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    Let u(x,t)=\phi(x+ct)+\psi(x-ct) where \phi and \psi are arbitrary functions. Find \frac{\partial u}{\partial x} and \frac{\partial u}{\partial t} in terms of the partial derivatives of \phi and \psi.

    I've written that u=u(\phi,\psi) and \phi=\phi(x,t), \psi=\psi(x,t) and using the chain rule I ended up with...

    \frac{\partial u}{\partial x}_t=\frac{\partial u}{\partial \phi}_\psi \frac{\partial \phi}{\partial x}_t + \frac{\partial u}{\partial \psi}_\phi \frac{\partial \psi}{\partial x}_t
    \frac{\partial u}{\partial t}_x=\frac{\partial u}{\partial \phi}_\psi \frac{\partial \phi}{\partial t}_x + \frac{\partial u}{\partial \psi}_\phi \frac{\partial \psi}{\partial t}_x

    Is that it? The question asks for an answer in terms ofpartial derivatives of \phi and \psi but I still have partial derivatives of u... I don't know how to get rid of that though since I don't know what \phi and \psi are. And my answer has nothing to do with the fact that \phi and \psi are functions of x+ct and x-ct so I'm getting the feeling I'm missing something... but I'm not sure what. Could someone give me a hint? Thanks!
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    (Original post by Plagioclase)
    Let u(x,t)=\phi(x+ct)+\psi(x-ct) where \phi and \psi are arbitrary functions. Find \frac{\partial u}{\partial x} and \frac{\partial u}{\partial t} in terms of the partial derivatives of \phi and \psi.

    I've written that u=u(\phi,\psi) and \phi=\phi(x,t), \psi=\psi(x,t) and using the chain rule I ended up with...

    \frac{\partial u}{\partial x}_t=\frac{\partial u}{\partial \phi}_\psi \frac{\partial \phi}{\partial x}_t + \frac{\partial u}{\partial \psi}_\phi \frac{\partial \psi}{\partial x}_t
    \frac{\partial u}{\partial t}_x=\frac{\partial u}{\partial \phi}_\psi \frac{\partial \phi}{\partial t}_x + \frac{\partial u}{\partial \psi}_\phi \frac{\partial \psi}{\partial t}_x

    Is that it? The question asks for an answer in terms ofpartial derivatives of \phi and \psi but I still have partial derivatives of u... I don't know how to get rid of that though since I don't know what \phi and \psi are. And my answer has nothing to do with the fact that \phi and \psi are functions of x+ct and x-ct so I'm getting the feeling I'm missing something... but I'm not sure what. Could someone give me a hint? Thanks!
    ∂u/∂x = φ'(x+ct) + ψ'(x-ct)
    ∂u/∂t= c φ'(x+ct) - c ψ'(x-ct)

    PS this is the general equation of a one dimensional wave, transversing with speed c
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    (Original post by TeeEm)
    ∂u/∂x = φ'(x+ct) + ψ'(x-ct)
    ∂u/∂t= c φ'(x+ct) - c ψ'(x-ct)

    PS this is the general equation of a one dimensional wave, transversing with speed c
    Oh wow, I made a right mess of that then. Thank you! Rest of the question makes a lot more sense now too...
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    (Original post by Plagioclase)
    Oh wow, I made a right mess of that then. Thank you! Rest of the question makes a lot more sense now too...
    My pleasure as always

    What course are you doing and what place if you do not mind asking.
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    i know this isn't helpful in the slightest but i admire you both immensely as i am horrific at maths and really admire anyone who can make head and tail out of something like that
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    (Original post by TeeEm)
    My pleasure as always

    What course are you doing and what place if you do not mind asking.
    Earth Sciences, Oxford. Our lecturer has gone through partial fractions extremely quickly (particularly bearing that most of the cohort hasn't done Further Maths) so we've very much had to "learn by doing" in the problem sheets - which I've actually found quite interesting, but there were a couple of questions I was stuck on and I'd like to work out how to do them before my tutorial.
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    (Original post by Plagioclase)
    "learn by doing"
    is quite normal in most top Universities.

    Good luck
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    (Original post by Plagioclase)
    Earth Sciences, Oxford. Our lecturer has gone through partial fractions extremely quickly (particularly bearing that most of the cohort hasn't done Further Maths) so we've very much had to "learn by doing" in the problem sheets - which I've actually found quite interesting, but there were a couple of questions I was stuck on and I'd like to work out how to do them before my tutorial.
    Read the maths lecture notes, they are very comprehensive imo (found here https://www0.maths.ox.ac.uk/courses/...28688/material). We are pretty much at the same stage of calculus as you are (infact, this question is on problem sheet 3)
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    (Original post by TeeEm)
    ∂u/∂x = φ'(x+ct) + ψ'(x-ct)
    ∂u/∂t= c φ'(x+ct) - c ψ'(x-ct)

    PS this is the general equation of a one dimensional wave, transversing with speed c
    Not wishing to muddy the waters, but I'd appreciate your thoughts on this.

    What is "φ'(x+ct)" here?

    My thinking would be \dfrac{\partial \phi}{\partial (x+ct)}, which is rather hidden using the apostrophe symbolism.

    Is that correct?
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    (Original post by ghostwalker)
    Not wishing to muddy the waters, but I'd appreciate your thoughts on this.

    What is "φ'(x+ct)" here?

    My thinking would be \dfrac{\partial \phi}{\partial (x+ct)}, which is rather hidden using the apostrophe symbolism.

    Is that correct?

    correct
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    (Original post by ghostwalker)
    Not wishing to muddy the waters, but I'd appreciate your thoughts on this.

    What is "φ'(x+ct)" here?

    My thinking would be \dfrac{\partial \phi}{\partial (x+ct)}, which is rather hidden using the apostrophe symbolism.

    Is that correct?
    Yes, my lecturer says that the prime denotes differentiation w.r.t the variable inside the function
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    (Original post by TeeEm)
    correct

    (Original post by Gome44)
    Yes, my lecturer says that the prime denotes differentiation w.r.t the variable inside the function
    Thanks to you both.
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    (Original post by TeeEm)
    ...
    A subsequent thought: We could write this as \dfrac{d \phi}{d (x+ct)} then, since "x+ct" is a single variable.

    Yes?
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    (Original post by ghostwalker)
    A subsequent thought: We could write this as \dfrac{d \phi}{d (x+ct)} then, since "x+ct" is a single variable.

    Yes?
    imagine the phi function is x^3 or u^3 ot v^3 ....

    then (x+ct)^3 will differentiate wrt t

    will differentiate to 3(x-ct)^2 times c
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    (Original post by TeeEm)
    imagine the phi function is x^3 or u^3 ot v^3 ....

    then (x+ct)^3 will differentiate wrt t

    will differentiate to 3(x-ct)^2 times c
    'Fraid you've lost me there. I'm not even clear whether you're agreeing or disagreeing with me.
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    (Original post by ghostwalker)
    'Fraid you've lost me there. I'm not even clear whether you're agreeing or disagreeing with me.
    If \phi(x+ct)= (x+ct)^2 then by the chain rule, with u=x+ct:

    \frac{d\phi}{dt} = \frac{d\phi}{du}\frac{du}{dt}= \frac{d u^2}{du}\frac{du}{dt}= \frac{d(x+ct)^2}{d(x+ct)}\frac{d  (x+ct)}{dt} = 2u \cdot c

    So this is just the usual use of the chain rule.
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    I'm going with

    \dfrac{\partial \phi}{\partial t}= \dfrac{d\phi}{d(x+ct)}\dfrac{   \partial(x+ct) }{\partial t} =c\dfrac{d\phi}{d(x+ct)}
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    (Original post by ghostwalker)
    'Fraid you've lost me there. I'm not even clear whether you're agreeing or disagreeing with me.
    agreeing
    I was trying to make it clear with an example because we never really write it in that notation
 
 
 
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