# What is more efficient in carrying energy: Longitudinal waves or transverse waves?

Let's look at the wave equation in the $\text{1D}$ case, for a transverse wave in a string:

$$\frac{\partial^2 \psi(x,t)}{\partial t^2}=v^2\frac{\partial^2 \psi(x,t)}{\partial x^2}\tag{1}$$

Or in shorthand:

$$\psi_{tt}=v^2\psi_{xx}$$

where $v$ is the propagation speed of the wave.

The solutions of $(1)$ can be found in the link provided. They are of the form:

$$\psi(x,t)=Ae^{i(\pm kx\pm\omega t)}$$

where:

$$\frac{\omega}{k}=v$$

The power transmitted by the string wave is given by:

$$\boxed{P=\frac12 \mu \omega^2 A^2 v}$$

where:

- $\mu$ is the mass of the string per unit length
- $\omega$ the angular velocity of the wave
- $A$ the amplitude of the wave

*All other things being equal*, longitudinal and transverse string waves transmit the same amount of power.

As Gert shows, the power in the wave is the same for both longitudinal and transversal waves on a string. However, the speed at which this power propagates differs for both cases.

The speed at which the power of a wave propagates is equal to the group velocity $v_\mathrm{g}=\frac{\partial \omega}{\partial k}$ of the wave. For elastic waves in an isotropic medium, the transversal wave speed is given by $v_\perp=\sqrt{c_{44}/\rho}$ and the longitudinal wave speed by $v_\perp=\sqrt{c_{11}/\rho}$ with $c_{ii}$ the stiffness constants and $\rho$ the mass density. Hence, the longitudinal wave is faster because $c_{11}>c_{44}$ and thus transports its power also faster through space.