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In the study of trasmission and reflection of waves in one dimension I do not understand completely the meaning of the conditions imposed.

Consider an impulse $\xi(x,t)$ moving on a rope linked with an other rope, of different type in $x=0$.

We have three wavefunctions.

$\xi_{\mathrm{incident}}(x,t)$

$\xi_{\mathrm{reflected}}(x,t)$

$\xi_{\mathrm{trasnmitted}}(x,t)$

The two conditions of continuity imposed are

$(1)$ Continuity of vertical displacement

$$\xi_{\mathrm{incident}}(0,t)+\xi_{\mathrm{reflected}}(0,t)= \xi_{\mathrm{transmitted}}(0,t) \tag{1}$$

$(2)$ Continuity of the vertical component of the force (tension of the rope $T$)

$$ -T \bigg(\frac{\partial \xi_{\mathrm{incident}}(x,t)}{\partial x}|_{x=0}+\frac{\partial \xi_{\mathrm{reflected}}(x,t)}{\partial x}|_{x=0}\bigg)=- T\bigg(\frac{\partial \xi_{\mathrm{transmitted}}(x,t)}{\partial x}|_{x=0}\bigg) \tag{2}$$

I'm totally ok with the calculation from that. What I do not understand completely is the sense of those conditions.

Firstly why in $(1)$ the sum $\xi_{\mathrm{incident}}(0,t)+\xi_{\mathrm{reflected}}(0,t)$ is considered and not just $\xi_{\mathrm{reflected}}(0,t)$? I mean when $\xi_{\mathrm{reflected}}$ is generated, $\xi_{\mathrm{incident}}$ has already desappeared. For example consider the picture. When I see $\xi_{\mathrm{reflected}}$, I do not see $\xi_{\mathrm{incident}}$ any longer. So what is the point of considering both $\xi_{\mathrm{incident}}$ and $\xi_{\mathrm{reflected}}$ togheter, interfering with each other?

enter image description here

Secondly $(2)$ is about the tension $T$. The magnitude of $\vec{T}$ is constant regardless any condition in all the two ropes. But here it is required that its vertical component is the same. What is the meaning of that? Can the horizontal component vary?

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Perhaps your first question is answered with the following diagram where the displacement of the incident (red) plus the displacement of the reflected (red) must equal the displacement of the transmitted (black)?

enter image description here

For your second question the answer is that the vertical acceleration of the rope which is proportional to the vertical component of the tension must be the same otherwise there will be a discontinuity across the boundary.

In the derivation of the wave equation for a string you consider a curved element of the string as shown below.

enter image description here

An assumption is made that the angles $\theta_1$ and $\theta_2$ are small so that the cosine of the angles ($\approx 1 - \frac{\theta^2}{2}$) are approximately one and the string element does not have a horizontal displacement so $T_1 \cos \theta_1 = T_2 \cos \theta_2 \Rightarrow T_1 \approx T_2 \approx T$

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