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When imposing a periodic boundary condition, the amplitude of the wavefunction at coordinate $x$ must match that at coordinate $x+L$, so we have: $$\Psi(x)=\Psi(x+L)$$ In your previous 'particle in a box' scenario, you mention that the general form of the wavefunction is given by a linear combination of sine and cosine with complex coeficients. It might be ...

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The second condition is saying that there is no discontinuity in the slope of the rope at the junction. In other words, there is no "kink" in the rope. Imagine if this assumption were to fail in the following way: $$\frac{\partial D_1}{\partial x}(0,t) = -1, \qquad \frac{\partial D_2}{\partial x}(0,t) = 1$$ Then near the origin, the rope would look ...

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The phase change happens because it is how waves behave. An additional link provides lecture notes. I know that u are not satisfied with this answer but you can compare this with mechanical waves in a string which gives better intuition by use of newtons laws.

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Its a method called co-ordinate grid transformations that is used to transform an arbitrarily shaped geometry into a square; in the computational domain. Grid transformations work as parametric transformations do, in co-ordinate geometry. what these transformations basically do is, they map the complex geometry (viz. a curve) into a simpler geometry (a line; ...

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The tension in the two cords is the same because they are tied together. For example if the tension in the thick cord was higher than the thin cord the thick cord would shrink and the thin cord stretch until the tensions were equal again. The frequency has to be the same in both cords because the phase of the wave has to match at the junction between the ...

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