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

Question

I was wondering, as my question suggests, that which kind of waves are more efficient in transporting energy, considering longitudinal and transverse waves. I checked out this post here, but didn't get a satisfactory answer.

What I think is that because for longitudinal waves, because the motion of particles is along the direction of energy transfer, it should be more efficient. This is because on the other hand for transverse waves, the elastic forces responsible for transfer of motion of one particle to the other would be at an angle and that seems less efficient.

In other words, for longitudinal waves both the force which transfers energy and the restoring force that pulls the particle back are not at an angle ( unlike transverse waves ).

I know that my reasoning is a bit vague which makes me unsure about this. I would be happy if someone can resolve this query of mine for a better understanding of the subject.

Thanks.

Answer 1

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.

Answer 2

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.