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bksree

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- TL;DR Summary
- Basic doubt on applying chain rule in DAlemberts soln to wave equation

In D Alembert's soln to wave equation two new variables are defined

##\xi## = x - vt

##\eta## = x + vt

x is therefore a function of ##\xi## , ##\eta## , v and t.

For fixed phase speed, v and given instant of time, x is a function of ##\xi## and ##\eta##.

Therefore partial derivative of x w.r.t y is (using the chain rule)

##\frac {\partial x} {\partial y} = \frac {\partial x} {\partial \xi} \frac {\partial \xi} {\partial y} + \frac {\partial x} {\partial \eta} \frac {\partial \eta} {\partial y} ##

But how to find ##\frac {\partial y} {\partial x}## ?

Here x (the independent variable) is function of ##\xi## and ##\eta##

(The chain rule says that if f(r,s) - dependant variable - then

##

\frac {\partial f} {\partial t} = \frac {\partial f} {\partial r}\frac {\partial r} {\partial t} + \frac {\partial f} {\partial s}\frac {\partial s} {\partial t} ##

TIA

##\xi## = x - vt

##\eta## = x + vt

x is therefore a function of ##\xi## , ##\eta## , v and t.

For fixed phase speed, v and given instant of time, x is a function of ##\xi## and ##\eta##.

Therefore partial derivative of x w.r.t y is (using the chain rule)

##\frac {\partial x} {\partial y} = \frac {\partial x} {\partial \xi} \frac {\partial \xi} {\partial y} + \frac {\partial x} {\partial \eta} \frac {\partial \eta} {\partial y} ##

But how to find ##\frac {\partial y} {\partial x}## ?

Here x (the independent variable) is function of ##\xi## and ##\eta##

(The chain rule says that if f(r,s) - dependant variable - then

##

\frac {\partial f} {\partial t} = \frac {\partial f} {\partial r}\frac {\partial r} {\partial t} + \frac {\partial f} {\partial s}\frac {\partial s} {\partial t} ##

TIA