Show that f(x,t)=(x-vt)^3+sin(x-vt) satisfies the one dimensional wave equation.

Question

Show that f(x,t)=(x-vt)^3+sin(x-vt) satisfies the one dimensional wave equation.

Answer 1

Hi, this is easy to show if you follow the steps below:

1. open the (x-vt)³ according to (a-b)³ formula,
2. Calculate df/dx (partial derivative) and d²f/dx² (partial second derivative), if everything is done correctly you get 6x+6vt-sin(x-vt)
3. Calculate df/dt (partial derivative) and d²f/dt² (partial second derivative), if everything is done correctly you get 6xv²+6v³t-v²sin(x-vt)
4. now substitute d²f/dx² and d²f/dt² you found in steps 2 and 3 into one dimensional wave equation (d²f/dx² -(1/v²) d²f/dt² =0, you will see that everything cancels out and you get 0, therefore that function f(x,t) satisfies the one dimensional wave equation.

Hope this helped, if something is unclear let me know.