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gerryliyana
 2 years ago
show that the expression u = sin (xvt) describing a sinusoidal wave, satisfies the wave equation. Show that in general u = f(x  vt) and u = f(x + vt) satisfies the wave equations, where f is any function with a second derivative
gerryliyana
 2 years ago
show that the expression u = sin (xvt) describing a sinusoidal wave, satisfies the wave equation. Show that in general u = f(x  vt) and u = f(x + vt) satisfies the wave equations, where f is any function with a second derivative

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wilson3
 2 years ago
Best ResponseYou've already chosen the best response.0slow down buddy . you wrote u=f(xvt) twice

gerryliyana
 2 years ago
Best ResponseYou've already chosen the best response.1i wrote once, u = f(x  vt) and u = f(x + vt)

Meepi
 2 years ago
Best ResponseYou've already chosen the best response.1The wave equation is: \[\frac{\partial^2 u}{\partial t^2} = v^2\frac{\partial^2 u}{\partial x^2}\] In your case, \(u = \sin(x  vt)\) And \[u_{tt} = \sin(x  vt) * v^2 \] \[u_{xx} = \sin(x  vt)\] So since \(u_{tt} = v^2 u_{xx}\), u satifies the wave equation

gerryliyana
 2 years ago
Best ResponseYou've already chosen the best response.1good job @Meepi i appreciate

Meepi
 2 years ago
Best ResponseYou've already chosen the best response.1No, \(u_{tt} = \sin(x  vt) * v^2\) and \(u_{xx} = \sin(x  vt)\) the v^2 is from the chain rule when you derive with respect to t in \(u_{tt}\) since \(u_{xx} = \sin(x  vt)\), \(u_{tt} = u_{xx} * v^2\)

gerryliyana
 2 years ago
Best ResponseYou've already chosen the best response.1isn't it \[\frac{ ∂^{2}u }{ ∂t^{2} } = \frac{ 1 }{ v^{2} } \frac{ ∂^{2}u }{ ∂x^{2} }\] ??? you typed \(\frac{ ∂^{2}u }{ ∂t^{2} } = v^{2} \frac{ ∂^{2}u }{ ∂x^{2} } \)

gerryliyana
 2 years ago
Best ResponseYou've already chosen the best response.1i thought \(\frac{ ∂^{2}u }{ ∂t^{2} } = \frac{ 1 }{ v^{2} } \frac{ ∂^{2}u }{ ∂x^{2} }\)

gerryliyana
 2 years ago
Best ResponseYou've already chosen the best response.1ohh my bad.., sorry

gerryliyana
 2 years ago
Best ResponseYou've already chosen the best response.1\[\frac{ ∂^{2}u }{ ∂x^{2} } = \frac{ 1 }{ v^{2} } \frac{ ∂^{2}u }{ ∂t^{2} }\]

Meepi
 2 years ago
Best ResponseYou've already chosen the best response.1It's alright your text book probably has the partial derivatives swapped

gerryliyana
 2 years ago
Best ResponseYou've already chosen the best response.1ur allright :), cool thanks :)

Meepi
 2 years ago
Best ResponseYou've already chosen the best response.1Also, for the general case, just use the chain rule as well to get \[u_{xx} = f''(x + vt)\] \[u_{tt} = f''(x + vt) * v^2\] Then \[u_{xx} = \frac{u_{tt}}{v^2}\]

Meepi
 2 years ago
Best ResponseYou've already chosen the best response.1Same can be done for u = f(x  vt)

gerryliyana
 2 years ago
Best ResponseYou've already chosen the best response.1ok.., i got it now :)

gerryliyana
 2 years ago
Best ResponseYou've already chosen the best response.1wanna help me with Legendre series ??

Meepi
 2 years ago
Best ResponseYou've already chosen the best response.1Haven't really done much of that but I can give it a shot :)

gerryliyana
 2 years ago
Best ResponseYou've already chosen the best response.1go to http://openstudy.com/study#/updates/5172d293e4b0f872395bc5be
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