Recent content by N00813

I looked over my notes again, and apparently d/dz = 1/2 (d/dx - i d/dy). From here, that would suggest \lim_{h \to 0} \frac{\mathrm{Re}(h)}h = \frac{1}{2} and \lim_{h \to 0} \frac{\mathrm{Im}(h)}h= \frac{-i}{2}. I can't think of how to get there, though.

Yes, from the question it appears they are different functions.

V = grad(u); so V_x = du/dx and V_y = du/dy df/dz = (d/dx - id/dy)(u+iv) = du/dx + i dv/dx - i du/dy + dv/dy Using the CR equations, du/dx = dv/dy and du/dy = -dv/dx. So df/dz = 2(du/dx) -2i(du/dy) = 2(V_x - iV_y).

Homework Statement f(z) = u(x, y) + iv(x, y) where z ≡ x + iy. Let the fluid velocity be V = ∇u. If f(z) is analytic, show that df/dz = V_x − iV_y Homework Equations V_x = du/dx V_y = idu/dy The CR equations du/dx = dv/dy, du/dy = -dv/dx. The Attempt at a Solution I...

Earlier in the question, it did say to use a complex exponential for y (=A exp(ikx-iwt)) to prove that time-averaged power = 1/2 Zw^2 A^2. I assumed that carried forwards. I suppose I'll have to ask my supervisor for this, then. Perhaps the answer really is zero.

There'd be no reflected and transmitted power then, would there? Since the wave is standing?

I know the string satisfies a wave equation. Evanescent waves would be my next guess, but beyond that I'm stuck.

The given constraint (and continuity of gradient) means that -ik(1-r) = -ik(t) = 0. Continuity of string means that 1 + r = t. If t = 0, then 1-r = 0 so r = 1. But then 1 + r = 2 =/= t = 0.

Homework Statement Given that a string is constrained such that dy/dx = 0 at x = 0 and unconstrained otherwise, what is the reflected and transmitted power? y is the deflection of the string from the x-axis. y_1 is incident wave, y_r is reflected and y_t is transmitted. Homework Equations...

Thanks! I suppose it makes it easier if I had used a test function, and then taken it away.

Homework Statement Given that \hat{p} = -i\hbar (\frac{\partial}{\partial r} + \frac{1}{r}) , show that \hat{p}^2 = -\frac{\hbar^2}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}) Homework Equations Above The Attempt at a Solution I tried \hat{p}\hat{p} =...

Thanks, I must have derped out last night. I'm thinking about turning the 2nd term into \lim_{h \to \infty} \frac{1}{h} \tilde{u}(k,t) and substituting the y = x + h into the first term, to give a final result of: \lim_{h \to 0} \frac{1}{h} ((e^{ik(0+h)}-e^{ik0}) \tilde{u}) = ike^{0}...

Homework Statement Using the formal limit definition of the derivative, derive expressions for the Fourier Transforms with respect to x of the partial derivatives \frac{\partial u}{\partial t} and \frac {\partial u}{\partial x} . Homework Equations The Fourier Transform of a function...

Ah, thanks. Finally figured it out!

How did you start off, then? Which part of the integrand did you choose to integrate and differentiate?