frequency establishes a standing wave on the string, but shouldn't the two reflected wave no matter what the frequency destructively interfere and give no wave?
So my question is:
- Does reflection of wave always produce a π phase shifted reflected wave?
- If it does then how does the reflected wave and the original wave interfere to give a standing wave?
I do understand how mathematically solving the wave equation with the proper boundary conditions gives the normal mode/ stationary wave solution and also know how the method of images is incorporated but yet I don't see why the virtual or image pulse should have the same phase while the reflected wave( hence the virtual wave) must be π shifted in case of rigid fixed ends.
Thank you for your time and help.