Transverse Waves

Pulse Equation

\[v_y = \frac{\partial y}{\partial t} = \mp v\frac{\partial y}{\partial x}\]

The pulse equation relates the y velocity of particles in the medium ($v_y=\frac{\partial y}{\partial t}$) with the speed ($v$) and slope of the wave ($\frac{\partial y}{\partial x}$). This suggests a given particle in a transverse wave is moving fastest when its vertical position is at equilibrium and the wave has the greatest slope.

Ideal Wave Equation

\[a_y = \frac{\partial^2 y}{\partial t^2} = v^2\frac{\partial^2 y}{\partial x^2}\]

The ideal wave equation relates the y acceleration of particles in the medium ($a_y=\frac{\partial^2 y}{\partial t^2}$) with the speed and curvature of the wave ($\frac{\partial^2 y}{\partial x^2}$). This suggests a given particle in a transverse wave has the greatest acceleration at the peak or trough.

Ideal Transverse Waves on a String

For ideal transverse waves on a string, we make the following assumptions and approximations:

• Small slope and small angles
• Constant tension throughout
• The string has mass, but there is no force of gravity or damping

We use $\mu$ for the mass of string per unit length and $\tau$ for tension along the string.

\[a_y = \frac{\partial^2 y}{\partial t^2} = \frac{\tau}{\mu}\frac{\partial^2 y}{\partial x^2}\] \[v = \sqrt{\frac{\tau}{\mu}}\]