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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.

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    $\begingroup$ I don’t understand what you mean by “efficiency”. For example, 100% of a light wave’s energy is transported by a transverse wave. Does that make it perfectly “efficient”? $\endgroup$ – G. Smith Jul 1 at 4:50
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    $\begingroup$ Along the same line, 100% of a sound wave's energy (in air) is transported by a longitudinal wave. The answer is probably medium-dependent, but a proof might make an interesting answer. $\endgroup$ – rob Jul 1 at 5:57
  • $\begingroup$ @G.Smith Yes, I apologize for it being a little unclear. Actually, I could not properly state what I was thinking. But, I guess power is what I meant because it would tell how fast the energy is transmitted, and that is what I mean by efficiency. $\endgroup$ – Abhinav Dhawan Jul 1 at 6:39
  • $\begingroup$ I've removed a number of comments that were attempting to answer the question and/or responses to them. (I left a few in place which were kind of close; those may be cleaned up later.) Please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. $\endgroup$ – David Z Jul 1 at 19:37
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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.

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    $\begingroup$ What is $\frac{\partial^2 \psi(y,t)}{\partial x^2}$ supposed to mean? You can’t differentiate a function of $y$ and $t$ with respect to $x$. $\endgroup$ – G. Smith Jul 1 at 5:48
  • $\begingroup$ @G.Smith Oopsie! Haste makes waste. Edited. Ta. $\endgroup$ – Gert Jul 1 at 6:29
  • $\begingroup$ @Gert Will the same formula for power work for longitudinal waves also? $\endgroup$ – Abhinav Dhawan Jul 1 at 6:36
  • $\begingroup$ You probably also mean $\psi_{tt}$. $\endgroup$ – Toffomat Jul 1 at 10:27
  • $\begingroup$ And $v^2$ should also be in the numerator, I think... $\endgroup$ – Abhinav Dhawan Jul 1 at 11:08
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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.

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  • $\begingroup$ Do you mean to say that second derivative of energy varies for longitudinal and transverse waves ? And I don't understand what c11 and c44 are...could you please elaborate... $\endgroup$ – Abhinav Dhawan Jul 1 at 9:19
  • $\begingroup$ The wave propagates with a specific speed and as the wave contains energy, also the energy propagates with this speed. The $c_{ii}$ constants are constants that characterize the stiffness of a material and determine the elastic wave speed. $\endgroup$ – Frederic Jul 1 at 9:30

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