### Description

## Title of Invention : THE CALCULATION METHOD OF WAVE REFLECTIVE INDEX ON INTERFACE

[0001]

The invention relate to both Classics light for duct calculation and basics wave theory for applying. The methods of this invention will be widely used that wave spread through the interface in various fields such as light, electromagnetic waves, sound waves, water wave and so on for duct calculation.

### BACKGROUND OF THE INVENTION

[0002]

About reflective index calculating problem

[0003]

Classic Fresnel equation calculated both water reflective index is 2％and glass reflective index is 4％in normal on interface, it has a big difference with reality fact. The paper we invent wave reflective index calculation method from normal incident wave on interface to solve reflective index calculating value that is little problem.

[0004]

About reflective critical angle calculating problem

[0005]

It is in connection with the defect of calculating reflection critical angle with Snell’s law, because just Snell’s law cannot calculate wave propagation from the air to the matter of reflective critical Angle problem, since now people have no other good ways to solve this problem, so this invent is a method to solve the practical problem. When calculating reflected critical angle with the Snell's law in tradition, delimits refraction angle is 90°, incidence angle which out of substance is reflective critical angle, therefore the light wave incident into substance has no reflected critical angle, all the wave energy can refract into substance. The concept above is wrong although it has been used many years by people. The wave refraction from water to the air can never coincide with the wave of the angle that refraction is 90°, it’s means people in water have never seen the target or the ship on the sea.

[0006]

The question was presented by 20 years before, when author engaged in marine engineering calculations, who found that the wave is going to be reflected besides the waterway then superpose and pile up with incident wave before the bulwark, and formed an unusual large wave, as a result of waterway excavated when the angle of both incident wave direction and the waterway is overmuch small. Meanwhile the wave in waterway will be reflected and stacked on the other side of the waterway too. Therefore the phenomenon of the wave been reflected and stacked will always exist whatever the wave travel from shallow water to deep water or opposites too.

[0007]

In the research of atmosphere duct soon after, when inversion temperature layer appear in the stable stratification structure of atmosphere, it is found that resulting in the electromagnetic wave reflected back to the sea level, where the electromagnetic wave launched from the ship radar which parallel to the horizontal plane intersects with the interface of inversion temperature layer in a very small angle (about 0.1°) cause of effect from the curvature of the earth. In order to achieve, the radar on ship can probe the target out of horizon, and this above is ducting effect. The ducting is made electromagnetic wave reflection that travel from Low-temperature air layer to high-temperature air layer, and the mirage in desert caused by waveguide that light wave travel from high-temperature air layer to Low-temperature air layer.

[0008]

The wave spread in ocean has the same effect of the phenomenon of both ocean acoustic waveguide and dead zone. When Thermocline appears seasonal at Coastal by cold water roll in china yellow &bohal sea in summer, cause of thermocline, the ship sonar on the surface of the sea is hard to probe the target which is at a few hundred meters away under the Thermocline water. The underwater sonar or the hydrophone is also hard to probe the sound of ship in a few hundreds away. The distance has a big difference with the rated detection distance 10-20 km by sonar.

[0009]

Whatever water wave or electromagnetic wave or sound wave, if there's wave travel speed difference interface exist, even if the difference is very small, reflected wave will appear when angle of both the incident wave and the interface is small enough, proved by experiments. No matter the wave travel from high wave velocity media to low wave velocity media or reverse, the reflected critical angle always exist.

[0010]

And for wave, as long as the angle of both the incident wave and the interface is small enough, the rebound wave of "stone skimming effect" exist both sides of the interface. And reflective region divide both absolute reflective region and relative reflective region.

[0011]

SUMMARY OF THE INVENTION

[0012]

The reflection wave being compressed wavelength three section calculation methods

[0013]

When normal component of (or m times) incident wavelength on interface equal to a quarter value of refractive wavelength in the medium, the incident wave energy would be all reflected. The reason when normal incident wavelength is compressed to a quarter value of refractive wavelength in the medium, the incident wave energy would be all reflected (or trapped) .

[0014]

When normal component of (or m times) incident wavelength on interface equal to a half value of refractive wavelength in the medium, a half of the incident wave energy would be reflected. The reason when normal incident wavelength is compressed to a half value of refractive wavelength in the medium, a half of the incident wave energy would be reflected

[0015]

When normal component of (or m times) incident wavelength on interface equal to resonance wavelength nearby three-quarter of value of refractive wavelength in the medium, the incident wave energy would begin to be reflected. The reason when normal incident wavelength is compressed to resonance wavelength nearby three-quarter value of refractive wavelength in the medium, the incident wave energy would begin to be reflected.

### BRIEF DESCRIPTION OF THE DRAWINGS

[0016]

For a more complete understanding of the present invention, and the advantages thereof, reference is now made to the following description taken in conjunction with the accompanying drawings, in which like numbers designate like parts, and in which:

[0017]

FIGURE 1 Reflective index and being compressed wavelength relation sketch map on interface；

[0018]

FIGURE 2 The phenomenon of light waves travel from water into air；

[0019]

FIGURE 3 The phenomenon of light waves travel from air into water；

[0020]

FIGURE 4 The application of ocean acoustic waveguide.

### DETAILED DESCRIPTION OF THE INVENTION

[0021]

1. The algorithms for absolute reflection critical angle

[0022]

When the wave speeds difference of both sides of the interface is small and refractive index close to 1, normal component of incident a wavelength can't satisfy a quarter refractive the wavelength in the medium, it is needed methods for the question. Following Stem wave study, we receive to enlighten.

[0023]

Absolute reflection critical angle calculation method:

[0024]

When refractive index

the normal component of m times incident wavelength on interface equal to a quarter value of refractive wavelength in the medium, this moment incident wave energy all reflection. The angle is absolute reflective critical angle in weak interface. Its expression

the formula is

Coefficient of wave individual number m is integer. Due to

so

It other is called generalized a quarter wavelength reflection critical Angle method.

[0025]

When Coefficient of wave individual number m＝1, refractive index

the normal component of incident wave velocity on the interface equal to a quarter value of refractive wave velocity in the medium, this moment incident wave energy all reflection. The angle is absolute reflection critical angle. Its expression c

_{in}
the formula is

； the c

_{in} is the incidence wave velocity； the c

_{refra} is the refraction wave velocity； the θ

_{abs} is the wave absolute reflection critical angle. Other name of the effect is called a quarter wavelength critical angle calculation method.

[0026]

The calculating conditions of this method is the incident wave which needs to propagate from high wave velocity medium to low wave velocity medium, that means c
_{in}＞c
_{refra}. Through calculating by this method the result of we get the absolute reflection critical angle is in side of high wave velocity medium (eg. air)

[0027]

If you want to calculate the absolute reflection critical angle of low wave velocity media (eg. substance) , first calculating high wave velocity media by the method in this paper, afterwards you can use Snell’s law to calculating since the reversibility of wave.

[0028]

When n＝1.25, the absolute reflection critical angle is smallest, the value is 78°27`, that means the angle of both the interface and the incidence wave is largest, the value is 11°33`；

[0029]

When n＝4, the value of absolute reflection critical angle is 88°25`, we can calculate out the value of the reflects the critical angle of wave which reverse out of the material is 14°29` by the Snell’s law. And this result is almost same as the result calculated by the Snell’s law when the angle of refraction is 90°

[0030]

When n＝1.0063, meanwhile m＝40 times wavelength, the value of absolute reflection angle is 89°38′, nearly to 90°.

[0031]

Absolute reflection critical angle calculation case detailed description

[0032]

This method in the experiment of light from the air into the water, can be applied to verify. Should have this method can be calculated, on water reflection absolutely critical Angle value is 79 ° 10 `. With middle school physics experiment equipment, but can also clear validation light waves into the water at 79 ° to 80 ° sliding hop by total reflection. When light incident Angle into water from 79 ° to 80 °, light wave image suddenness disappear in the water, without any trace.

[0033]

Now we compare the critical angles of two the result calculated by the Snell’s law (when the angle of refraction is 90°) and by this method to illustrate the method of this paper rationality. See the table 1 below.

[0034]

Table 1 List different refractive index two methods be compared to calculate the critical Angle

[0036]

Derived from table 1: there’s little differences between the results of these two theorems, when n＝1.25, m＝1, we got the largest difference value of 1°41′. It shows that this method and extreme conditions (non-existent) Snell law is close to the calculated results, proved this method has its rationality and practicability. At least this method is used to calculate is closer to the real objective facts.

[0037]

That proved the value of critical angle calculated by Snell’s law is closed to real threshold, that’s the reason of why there has no person to put forward any objection when the Snell’s law has been used widely in the past 300 years. But just cause of this little difference, there exist a reflection critical angle when the light wave spread from the air into substance, not all parts of the light wave energy can refract into substance.

[0038]

2. The algorithms for Relative (or resonance) reflection critical Angle

[0039]

When the wave speeds difference of both sides of the interface is small and refractive index close to 1, normal component of incident a wavelength can't satisfy a quarter refractive the wavelength in the medium.

[0040]

Resonance critical angle calculation method:

[0041]

When wave refractive index

the normal component of m times incident wavelength on interface take part in resonating, the wave begin to resonate in the interface, this moment incident angle is relative reflection critical angle. The calculation method is

Coefficient of wave individual number m is integer, due to

so

The θ

_{resonance} is wave relative reflection critical angle

[0042]

Proof: any wave pass through interface of the medium follows the Snell’s law, and the resonant wave on interface follows that the normal components of m times the incidence wave velocity equal to the normal components of refraction wave the velocity. Its expression are two equations by both

and mc

_{in}cosθ

_{in}＝c

_{refra}cosθ

_{refra}. To derived solving two equations to eliminate refraction term, last resolution to obtain

It is over！

[0043]

The method can be calculated relative reflection critical angle of both the interface in natural. This theorem is also known as generalized resonance critical angle method.

[0044]

When Coefficient of wave individual number m＝1, n≥1.25, wave begin to reflection , this moment resonance reflection critical angle calculating method is tgθ
_{resonance}＝n, this is called resonance critical angle calculation method.

[0045]

The ultra weak interface calculation method

[0046]

For ultra weak interface waveguide, m ＝ 8, n ＝ 1.0313, it is absolutely critical Angle is: 88°16`, and relative critical Angle is: 88°17`, the value of relative critical Angle is just more than the value of absolutely critical Angle. For the ultra weak interface (n≤1.0313) we can’ t consider the existence of relative critical Angle, only calculate absolute critical Angle.

[0047]

Absolute and relative reflection critical angle calculation case detailed description

[0048]

Table2 Different refractive index both absolute and relative reflection critical angle compared by two methods in this paper calculating

[0049]

[Table 0001]

Material |
m＝2 |
m＝1 |
Ice |
Water |
Glass |
Crystal |
Diamond |
m＝1 |

refractive index (n) |
1.125 |
1.25 |
1.309 |
1.333 |
1.5 |
2 |
2.417 |
4 |

Absolute reflection critical angle out r of matte |
62°03 |
51°27 |
48°34` |
47°20` |
41°10` |
29°45` |
24°17` |
14°29 |

Relative reflection critical angle out of matter |
60°33 |
38°40 |
37°17 |
36°43 |
33°42 |
26°34 |
22°29 |
14°04 |

Difference both absolute and angle out of material |
1°30 |
12°47 |
11°17 |
10°37 |
7°28 |
3°11 |
1°48 |
0°25 |

Into the material absolutely critical angle |
83°33 |
78°27 |
79° |
79°10` |
80°40` |
82°49` |
83°40` |
88°25 |

Into the material relative reflection critical angle |
78°24 |
51°21 |
52°27 |
52°50 |
56°29 |
63°26 |
67°30 |
85°35 |

Into the material difference of critical angle |
5°18 |
27°06 |
26°33 |
26°20 |
24°11 |
19°23 |
16°10 |
2°50 |

[0050]

Derived from table 2: When the wave through out of the substance , absolute and relative reflection critical angle are reducing when the refractive index is increasing. The difference value make a inflection point at the time refractive index is 1.25. When the wave travel into the material, the absolute and relative reflection critical angle make a inflection point, the value of both the angles in refractive index 1.25 reaches its maximum value. This shows whether refractive index increases or decreases begin at the turning point, two critical angles of convergence, with the refractive index increases or decreases up to a certain value are two critical angle approximations.

[0051]

3. The algorithms for symmetry point of refractive-reflective incident wave angle

[0052]

When refractive index

the normal component of m times incident wavelength on interface equal to a half value of refractive wavelength in the medium, this moment incident wave angle is wave energy symmetry reflective critical angle in weak interface. Its expression

the calculation method is

where refractive index

where Coefficient of wave individual number m is integer. For

so

It other is called generalized symmetry point of refractive-reflective wave energy Angle calculation method or inflection point calculation method.

[0053]

When Coefficient of wave individual number m＝1, refractive index

the normal component of incident wave velocity on the interface equal to a half value of refractive wave velocity in the medium, this moment reflection wave energy equal to refraction wave energy.

the calculation method is

This method is also known as symmetry point wave energy angle calculation method of refractive-reflective.

[0054]

4. The methods for wave reflective index normal incident wave on interface

[0055]

4.1 Wave coefficient of resonance calculation method

[0056]

In accordance with both absolute reflection critical angle and symmetry refractive-reflective wave energy angle expression form， resonance critical angle can be expressed:

[0057]

Because of the know relative reflection critical angle calculation method: tgθ

_{resonance}＝n, coefficient of resonance z

_{h} could be determined by z

_{h}＝16 (ncos (arctg (n) ) -0.75) .

[0058]

4.2 Wave refractive index calculation method

[0059]

Assumption wave reflective index from value of zero to value of 0.5 has presented linear increase, following rate of the wavelength from resonance point to a half wavelength point.

[0060]

When wave refractive index in2≥n≥1, Wave reflective index R

_{f} could expressed:

When R

_{f}＝0， so calculating to n＝1.2961, when wave refractive index n≤1.2961, wave energy has no reflected in normal incident wave on interface and the wave transmission the interface refracted in medium.

[0061]

Assumption wave reflective index from the value of 0.5 to the value of 1 has presented linear increase, following rate of the wavelength from a half wavelength point to a quarter wavelength point.

[0062]

When wave refractive index in 4≥n≥2, Wave reflective index R
_{f} could expressed:

[0063]

[Math. 0001]

[0064]

When wave refractive index in n≥4, Wave reflective index R
_{f} could expressed: R
_{f}＝1.

[0065]

table 3, Wave reflective index calculation insults by the method for common substances following refractive index

[0067]

As shown in table 3, along with Refractive index (n) growing, the value of both Coefficient of resonance and Wave reflective index increase.

[0068]

About reflective index calculating method technical effect

[0069]

Classic Fresnel equation calculated both water reflective index only is 2％and glass reflective index only is 4％in normal on interface, it is little that has a big difference with reality fact. The invention we found wave reflective index calculation method to solve reflective index calculating more little problem from normal incident wave on interface. To calculate results of both water reflective index is 9％and glass reflective index is 25％in normal on interface by this invent reflective index method.

[0070]

About reflective critical angle calculating method technical effect,

[0071]

The invent methods for absolute and Relative reflection critical Angle may calculate not only out of the medium, but also into the medium absolute and Relative reflection critical Angle, this is not calculated by Snell’s law for the critical Angle. into medium To compare the critical angles of two the result calculated by the Snell’s law (when the angle of refraction is 90°) and by the absolute reflective critical angle method to illustrate the method of this paper rationality for out of the substance. There’s little differences between the results of these two methods. It shows that this method and extreme conditions (non-existent) Snell’s law is close to the calculated results, proved this method has its rationality and practicability. At least this method is used to calculate closer to the real objective facts. That proved the value of critical angle calculated by Snell’s law is closed to real threshold, that’s the reason of why there has no person to put forward any objection when the Snell’s law has been used widely in the past 300 years. But just cause of this little difference, there exist a reflection critical angle when the light wave spread from the air into substance, not all parts of the light wave energy can refract into substance.

[0072]

Invent content to apply example

[0074]

The phenomenon of light waves travel from water to air for Fig 2:

[0075]

When m＝1, n＝1.3333, when the angle of incidence wave in water is less than 36°43′， all the parts of wave energy can pass through the interface and then refracts to the air. When the angle of incidence wave in water is between 36°43′and 47°20′, it is resonance range, at this moment the light wave energy has been separated, some energy of wave has been reflected back to water and some has been refracted to the air. Along with the angle growing, the energy of reflected wave increases linearly until the angle is 47°20′total reflection occurred.

[0076]

When the angle of incidence in water is larger than 47°20′, the light energy will be all reflected to trapping back into the water.

[0077]

The phenomenon of light waves travel from air to water for Fig 3:

[0078]

By the methods when m＝1, n＝1.3333, when the angle of incidence wave in air is less than 52°50′， all the parts of wave energy can pass through the interface and then refracts to the water.

[0079]

When the angle of incidence wave in air is between 52°50′and 79°10′, it is resonance range, at this moment the light wave energy has been separated, some energy of wave has been reflected back to air and some has been refracted to the water. Along with the angle growing, the energy of reflected wave increases linearly until the angle is 79°10′total reflection occurred.

[0080]

When the angle of incidence in air is larger than 79°10′, the light energy will be all reflected to trapping back into the air.

[0081]

The above knowable, in any form or transition zone interface waveguide phenomenon, there are two trapping area, is an absolute wave energy trapping area, is a relative or part of a wave energy trapping area. Trapping wave can depend on relationship between the critical Angle and resonance Angle.

[0082]

Second apply case :

[0083]

The application of ocean acoustic waveguide for Fig 4.

[0084]

In China Huang sea and Bo sea have a cold water mass that is position in than 50 meters water depth of the ocean underwater sea in the summer, there were the seasonal thermocline comes up.

[0085]

Test area is: 124E, 38N nearby. In mid-august, the water temperature up of thermocline layer is warm water T＝24. (℃) , deep of water is 5. meters, the water temperature under thermocline layer is cold water T＝8. (℃) , deep of water is 45. meters. The application Mackenzie (1981) (References 8) of formula to calculate acoustic velocity, with its corresponding to sound velocity up of thermocline layer and under thermocline layer ratio is:

can take the m ＝ 5. So the absolute reflection of detecting underwater target from the ship critical Angle is 87°16`, relative reflection critical Angle is 86°37`, by the methods, ship hull sonar sound ray with an Angle that thermocline layer interface: absolute reflection Angle is: 02°44`, relative reflection Angle is 03°23`. Application of tangent function, corresponding as follows: 0.04774 and 0.05912, and the actual distance that thermocline layer to the surface of the water depth is 25 meters, the horizontal distance corresponding as follows: 523.67 meters and 422.87 meters. When the horizontal distance less than 423 meters from ship to underwater target, the target can be detected clearly； When the distance increase from 423 meters position to 524 meters position, target signal got weaken by degree until to be in a position of 524 meters target signal shall all be lost. Therefore the fact of measurement results of the actual sea trial is in the 500 meters when signal began an obvious weaken, about 600 meters position nearby underwater target signal is all lost. Is the theory and the actual match very well！ Sea test case we guarantee many times like this, for under water target most deleted distance of surface ship sonar is little than one kilometer in summer the sea, most of the test data and the theorem in this paper the calculated results are in good agreement.

[0086]

If it is not the measure for thermocline strength in actual sea but using history month average data such as WOA13, its horizontal distance error rate are within 40-50％, but it may be use. This fully shows that this theorem of practicability.

[0087]

If we do not consider the thermocline influence on sound propagation, the thermocline disappear normal in winter, sonar of sea surface ship detected range is more than 10 kilometers. If considering the thermocline influence on sound propagation such as summer, by no method in this paper, and using the Snell theorem it cannot be calculated the incident wave Angle from the upper warm water into the lower cold water. The algorithm in this article is applied to calculate the critical Angle of the incident wave from high-speed to low-speed meson layer. Although the critical Angle is small, but its role is very big, decided to the sonar of the surface ship detected range in the presence of thermocline. Can be seen from this case, it is surface ships antisubmarine detection and the strong demand for underwater target search, we need to for this kind of reality and difficult problem in its work.

[0088]

Third apply case—for Anti-reflection coating calculating

[0089]

When wave refractive index in2≥n≥1, Wave reflective index R

_{f} could expressed:

When R

_{f}＝0， so calculating to n＝1.2961, when wave refractive index n≤1.2961, wave energy has no reflected in normal incident wave on interface and the wave transmission the interface refracted in medium.

[0090]

Anti-reflection coating may expressed:

when R

_{f}＝0, so n

^{1}＝1.2961, therefore n

_{coating}＝1.1574.

[0091]

When 1.1574≤n
_{coating}≤1.2961 no reflective wave energy appear.

[0092]

So average value is:

Anti-reflection coating best refractive index is 1.2268, for no reflection wave energy. The results is no difference with the results of refractive index 1.225 of the test data of people known.

### Claims

[Claim 1]

The calculation method of wave reflective index on interface: when wave refractive index in 2≥n≥1, normal on interface wave reflective index R

_{f} calculation method:

where z

_{h} is coefficient of resonance； when wave refractive index in 4≥n≥2, normal on interface wave reflective index R

_{f} calculation method: [Math. 0001]

when wave refractive index in n≥4, normal on interface wave reflective index R

_{f} calculation method: R

_{f}＝1, incident wave energy is all reflected.

[Claim 2]

The calculation method of wave reflective index on interface of claim 1 wherein said coefficient of resonance z

_{h} calculation method: in accordance with both absolute reflection critical angle and symmetry point of refractive-reflective incident wave angle expression form， resonance critical angle can be expressed:

because of the know resonance critical angle calculation method tgθ

_{resonance}＝n, to obtain Coefficient of resonance z

_{h}＝16 (ncos (arctg (n) ) -0.75) .

[Claim 3]

The calculation method of wave reflective index on interface of claim 2 wherein said absolute reflection critical angle calculation method: when from high wave velocity medium to low wave velocity medium, wave absolute reflection critical angle calculation method is

where refractive index

coefficient of wave individual number m is integer, for [Math. 0002]

so [Math. 0003]

when coefficient of wave individual number m＝1, refractive index

absolute reflective critical angle calculation method is

when from low wave velocity medium to high wave velocity medium absolute reflection critical angle calculation method is first calculating high wave velocity media by the method in this invention, afterwards you can use Snell’s law to calculating absolute reflection critical angle in low wave speed medium for the reversibility of wave.

[Claim 4]

The calculation method of wave reflective index on interface of claim 2 wherein said resonance reflection critical Angle calculation method: when wave refractive index:

resonance reflection critical Angle calculation method is

coefficient of wave individual number m is integer, for

so

the θ

_{resonance} is wave relative reflection critical angle； when coefficient of wave individual number m＝1, refractive index:

resonance reflection critical angle . calculation method is .tgθ

_{resonance}＝n.

[Claim 5]

The calculation method of wave reflective index on interface of claim 2 wherein said symmetry point of refractive-reflective incident wave angle calculation method: when wave refractive index

symmetry point of refractive-reflective incident wave angle calculation method is

where coefficient of wave individual number m is integer, for

so

when coefficient of wave individual number m＝1, refractive index

symmetry point of refractive-reflective incident wave angle calculation method is

### Drawings