HW 4: Wave Equation and Classical Waves

Question-1¶

Plug the following expression into 1D wave equation demonstrating that function $u(x,t)$ obeys wave equation and is a general solutions expressed as waves going in opposite directions via functions which depending on $x\pm vt$

$u(x,t) = f(x+vt)+g(x-vt)$

Question-2¶

• Show mathematically that a linear combination of two traveling waves going in the opposite direction produces a standing wave. E.g illustrate this using simple sin or cos peridic waves.

• Can one express traveling waves via standing waves?

Question-3¶

• Which of the following expressions correspond to traveling waves?

• Pick one traveling wave and show that it obeys the wave equation then identify the wavelength period and velocity of that wave.

• $u(x,t)=(ax-bt)^2$

• $u(x,t)=(ax+bt+c)^2$

• $u(x,t)=\frac{1}{ax^2+b}$

• $u(x,t)=e^{-(a^2x^2-b^2t^2-2abtx)}$

Question-4¶

Find the general solution to the following ODEs:

• $\frac{d^2 y}{dt^2}-4\frac{dy}{dt}+3y=0$

• $\frac{d^2 y}{dx^2}+6\frac{dy}{dx}=0$

Question-5¶

• Solve the following wave equation by applying boundary condition $\psi(0)=\psi(a)=0$ and find expression of $\psi(x)$ where coeffficient $k$ will depend on an integer number as a result of satisfying boundary conditions