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I have three doubts in regard to waves on a string which I will try to make as clear as possible.
For this purpose, I have considered a general wave:
$$y_i=y_0\sin(\omega t  kx)$$
(1) If a wave pulse:
$$y = y_0 e^{\frac{1}{T^2} \left(t\frac xv \right)^2}$$
is incident against a rigid boundary, such as a fixed wall to which the string is attached, why does it get inverted when it is reflected? I know that as the wave hits the boundary, considering the wave pulse travelling from left to right, the left part pulls the element of string fixed to the wall upwards, and the wall exerts an equal and opposite force on it, thus preventing the element from moving. Due to this force, the left part of string is pulled downwards upto its mean position, beyond which it (according to me) moves downwards due to inertia. My question is, at this stage, won't the left part of the string pull the said element downward again? Wouldn't the wall again generate a new pulse in the upward direction? Since this doesn't really happen, what is the correct reasoning behind the inversion of the pulse? Also if possible, could similar reasoning be applied for reflection against a nonrigid (soft) boundary? (View attached image)
(2) Now considering the initial sine wave, if it reflects against the rigid boundary, what will be the equation of the reflected wave? As far as I know, it must be:
$$y_r=y_0\sin(\omega t + kx + \pi)$$
or,
$$y_r=y_0\sin(\omega t + kx)$$
Is this correct? And what about reflection against a nonrigid (soft) boundary? Is it the following?
$$y_r=y_0\sin(\omega t + kx)$$
Also, how do we obtain these equations (short proofs or conceptual proof)?
(3) In our textbook, the standing waves were explained as superposition of two waves travelling in the opposite direction, where one was the incident wave (given as ##y_i=y_0\sin(kx  \omega t)## in the book), and the other was reflected from either a hard boundary or a soft one. But in both cases, the equation of standing wave used is exactly the same, i.e.:
$$y=2y_0\sin(kx)\cos(\omega t)$$
where component waves were given as ##y_i=y_0\sin(kx  \omega t)## and ##y_r=y_0\sin(kx + \omega t)##. This meant that my equation for reflection was wrong too, as ##y_r## is different in this case! Also for soft boundary the exact same equation of standing wave was used, meaning the reflected wave was also the same in both cases! Clearly this isn't possible. So what is the actual mathematical derivation of standing waves on string fixed at both ends, and string fixed at one end only?
Finally, can we apply all these results for a longitudinal pressure waves (i.e. sound)?
Thanks in advance!
For this purpose, I have considered a general wave:
$$y_i=y_0\sin(\omega t  kx)$$
(1) If a wave pulse:
$$y = y_0 e^{\frac{1}{T^2} \left(t\frac xv \right)^2}$$
is incident against a rigid boundary, such as a fixed wall to which the string is attached, why does it get inverted when it is reflected? I know that as the wave hits the boundary, considering the wave pulse travelling from left to right, the left part pulls the element of string fixed to the wall upwards, and the wall exerts an equal and opposite force on it, thus preventing the element from moving. Due to this force, the left part of string is pulled downwards upto its mean position, beyond which it (according to me) moves downwards due to inertia. My question is, at this stage, won't the left part of the string pull the said element downward again? Wouldn't the wall again generate a new pulse in the upward direction? Since this doesn't really happen, what is the correct reasoning behind the inversion of the pulse? Also if possible, could similar reasoning be applied for reflection against a nonrigid (soft) boundary? (View attached image)
(2) Now considering the initial sine wave, if it reflects against the rigid boundary, what will be the equation of the reflected wave? As far as I know, it must be:
$$y_r=y_0\sin(\omega t + kx + \pi)$$
or,
$$y_r=y_0\sin(\omega t + kx)$$
Is this correct? And what about reflection against a nonrigid (soft) boundary? Is it the following?
$$y_r=y_0\sin(\omega t + kx)$$
Also, how do we obtain these equations (short proofs or conceptual proof)?
(3) In our textbook, the standing waves were explained as superposition of two waves travelling in the opposite direction, where one was the incident wave (given as ##y_i=y_0\sin(kx  \omega t)## in the book), and the other was reflected from either a hard boundary or a soft one. But in both cases, the equation of standing wave used is exactly the same, i.e.:
$$y=2y_0\sin(kx)\cos(\omega t)$$
where component waves were given as ##y_i=y_0\sin(kx  \omega t)## and ##y_r=y_0\sin(kx + \omega t)##. This meant that my equation for reflection was wrong too, as ##y_r## is different in this case! Also for soft boundary the exact same equation of standing wave was used, meaning the reflected wave was also the same in both cases! Clearly this isn't possible. So what is the actual mathematical derivation of standing waves on string fixed at both ends, and string fixed at one end only?
Finally, can we apply all these results for a longitudinal pressure waves (i.e. sound)?
Thanks in advance!
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