Chapter 16, Problem 16.4CYU

Check Your Understanding The wave equation $\frac{{\partial }^{2}y\left(x,t\right)}{\partial x^2}=\frac{1}{v^2}\frac{{\partial }^{2}y\left(x,t\right)}{\partial t^2}$ works for any wave of the

form $y\left(x,t\right)=f\left(x\mp vt\right)$ . In the previous section, we stated that a cosine function could also be used to model a simple harmonic mechanical wave. Check if the wave

   $y\left(x,t\right)=0.50\text{m}\cos \left(0.20\pi {\text{m}}^{-1}x-4.00\pi {\text{s}}^{-1}t+\frac{\pi }{10}\right)$

is a solution to the wave equation.

Any disturbance that complies with the wave equation can propagate as a wave moving along the x-axis with a wave speed v. It works equally well for waves on a string, sound waves, and electromagnetic waves. This equation is extremely useful. For example, it can be used to show that electromagnetic waves move at the speed of light.