Energy of Waves

Waves on a String

Kinetic Energy

Kinetic energy density is

\[u_k = \frac{1}{2}\mu\left(\frac{\partial y}{\partial t}\right)^2\]

where $\mu$ is the mass per unit length of the string. Notice how this follows the kinetic energy equation

\[K = \frac{1}{2}mv^2\]

Where $\frac{\partial y}{\partial t}$ is the transverse velocity.

Be careful that first equation is the kinetic energy density so to get the total kinetic energy, you need to find the velocity for each segment and multiply.

Potential Energy

Potential energy density is

\[u_p=\frac{1}{2}\tau\left(\frac{\partial y}{\partial x}\right)^2\]

where $\tau$ is the tension.

This makes sense because the slope of the string, $\frac{\partial y}{\partial x}$ is proportional with the stretch of the string which stores energy.

For waves in only one direction, $u_p = u_k$.

Power

\[P(x, t) = -\tau\frac{\partial y}{\partial x}\frac{\partial y}{\partial t}\]

Sound Waves

\[u_k = \frac{1}{2}\rho\left(\frac{\partial s}{\partial t}\right)^2\] \[u_p = \frac{1}{2}B\left(\frac{\partial s}{\partial x}\right)^2\]