Recent content by jonny997

  1. J

    Separation of variables: Wave equation governing a string with fixed ends

    Sorry I’ll try to explain it better. We can write the spatial part of the solution as ##\varphi (x) = A\sin{nx}## and the temporal part as ##\psi (t) = B\cos{nt} + C\sin{nt}##, with ##n \in \mathbb{Z}##. Part of the full solution then is $$ u_{n}(x,t) = \varphi (x)\psi (t) = (AB\cos{nt} +...
  2. J

    Separation of variables: Wave equation governing a string with fixed ends

    Okay thank you. I’m still a bit confused though… In the book I’m following m is restricted further and only m = 1,2,3,… is considered. If m = 0 the solution vanishes. I’m not understanding why the case in which m<0 resolves to that of m>0 when the coefficients for cosine differs by a -ve sign...
  3. J

    Separation of variables: Wave equation governing a string with fixed ends

    Is it necessary to write ##\sqrt{(-m^2)}=a+ib##? Can I not just write ##e^{imx} = e^{-imx}## This is what I got writing the exponent out as a + ib: $$ e^{(a+ib)\pi} = e^{-(a+ib)\pi} \rightarrow e^{2\pi(a+ib)} = 1 \rightarrow e^{2\pi a}(\cos{2\pi b} + i\sin{2\pi b}) = 1 $$ Then I guess a has...
  4. J

    Separation of variables: Wave equation governing a string with fixed ends

    Hey, thanks for the response. I guess eq. (1) implies that m in the exponent has to be an integer?
  5. J

    Separation of variables: Wave equation governing a string with fixed ends

    I am trying to follow through a derivation of the solution to the wave equation governing a string with fixed ends via separation of variables but I am stuck at the step which concludes ‘m’ must be a natural number 1,2,3,… etc. as opposed to an integer. After analysing each case: m > 0, m = 0...
  6. J

    Max inversion temperature for a gas (Dieterici’s equation of state)

    Ahhh okay, thank you. For some reason it didn’t click that I could solve for V using ## T = \frac{2a}{R}\left(\frac{1-\frac{b}{V}}{b}\right) ## and then sub that into the other equation 😬 I’ve got it now
  7. J

    Max inversion temperature for a gas (Dieterici’s equation of state)

    Hey. Thanks for the response. Do you have any idea how I would go about obtaining the expression my lecturer has provided? Namely, $$ P_{inv} = \left[\frac{2a}{b^2} - \frac{RT}{b}\right]e^{\frac{1}{2}-\frac{a}{RTb}}$$ If I am to rewrite ## T = \frac{2a}{R}\left(\frac{1-\frac{b}{V}}{b}\right) ##...
  8. J

    Max inversion temperature for a gas (Dieterici’s equation of state)

    The notes my lecturer has provided state that the maximum temperature can be found taking p = 0 in the inversion curve formula, given as: I’m not sure how to obtain this?? These are the formulas: This is my attempt at a solution : Not sure if this approach is right?
  9. J

    Fluid mechanics concept help please -- Pressure versus depth

    So in that case we would be taking the surface of the fluid to be the origin and measuring downwards?
  10. J

    Fluid mechanics concept help please -- Pressure versus depth

    Ohhh okay, that makes everything a lot clearer... So dp is the difference in pressure between the top and bottom portions of the slap, therefore p_top = p_bottom + dp So the force from the pressure at the top of slab acts downwards and is given by (p+dp)A
  11. J

    Fluid mechanics concept help please -- Pressure versus depth

    I think its mainly the (p+dp)A force I'm not understanding. Specifically the dp part, I'm not sure exactly what that is supposed to represent or why its acting downwards. I thought there would be more pressure at the bottom?
  12. J

    Fluid mechanics concept help please -- Pressure versus depth

    dw is the weight of the slab of fluid and the negative sign implies that it acts downwards. (p+dp)A is the force on the slab due to the fluid above it, it also acts downwards, hence the negative sign. pA is the force acting upwards on the slab and so its positive... The net force acting on...
  13. J

    Fluid mechanics concept help please -- Pressure versus depth

    I'm still kinda confused... :cry: I'm not sure where the (p+dp)A came from... How do you work out that p+dp is less than p from the drawing?
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