# I d'Alembert's solution to the wave equation

#### bksree

Summary
It is concluded that
df/d zeta = df/dx and df/d eta = dg/dx
Will someone explain how this conclusion is made while differentiating
Hi
On page 81 of the book "A student's guide to waves by Fleisch and Kinneman a conclusion is made while differentiating D Alembert's solution to the wave equation.
Will someone explain this please ? The details are in the attachment

TIA

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#### vanhees71

Gold Member
I find the introduction of $\xi$ and $\eta$ as confusing. It's simply the chain rule of differentiation:
$$\partial_t y(t,x)=\partial_t [f(x+v t)+g(x+v t)]=v f'(x+v t)-v g'(x+v t),$$
where $f'$ is the derivative of the function $f:\mathbb{R} \rightarrow \mathbb{R}$ wrt. to its argument.

#### bksree

Hi
Thank you.
My doubt is :
How is it concluded that (all partial derivatives) df/dξ = df/dx and df/dη = dg/dx

TIA

#### vanhees71

Gold Member
Of course you have
$$f'(x+v t)=\partial_x f(x+v t)$$
etc...

#### bksree

Is this what you mean ?
∂ f/∂ ξ = ∂ f/∂ (x+vt)
= ∂f /∂ x at t = 0

TIA

Last edited:

#### vanhees71

Gold Member
It's valid for any $t$. Note that the independent two variables in the problem are $t$ and $x$, and thus $\partial_x$ means the derivative of a function wrt. $x$ with $t$ hold constant. It's all just the chain rule. Writing
$$f(t,x)=f[\xi(t,x)]$$
you have
$$\partial_t f(t,x)=\frac{\mathrm{d}}{\mathrm{d} \xi} f[\xi(t,x)] \partial_t \xi, \quad \partial_x f(t,x)=\frac{\mathrm{d}}{\mathrm{d} \xi} f[\xi(t,x)] \partial_x \xi.$$
Now take
$$\xi(t,x)=x+v t.$$
NB: Note that there's a comfortable LaTeX editor in PF (using mathJax), which gives much better readable math:

"d'Alembert's solution to the wave equation"

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