The Average Sound Energy Spectrum Produced by a Single Raindrop

The characteristics of underwater acoustic radiation generated by a single raindrop impacting the water surface play a fundamental role in the study of rainfall noise. However, the underwater sound signal generated by a single raindrop has great randomness in the amplitude of sound pressure, frequency range and energy size, so that the uncertainty of the sound energy spectrum curve of a single raindrop is relatively large. In this paper, by generating artificial rainfall containing only one size of raindrop, the acoustic radiation generated by artificial rainfall composed of about 136,000 raindrops with an equivalent diameter of 0.9 mm was measured by reverberation method. A smooth, continuous sound energy spectrum of single raindrop covering the full frequency band is obtained.


Introduction
The underwater acoustic radiation generated by rainfall at sea is one of the important components of marine environmental noise, which has an important impact on marine environmental noise.Marine environmental noise is the interference background of various water acoustic equipment, which directly affects the performance of the sound system.Accurate measurement of rainfall intensity and rainfall amount is the basis of rainfall research.But measuring rainfall intensity directly in the ocean is very difficult.Fortunately, there is a noticeable underwater noise when it rains [1]- [3].Many researchers have tried to measure rainfall intensity by measuring its underwater noise.Rainfall is made up of individual raindrops.The characteristics of underwater acoustic radiation generated by a single raindrop impacting the water surface play a fundamental role in the study of rainfall noise.
Franz [4] used a single raindrop splash on the water surface to generate underwater acoustic radiation experiment.It was concluded that the primary impact sound and bubble pulsation radiation sound play the main roles in rain-induced noise.Longuet-Higgins [5] and Oguz[6] pointed out that raindrops with a particle size of 0.8mm-1.1mmwill 100% produce bubbles when they hit the smooth water surface vertically.These bubbles are called "type I bubbles" by Medwin [7].The final velocity of a large rain falling perpendicularly onto the surface of a calm water will produce an impact sound and a strong bubble radiation sound，these bubbles are called "type II bubbles".Nystuen [8] proposed a new sound source for underwater noise generated by raindrop sputtering--a bubble formed when the raindrop hits the water surface twice.These bubbles are called "type III bubbles".
Even for the impact sound produced by raindrops of the same size, its sound pressure amplitude and sound energy will show certain randomness [9].The sound pressure amplitude, pulsation frequency and sound energy of bubble pulsation produced by raindrops of the same size show great randomness Therefore, the sound energy spectrum of a single raindrop needs to be described statistically.
At present, the average sound energy spectrum of a single raindrop is generally used in the model proposed by Nystuen [8] in 1995.Nystuen study gives the sound energy spectrum curves of four particle size raindrops.Then the sound energy spectrum curve is generalized to the sound energy spectrum of small, medium, large and very large raindrops.Since then, the sound energy spectra of single raindrops in almost all literature have been quoted from Nystuen's data.However, there are two shortcomings in the present use of a single-type acoustic energy.(1) The sound energy spectrum curve is not smooth and uncertain.There are two reasons.One reason is that Nystuen didn't use enough samples, even if some curves use data of more than 400 drops, it is still not enough to get a smooth curve; Another reason is the randomness of individual raindrop energy data, The amplitude and frequency of the radiated sound energy are randomly distributed over a wide range, whether it is the impact sound or the bubble sound.When the sound energy quantity is weighted according to the probability, the sound flow of the sound volume is fluctuating with the frequency.(2) There is no data of the sound energy spectrum curve in the frequency band above 25 kHz.Because the frequency range of shocks and bubbles Nystuen used for weighted averages did not exceed 25 kHz.In fact, when rainfall occurs, there is also rainfall noise in the frequency band above 25 kHz.
To sum up, the underwater sound energy spectrum generated by a single raindrop impacting the water surface currently used has an uneven curve and large uncertainty, and does not cover the full frequency band.It is necessary to develop a method to give a smooth and continuous underwater acoustic energy spectrum curve of a single raindrop over the whole frequency band.
In this paper, a new method is proposed to measure the sound energy spectrum generated by single raindrops.Firstly, a sufficient number of raindrops of a single size are produced at once.And then measured the sound energy produced by all the raindrops.Finally, the sound energy spectrum of a single raindrop is calculated from the total sound energy spectrum.

Producing artificial rainfall
In this paper, artificial rainfall generation system is used to generate artificial rainfall.The artificial rainfall generation system (shown in figure 1) consists of a water storage tank 1, a centrifugal pump 2, a flowmeter 3, a control panel 4, a sprinkler head 5, a filter paper 6, a reverberation tank 7, a pipeline 8, and an in-line pump 9.The water storage tank 1 is filled with tap water.Centrifugal pump 2 is placed in water storage tank 1. Flow meter 3 is connected to centrifugal pump 2 through pipe 8.Control panel 4 is connected to sprinkler 5 and flowmeter 3 by circuit.Control panel control sprinkler switch, record flow data.Filter paper 6 spread on hardwood, exposed to artificial rainfall during rainfall.The reverberation water tank 7 is under the sprinkler head and is used to collect the artificial rainfall emitted by the sprinkler head 5. In-line pump 9 can pump water from reverberation tank 7 to storage tank 1 to achieve water recycling.Among them, sprinkler system 5 contains eight groups of sprinkler heads, and each group contains five sprinkler heads.The sprinkler system has a total of 40 electronic control switches, each nozzle can be opened and closed individually, or can be opened and closed in any combination.Each nozzle produces raindrops of the same size, and the height of the nozzle from the water surface is adjusted to the maximum height of 6 m.Research points out that when the falling height of raindrops is above 4.3 m, large raindrops can reach 80% of the final velocity [10].In order to make the rainfall conform to the natural conditions, the height of the sprinkler from the water surface is adjusted to the maximum height of 6 m.And because the water is pumped to the nozzle by the pump, the raindrop has a certain initial velocity after ejecting from the nozzle.Therefore, it can be assumed that the speed of raindrops hitting the surface of the water reached the final speed of natural rainfall conditions.The results obtained by artificial rainfall in this paper can represent the actual rainfall to a certain extent.

Measuring artificial rainfall noise
In this paper, artificial rainfall noise measurement system is used to measure rainfall noise.The artificial rainfall noise measurement system (shown in figure 2) consists of a reverberation tank 7, a hydrophone array 10, a measurement amplifier 11, a data collector 12, and a computer 13.Hydrophone array 10 is placed in reverberation tank 7.After opening the sprinkler 5 to generate artificial rain, the hydrophone array 10 moves gently in the reverberation tank 7 for spatial averaging.The underwater acoustic signal generated by the rainfall collected by the hydrophone array 10 is amplified by the measuring amplifier 11, transmitted to the data collector 12, and then entered into the computer 13 for storage.
The measurement method used in this paper is the reverberation pool method.The reverberation pool method is to measure the spatial average sound pressure level in the reverberation control area far away from the sound source through spatial average, and then calibrate it through the reverberation pool to obtain the radiated sound power of the sound source [11].The expression of the reverberation pool method for measuring the radiated sound power of a sound source is as follows: where, (dB  0.67 × 10 −18 W)represents the sound power level of the sound source.⟨⟩ (dB  1Pa)represents the average sound pressure level of the measured space in the reverberation control area of the reverberation pool.10 lg  represents the correction from reverberation field to free field, and is the difference between the average sound pressure level measured in the reverberation control area in the reverberation pool and the sound power level radiated by the sound source.This quantity reflects the characteristics of the reverberation pool itself, and its value is independent of the sound source, and can be calibrated by comparison method.The radiated sound power of the sound source can be measured according to formula (1) by using the reverberation pool method.The total underwater radiated sound power  generated by artificial rainfall was measured in the reverberation pool.If the rainfall occurrence area is determined, the sound power radiated by rainfall per unit area can be obtained: is the intensity of artificial rainfall noise source corresponding to rainfall intensity.This method of measuring the power of underwater acoustic radiation generated by rainfall is called reverberation pool measurement method.It is assumed that the artificial rainfall above the pool falls uniformly in the area of the reverberation tank.The sound power level generated by artificial rainfall per unit area can be calculated according to formula (2):

Measuring the number of raindrops
In the study of rainfall noise, the number of raindrops is calculated by using the empirical formula of raindrop size distribution.In actual rainfall, assuming that the number of raindrops with particle size falling to the water surface per unit time is, then: where,   (  ) is the final speed of the raindrop hitting the water surface; (  ) is the size distribution of raindrops in rainfall, that is, the number of raindrops of a certain size in the unit space volume.(  ) is generally calculated by measuring rainfall intensity and then according to Marshall-Palmer [12] distribution formula: where, the constant  0 =8000#/ 3 mm =800#/ 3 0.1 mm , = 4.1 −0.2 , is rainfall intensity.Beard [13] shows that in convective rain, there are more raindrops with large particle size than equation (5).In artificial rainfall, the size distribution of raindrop is often inconsistent with that of natural rainfall.For artificial rainfall, the raindrop size is obviously different from the continuous distribution of actual rain particle size because the raindrop nozzle producing raindrop is limited and fixed.In addition, the distribution of the number and proportion of raindrops of each particle size in artificial rainfall is also inconsistent with the actual rainfall.
In the artificial rainfall experiment carried out in the pool, the actual particle size distribution of raindrops can be measured by using the stain method.When the flow rate (that is, the volume of water that flows through the pipe of the artificial rainfall system and falls to the pool per unit time) is determined, the actual number of raindrops of each particle size can be calculated.
Supposing that the number of raindrops with diameters 1 、 2 、 3 、…、  as a percentage of the total number is %、%、%、…、%.Then the volume fraction   of a raindrop with diameter   is: where, the volume of raindrops with diameter   is  0 =   3 /6, formula (6) can be expressed as If the flow rate  is known, the total volume   of raindrops with diameter   falling on the water surface of the pool in one second is   =   (8) The total number   of raindrops of diameter   falling on the water in one second is: Finally, the total number   of raindrops with a particle size of   is:   =   01 + 02 + 03 +⋯+ 0  (10) According to formula (10), the number of raindrops of different diameters can be calculated as long as the proportion of the number of raindrops of each diameter and the flow rate are known.If the artificial rainfall contains only one size of raindrop, then the number of raindrops becomes: where,  0 is the volume of a single raindrop of this single particle size raindrop.

Calculating the sound energy spectrum of a single raindrop
Assuming that the influence of raindrop interaction on rain-induced noise during precipitation can be ignored, then the total sound energy spectrum generated by raindrops can be approximated as the linear superposition of noise field generated by sputtering of these single raindrops [7].If the total number of raindrops is measured by certain technical means, then the sound energy spectra of raindrops with the corresponding size can be calculated.If the total sound energy spectrum density   (,   ) and the number  of raindrops produced by a large number of single particle size raindrops are measured, the sound energy spectrum of a single raindrop  0 (,   ) can be obtained by the following method.
Assuming that falling raindrops follow an independent distribution on the surface of the water, then the energy spectral density generated by the number of single particle size raindrops can be expressed as: (,   ) =  0 (,   ) (12) where,  0 (,   ) is the average energy spectrum (J/Hz) of a single raindrop of diameter   .
The energy spectral density level generated by single particle size raindrops with number N is [7]   (,   ) = 10lg  (,   ) (13) Substituting ( 12) into (13) yields: (,   ) = 10lg + 10lg 0 (,   ) (14) where, 10lg 0 (,   ) is the sound energy spectral density level  0 () of single raindrops with diameter    0 (,   ) =   (,   ) − 10lg (15) According to equation (15), as long as the sound energy spectrum of a large number of raindrops with the same particle size and the number of raindrops are measured, the sound energy spectrum of a single raindrop with the same particle size can be obtained.

The underwater noise of rainfall produced by a single sprinkler
According to the actual situation, we choose two groups of sprinkler heads in the sprinkler system for experiment.All the rain generated by the middle two sets of sprinkler heads goes into the tank, and no rain falls outside the pool.For convenience, ten of these two groups of nozzles are named 1 、 2 、 3 、 ICFOST-2023 Journal of Physics: Conference Series 2718 (2024) 012099 IOP Publishing doi:10.1088/1742-6596/2718/1/0120996  4 、 5 and  1 、 2 、 3 、 4 、 5 respectively.When the ten sprinkler heads are opened separately to produce rain, the flow rate  measured by the electronic flowmeter is shown in Table 1.m /h 0.32 0.57 0.33 0.37 0.73 0.25 0.35 0.37 0.56 0.73

Table1. Rainfall type and flux produced by different nozzles
The sound power level of rainfall radiation generated by each sprinkler was measured in the water tank by reverb method after opening A and B sprinkler heads separately.The result is shown in figure 3. It can be found that when rainfall occurs, there is a noticeable underwater noise.Even  1 with the least rainfall intensity has an acoustic power spectrum that exceeds the background noise by 18 dB near 15 kHz.The spectrum of each type of rainfall has a broadband peak near 15 kHz.When the frequency exceeds 15 kHz, rainfall noise decreases with the increase of frequency.

Particle size distribution of rainfall produced by a single sprinkler
When the ten sprinkler heads are turned on individually to produce rain, the sound power level of the rain noise and the corresponding flow rate can be measured.The actual raindrop size was also measured using the stain method.Then the actual particle size distribution of raindrops is obtained.IOP Publishing doi:10.1088/1742-6596/2718/1/0120997 by this nozzle are not all the same size.The measured diameter of the stain can be converted to the diameter of the raindrop.Then, the actual raindrop particle size distribution of the sprinkler during rainfall is obtained, as shown in figure 4(b).From the particle size distribution diagram, it can be seen that the raindrops produced by  1 nozzle are very small raindrops and small raindrops.The largest proportion of particle size is 0.3 mm to 0.5 mm of very small raindrops.Small raindrops (0.8mm and 0.9mm) make up only 12% of the total.
Table 2 shows the percentage of the number of raindrops of different particle sizes in the rainfall of different flow generated by each sprinkler head of two groups A and B, that is, the distribution of raindrop particle sizes of rainfall.The corresponding radiated sound power of rainfall with different particle size distributions in table 2 is shown in figure 3.In this paper, the radiated sound power of rainfall with actual particle size distribution is measured for the first time.

Sound power level of single raindrop noise
When the particle size distribution and flow rate are known, the specific number of raindrops of each particle size falling into the tank per unit time can be calculated according to formula (10).Table 3 shows the number of raindrops of each size produced by the  1 nozzle.It is found that for the rainfall generated by  1 sprinkler, the number of raindrops of each particle size is between tens of thousands and hundreds of thousands.Since very small raindrops with particle size less than 0.8mm do not produce bubbles, and the extremely weak impact sound generated can be ignored compared with the first type of bubble sound, only 0.8mm and 0.9mm raindrops contribute to the sound field in the artificial rain generated by the sprinkler head.Because the acoustic characteristics of the first type of bubbles produced by small raindrops are basically the same, 0.8mm and 0.9mm raindrops can be regarded as the same type of raindrop, that is, equivalent to 0.85mm raindrops.Then the rainfall noise generated by the sprinkler  1 can be considered as the rainfall noise generated by the rainfall containing only a single particle size raindrop.The diameter of all raindrops in the rain generated by the  1 nozzle can be equivalent to 0.85 mm.
According to Table 3, the number of raindrops with a diameter of 0.85mm is 1.361 × 10 5 .The sound power level of a single raindrop with an equivalent particle size of 0.85 mm can be calculated according to equation (15): where,   (, 0.85) is the sound power level radiated by all raindrops in the rain generated by the  1 nozzle.The sound power level generated by a single raindrop with an equivalent particle size of 0.85 mm obtained by the above formula is shown in figure 5.According to the sound power level data given in figure 5, it can be obtained that the total sound power level generated by a single raindrop with an equivalent particle size of 0.85 mm is 74.5 dB.

Conclusion
In this paper, the total sound power spectrum of about 136,000 raindrops with equivalent particle size of 0.85 mm is measured by reverberation method in a reverberation tank.According to the experimental results, the average sound energy spectrum of a single 0.85 mm raindrop is obtained for the first time.
The average sound energy spectrum curve is smooth and continuous, basically covering the whole frequency band.The problems of discontinuity, unsmoothness and missing of high frequency data in the single raindrop sound power spectrum curve obtained by average weighting method are solved in this paper.The following conclusions are drawn in this paper.
(1) In the experiment in this paper, the number of raindrops with a particle size of 0.9 mm hitting the water surface per unit time reached 136,000.The problem of insufficient experimental samples for single raindrops in the past has been solved.
(2) The sound energy spectrum curve of 0~50 kHz was obtained in this paper.In the past, the problem that the sound energy spectrum curve has no data in the band above 25 kHz has been remedied.
(3) In this paper, a continuous smooth sound energy spectrum curve of almost all frequency bands is obtained.The problem that the sound spectral curve of single raindrop fluctuates greatly and is not smooth and continuous with the change of frequency is solved.

Figure 2 .
Figure 2. Artificial rainfall noise measurement system

3 .
(a) Group A sprinkler (b) Group B sprinkler Figure Rainfall noise generated by single nozzles(The A0 and B0 curves represent background noise) (a) The distribution of color spots (b) Distribution of probabilities

Figure 4 .
Figure 4. Raindrop size distribution in artificial rainfall Figure 4(a) shows the stain produced by rain falling on the filter paper after  1 sprinkler head is turned on separately.It can be found that although only one nozzle is turned on, the raindrops produced

Figure 5 .
Figure 5.The sound source level generated by single raindrops of 0.85 mm

Table 2 .
Percentage of raindrops of different particle sizes

Table 3 .
The number of raindrops of different diameters