Solving the Damped Wave Equation: A Study of u(x,t) and its Derivatives

[Buddy711]

For a traveling wave


$$u(x,t) = u(x-ct)$$

How is the relation below hold?
$$u_{x}u_{xt}=-u_tu_{xx}$$

I don't understand why there is (-) sign .

Thanks in advance !

PS.
Here is the URL of the book I am having trouble with
https://www.amazon.com/dp/0198528523/?tag=pfamazon01-20

of the book and it is written on page 436 for the last equation.
You can search with this keyword :
"consider the damped wave equation"

 

[JJacquelin]

I don't understand why there is (-) sign

Me too. I think that there should be no (-).

 

[LCKurtz]

I think you have misunderstood what the book is doing. First it differentiates the energy integral with respect to t:

$$E(t) = \int_{-\infty}^\infty \frac 1 2 (u_x^2+u_t^2)\ dx$$
This gives
$$E'(t) = \int_{-\infty}^\infty u_xu_{xt}+u_tu_{tt} dx$$

Now look at the first term, rewriting u_xt as u_tx:
$$\int_{-\infty}^\infty u_xu_{tx}\ dx$$
Integrate it by parts letting
U = u_x, dV = u_txdx
dU = u_xxdx, V = u_t
This gives
$$\int_{-\infty}^\infty u_xu_{tx}\ dx = u_xu_t|_{-\infty}^{\infty} -\int_{-\infty}^{\infty} u_tu_{xx}\ dx$$

I'm guessing from the context (that's where you come in) that the part in the evaluation bracket goes to zero. If so, substitute this term back in for the first term in the integral:

$$E'(t) = \int_{-\infty}^\infty -u_tu_{xx}+u_tu_{tt} dx=<br /> \int_{-\infty}^\infty u_t(-u_{xx}+u_{tt}) dx$$