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The mechanical displacements in piezoelectric materials carry along macroscopic electric fields, allowing tunneling of acoustic waves across a vacuum gap beyond the charge-charge interaction distance. However, no rigorous proof of complete acoustic wave tunneling has been presented, and the conditions to achieve complete tunneling have not been identified. Here, we demonstrate analytically the condition for such phenomenon for arbitrary anisotropic crystal symmetries and orientations, and that complete transmission of the incoming wave occurs at the excitation frequency of leaky surface waves. We also show that the complete transmission condition can be related to the surface electric impedance and the effective surface permittivity of the piezoelectric material, relevant to realize the complete tunneling experimentally. We support our findings with numerical results for the maximum power transmittance of a slow transverse wave tunneling between identical ZnO crystals. The results show that complete tunneling can be achieved for a large range of orientations.
Acoustic waves (acoustic phonons) are deformations or vibrations propagating through a material medium. As such, they do not exist in vacuum, leading to the initial conclusion that it is impossible for the vacuum to transmit the energy of an acoustic wave between two separated media. However, at the atomic scale the vibrations of the nuclei can propagate via their electrical interactions through vacuum. Thus, a question can be raised, whether acoustic phonons can also be transmitted across larger than atomic scale vacuum gaps through some electromagnetic mechanism. This is a relevant question, as with the advances in experimental techniques, nanometer to sub-nanometer scale vacuum gaps can be achieved1,2,3,4. The possibility of such acoustic phonon tunneling, as it is often called in the literature, has attracted a considerable amount of theoretical work in recent years to investigate possible mechanisms of the effect such as Casimir and van der Waals forces, particularly in the context of near-field heat transfer5,6,7,8,9,10,11,12,13,14,15,16,17,18.
In this work, we focus on the power transmittance of acoustic wave tunneling. We use the general formalism developed for piezoelectric acoustic wave tunneling23 to analytically prove the existence of the complete tunneling phenomenon between two vacuum separated identical solids. In addition, a resonant tunneling condition is also derived, corresponding to the excitation of leaky surface waves. We also propose that this condition could be checked experimentally. Further discussion of the results are presented with a few numerical examples for ZnO crystals. In particular, we find our results differ from those obtained before5.
Furthermore, if we assume that the two solids consist of the same material with identical crystal orientations, two additional relations that link the single surface coefficients of the two solids can be found by exploiting the completeness of the eigensolutions of the scattering problem (see Supplementary Note 2 for the derivations). The first one relates the reflection coefficients \(\barr_\rmV^(i)\) of the two solids as:
where the equality is satisfied when there exists only one transmitted bulk wave mode in solid 2 and the mode is the same as the incident wave in solid 1. By applying the relations (2) and (3), PΣ can then be simplified to:
Our numerical formalism can also be applied to the particular case studied with a simplified model before5, the details of which can be found in Supplementary Note 5. We do not find complete tunneling for the incoming modes and the crystal orientation in question, in contradiction to previous work5.
In conclusion, we have analytically and numerically proven it is possible for acoustic waves to completely tunnel across a vacuum gap between two piezoelectric solids, up to gap sizes of about a wavelength. We showed that such complete tunneling, with unity power transmittance, is possible only if one transmitted partial bulk mode is excited, it being the same mode as the incident wave. We derived a simple resonance tunneling condition for the complete tunneling effect, Eq. (8), and proved its validity and range of applicability with numerical examples for arbitrarily rotated ZnO crystals. As this is a strong and not a rare effect, it could have an impact in future acoustic wave devices, as well as in other application areas concerning phonons, such as controlling heat transport, optomechanics and quantum information science.
An incident plane wave is scattered into a linear combination of partial waves at an interface, which are either reflected or transmitted. The general solutions of such partial waves that satisfy the governing equations take the expressions34,35,36:
In this study, we solved these governing equations under the framework of extended Stroh formalism23, in which Eq. (10) is combined and rearranged into an eight-dimensional eigenvalue problem37,38 in the form of:
The time-averaged transmitted power flow density in the direction normal to the surfaces from solid 1 to 2 can be expressed by the real part of the piezoelectric Poynting vector in the normal direction34:
Knowing how to complete The Hidden Truth in Wuthering Waves will put you one step closer to completing the Huanglong I Trophy set. However, this isn't actually the name of the quest you need to complete!
As with most things in Wuthering Waves, earning this Trophy isn't as simple as it first appears. You'll need to head to Wuming Bay and complete a rather long side quest to get your hands on it. Don't worry though, we're here to help you every step of the way.
To start A True Arena Champion side quest in Wuthering Waves you need to head to the small island between Shattered Blocks and Remnant Residence in Wuming Bay south east of Jinzhou. We've marked the exact location on the map below:
When you get to this island, you'll be instructed to rescue the trapped people. Take down the Tacet Discords here and then speak to Zeyuan (the person with the marker above their head). They'll then tell you about the Seaside Arena, agree to help them and you'll start the A True Arena Champion side quest.
Speak to Zeyuan again here, then walk in through the gate and speak to the person on the right. Then, from the gate, head forwards and speak to the person at the top of the ramp here. Once you've done all of this, check your map for a marker and follow it to speak to Zeyuan to get your Camp Pass.
Then, open your map to check for the next quest marker. Follow this and speak to the Receptionist here who'll reveal you'll need to earn three badges to register for the Seaside Arena. To do this, you'll need to complete three trials...
Speak to the Receptionist to begin your second trial which is all about speed. At this point, we highly recommend changing your team to include a Resonator that uses ranged weapons or guns, like Chixia, as it makes this trial far easier - though it is possible to do it with Rover.
The trainer for the third trial can only be found after 5pm in-game time. If you're too early, time skip to the correct time period and then head to the marker on the map at the tip of Remnant Residence:
Once you've defeated them, open your map to see the quest marker has moved elsewhere on Remnant Residence. Head to this marker to speak to the Receptionist to get your Strength Badge.
For each round we recommend focusing your attacks on the Echoes first and then turning your attention to the human enemies. Make sure to keep an eye on both enemies in each round and dodge any incoming attacks they send your way.
In the first round you'll face the Champion and two Rocksteady Guardians. These enemies will gain a shield once they enter the Beserk State. When you break their shield, the small white bar underneath their health bars, their Vibration Strength will be significantly lowered. They do pack a punch though, so we recommend luring them away from the Champion and each other as much as you can when facing them.
We found it easiest to target each Guardian alone with basic, intro and outro attacks. However, using Resonance Liberation and Forte Circuit skills when they're together will highly damage both of them.
Their incoming attacks are quite predictable, so if you see them lifting their arms or charging up you need to get away from them or dash and dodge to the side to avoid impact. Remember to watch for the Champion, he's got a habit of sneaking up on you while you deal with his Guardians.
The six echoes are connected to the Champion and, luckily, most of them come into the field with an already damaged health bar. Focus all of your energy on taking out all six Echoes while dodging their incoming attacks. The Champion does seem to move faster in this round, so make sure you keep an eye out for him.
Now that you've got the Trophy, head into your 'Trophy' tab in the pause menu and navigate to the Huanglong I Trophy section. All you've got to do now is select 'Claim' to tick this off your to-do list!
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