Source: https://jmrplens.github.io/phonometry/vibration/immission/railway-vibration/ # Vibration next to a railway (DIN 45672) A train passing a house puts vibration into the ground for twenty seconds and then leaves, and the question is rarely what those twenty seconds were. It is whether the vibration at the building is more than at the track wall, whether the mat under the ballast changed anything, or whether the freight trains of the night are worse than the passenger trains of the day. Every one of those compares two numbers, and two numbers can only be compared if they were formed the same way. That is what DIN 45672 is for. Part 1 says how to measure next to a railway: where the transducers go, which directions, which trains, what the report has to contain. Part 2 says how to **reduce** the record once it exists, and it is the part this page implements, with the meter of [DIN 45669-1](https://jmrplens.github.io/phonometry/vibration/immission/vibration-meter/) in its 4 Hz to 315 Hz railway working range. Neither part says what the result may be: the assessment is DIN 4150-2 for people and DIN 4150-3 for buildings. ## 1. Three stretches of one record The first thing Part 2 does is cut the record into three stretches, and every quantity after that carries the index of the one it was read over (Clause 5). - **`T₁`**, about four seconds with the largest vibration in its middle. It is where the characteristic values of the passage are read. A short or fast train often needs less, and then `T₁` may be as long as `T₂` and no longer. - **`T₂`**, the passage itself: the time a train takes to cross an imaginary measuring section. Experience, the standard says, puts its start where the amplitude reaches about a quarter of the most frequent maxima of the passage and its end where it falls back to the same value. That is a judgement about the record, so the library asks for it rather than guessing it. - **`T₃`**, the whole event, quiet approach and departure included. It is the one the energy of the passage is formed over. A thirty-second time axis carrying the velocity of one train passage in millimetres per second. A faint grey trace is the velocity itself, quiet for six seconds, rising over three seconds to a pulsing envelope of about 0.3 to 0.45 millimetres per second peak that holds until 21 seconds and falls away by 24. A blue curve over it is the running r.m.s. of Formula (1), rising with the envelope to about 0.2 and pulsing with the bogies. A red dashed line marks its maximum at 0.218 millimetres per second. Above the record, three double-headed arrows mark the stretches: T1 over four seconds around 15.5 seconds, T2 from 6.75 to 23.25 seconds, and T3 over the whole thirty
Figure code ```python import numpy as np from phonometry import vibration fs_hz = 2048 t = np.arange(30 * fs_hz) / fs_hz rng = np.random.default_rng(7) envelope = np.clip((t - 6) / 3, 0, 1) * np.clip((24 - t) / 3, 0, 1) bogies = 1 + 0.5 * np.sin(2 * np.pi * 1.6 * t) ** 2 record = envelope * bogies * ( 0.20 * np.sin(2 * np.pi * 40 * t) + 0.08 * np.sin(2 * np.pi * 63 * t + 1.0) + 0.03 * rng.standard_normal(t.size) ) passage = vibration.evaluate_train_passage(record, fs_hz, t2_s=(6.75, 23.25)) passage.plot() ```
The record above is synthetic and shaped like a real one: the envelope of a train arriving and leaving, a pulse per bogie, a sleeper-passing tone at 40 Hz, a floor resonance at 63 Hz and some broadband noise. Its `T₂` starts where the envelope reaches a quarter, 6,75 s, and ends where it falls back, 23,25 s: ```python import numpy as np from phonometry import vibration fs_hz = 2048 t = np.arange(30 * fs_hz) / fs_hz rng = np.random.default_rng(7) envelope = np.clip((t - 6) / 3, 0, 1) * np.clip((24 - t) / 3, 0, 1) bogies = 1 + 0.5 * np.sin(2 * np.pi * 1.6 * t) ** 2 record = envelope * bogies * ( 0.20 * np.sin(2 * np.pi * 40 * t) + 0.08 * np.sin(2 * np.pi * 63 * t + 1.0) + 0.03 * rng.standard_normal(t.size) ) passage = vibration.evaluate_train_passage(record, fs_hz, t2_s=(6.75, 23.25)) for name, (start, end), rms in zip( ("T1", "T2", "T3"), passage.intervals_s, passage.interval_rms_mm_s ): print(f"{name}: {start:5.2f} s to {end:5.2f} s, r.m.s. {rms:.4f} mm/s") # T1: 13.47 s to 17.47 s, r.m.s. 0.1945 mm/s # T2: 6.75 s to 23.25 s, r.m.s. 0.1783 mm/s # T3: 0.00 s to 30.00 s, r.m.s. 0.1324 mm/s print(f"v_max {passage.peak_velocity_mm_s:.3f} mm/s") # 0.481 mm/s print(f"v_Fmax {passage.running_rms_max_mm_s:.3f} mm/s") # 0.218 mm/s print(f"KB_Fmax {passage.kbf_max:.3f}") # 0.216 ``` Before anything is read, the record goes through the band limitation of the DIN 45669-1 railway range, as it does inside the meter. That is why a record has to be the velocity the transducer gives, and why `KB_Fmax`, the number DIN 4150-2 judges people by, comes out of the same call as the rest. ## 2. The running r.m.s., and the fourteen per cent that is seven The running r.m.s. of Formula (1) is the exponential average a sound level meter calls Fast, `τ = 125 ms`, and its level against `v₀ = 5·10⁻⁸ m/s` is the velocity level of Formula (2). Beware the reference: DIN 45672-2 defines this 5·10⁻⁸ m/s itself, without naming a source. It is the 50 nm/s that note b of ISO 1683:2015 Table 3 says is also used for structure-borne sound, not the 1 nm/s of the main row of that table, so the same vibration is 34 dB apart under the two. ```python import numpy as np from phonometry import vibration fs_hz = 2048 t = np.arange(3 * fs_hz) / fs_hz one_mm_s_rms = np.sqrt(2) * np.sin(2 * np.pi * 40 * t) level = vibration.running_velocity_level(one_mm_s_rms, fs_hz) print(f"{level[-fs_hz:].mean():.2f} dB") # 86.02 dB ``` The average starts from rest, so the first stretch of any record reads low, and Clause 4 says why that matters: the averaging has to be running before the train arrives. It quotes the cost as 14 % after `2τ` and 2 % after `4τ`, and those are the shortfalls of the **mean square**, `e⁻²` and `e⁻⁴`. The running r.m.s. the sentence is about, and the one its own Figure 3 draws, is short by 7 % and 0,9 %. The advice stands; the size of the error is overstated by a factor of two, and the [errata page](https://jmrplens.github.io/phonometry/reference/errata/) has the detail. ## 3. The event value, and an hour of trains The quantity a passage is summarised by is its **event value** (Formula (8)): the interval r.m.s. of the whole event referred to one hour, ```text v_E = ṽ₃ · √(T₃ / 3600 s) ``` which is the constant velocity that, held for an hour, carries the energy the passage carried. Its level (Formula (9)) is the same thing in decibels, and because each event value is already the energy of a passage spread over the same hour, the passages of an hour add in square (Formula (10)): ```python from phonometry import vibration one = vibration.event_velocity(0.1324, 30.0) print(f"one passage: v_E = {one:.5f} mm/s") # 0.01209 mm/s print(f"L_vE = {vibration.event_velocity_level(0.1324, 30.0):.2f} dB") # 47.67 dB six = vibration.combined_event_velocity([one] * 6) print(f"six in an hour: {six:.5f} mm/s") # 0.02961 mm/s, sqrt(6) times one ``` That is the number to compare between two hours of traffic, and it is why `T₃` includes the quiet ends: they carry no energy, and the hour does not care how long the record ran. ## 4. Third octaves: the spectra of Figure 6 Clause 7.2 analyses the passage in third octaves from 4 Hz to at least 315 Hz, and prints two spectra as the result: the **interval level** over `T₂` (Formula (6)), the running r.m.s. of each band averaged over the passage, and the **maximum level** over `T₃` (Formula (7)), the largest value the running r.m.s. of each band reaches. The gap between them is the crest of the band: wide where the vibration comes in bursts, narrow where it is steady. Third-octave bands from 4 to 315 hertz on an evenly spaced axis, velocity level in decibels re 5 times ten to the minus eight metres per second. A blue line with round markers is the maximum level over T3 and a green dashed line with square markers is the interval level over T2, both low and flat around 35 to 45 decibels except for two peaks: about 72 and 70 decibels at 40 hertz, and about 64 and 62 at 63 hertz. Red crosses at every band from 4 to 315 hertz are the same T2 spectrum obtained from the narrow-band density through Table 1; they sit on the green curve almost everywhere and fall about 5 decibels below it at 50 hertz, between the two tones
Figure code ```python import numpy as np from phonometry import vibration fs_hz = 2048 t = np.arange(30 * fs_hz) / fs_hz rng = np.random.default_rng(7) envelope = np.clip((t - 6) / 3, 0, 1) * np.clip((24 - t) / 3, 0, 1) bogies = 1 + 0.5 * np.sin(2 * np.pi * 1.6 * t) ** 2 record = envelope * bogies * ( 0.20 * np.sin(2 * np.pi * 40 * t) + 0.08 * np.sin(2 * np.pi * 63 * t + 1.0) + 0.03 * rng.standard_normal(t.size) ) passage = vibration.evaluate_train_passage(record, fs_hz, t2_s=(6.75, 23.25)) ax = passage.plot_spectrum() t2 = slice(round(6.75 * fs_hz), round(23.25 * fs_hz)) spectrum = vibration.narrowband_psd(passage.velocity_mm_s[t2], fs_hz) centres, rms = vibration.third_octaves_from_narrowband( spectrum.frequencies, spectrum.psd ) narrow = 20 * np.log10(rms / vibration.VELOCITY_LEVEL_REFERENCE_MM_S) positions = [list(centres).index(band) for band in passage.band_centres_hz] ax.plot(range(len(positions)), narrow[positions], "x", label="T2 from the narrow band") ax.legend() ```
```python import numpy as np from phonometry import vibration fs_hz = 2048 t = np.arange(30 * fs_hz) / fs_hz rng = np.random.default_rng(7) envelope = np.clip((t - 6) / 3, 0, 1) * np.clip((24 - t) / 3, 0, 1) bogies = 1 + 0.5 * np.sin(2 * np.pi * 1.6 * t) ** 2 record = envelope * bogies * ( 0.20 * np.sin(2 * np.pi * 40 * t) + 0.08 * np.sin(2 * np.pi * 63 * t + 1.0) + 0.03 * rng.standard_normal(t.size) ) passage = vibration.evaluate_train_passage(record, fs_hz, t2_s=(6.75, 23.25)) bands = list(passage.band_centres_hz) for band in (40.0, 63.0): i = bands.index(band) print( f"{band:g} Hz: L_vF2 {passage.band_interval_levels_db[1, i]:.1f} dB," f" L_vFmax {passage.band_max_levels_db[i]:.1f} dB" ) # 40 Hz: L_vF2 70.4 dB, L_vFmax 72.0 dB # 63 Hz: L_vF2 62.4 dB, L_vFmax 64.3 dB print(f"sum level {vibration.band_sum_level(passage.band_interval_levels_db[1]):.1f} dB") # 71.1 dB, Formula (B.3) ``` ## 5. Narrow band, and what Table 1 is Third octaves are coarse where a floor resonates, so Clause 7.3 also asks for the one-sided **power spectral density** of Formula (12), blockwise with a Hanning window, at least half a block of overlap and a linear average of the blocks. The resolution it recommends is 1,25 Hz, fine enough for the resonance of an ordinary floor, which at the 500 Hz and 400 lines of a common analyser is a block of 0,8 s. The lines add back to the mean square of the stretch, which is Formula (14). To compare a narrow-band result with a third-octave one, Formula (23) adds the lines of each band back together, and Table 1 says how many lines each band takes: 1 at 4 Hz, 6 at 31,5 Hz, 58 at 315 Hz. The standard gives the counts "for comparability" and does not derive them. They are what a sixth of an octave either side of the nominal centre holds, everywhere but at 10 Hz and 12,5 Hz, where those edges would give one line and three and the table gives two and two. The table says how many lines and not which, so the library states its choice: the `Kₙ` lines nearest the nominal centre on a logarithmic axis, which are exactly the lines inside the nominal edges wherever the count is what the edges hold, and 10 and 11,25 Hz, then 12,5 and 13,75 Hz, where it is not. It is a rule of nominal bands and behaves like one: nominal edges do not meet, so six lines fall between two bands (71,25 Hz, 141,25 Hz and 142,5 Hz, and 353,75 Hz to 356,25 Hz) and five are counted in two (178,75 Hz, 223,75 Hz, and 446,25 Hz to 448,75 Hz). A tone in one of the gaps is in no band at all, which is worth knowing before a spectrum is converted. ```python import numpy as np from phonometry import vibration fs_hz = 2048 t = np.arange(30 * fs_hz) / fs_hz rng = np.random.default_rng(7) envelope = np.clip((t - 6) / 3, 0, 1) * np.clip((24 - t) / 3, 0, 1) bogies = 1 + 0.5 * np.sin(2 * np.pi * 1.6 * t) ** 2 record = envelope * bogies * ( 0.20 * np.sin(2 * np.pi * 40 * t) + 0.08 * np.sin(2 * np.pi * 63 * t + 1.0) + 0.03 * rng.standard_normal(t.size) ) passage = vibration.evaluate_train_passage(record, fs_hz, t2_s=(6.75, 23.25)) t2 = slice(round(6.75 * fs_hz), round(23.25 * fs_hz)) spectrum = vibration.narrowband_psd(passage.velocity_mm_s[t2], fs_hz) print(f"block {spectrum.nperseg / fs_hz:.2f} s") # 0.80 s centres, rms = vibration.third_octaves_from_narrowband( spectrum.frequencies, spectrum.psd ) levels = 20 * np.log10(rms / vibration.VELOCITY_LEVEL_REFERENCE_MM_S) for band in (40.0, 50.0, 63.0): print(f"{band:g} Hz from narrow band: {levels[list(centres).index(band)]:.1f} dB") # 40 Hz from narrow band: 70.5 dB (70.4 dB from the band filter) # 50 Hz from narrow band: 39.2 dB (44.7 dB from the band filter) # 63 Hz from narrow band: 62.5 dB (62.4 dB from the band filter) ``` The two routes agree where the energy is and part where there is little of it, at 50 Hz between two tones: a band filter's skirts let in some of the neighbouring 40 Hz and 63 Hz and a Hanning line does not. The standard says as much, and says that the comparison is only fair at the resolution Table 1 is written for, which is why the library refuses a spectrum spaced otherwise. The level of a density against `(5·10⁻⁸ m/s)²/Hz`, as its Figure 7 draws it, is `spectral_density_level`, and a density read off an analyser in V²/Hz is turned into (mm/s)²/Hz by dividing by the square of the transducer coefficient and of the gain, which is all Annex A says. ## 6. Before, after, and across passages Part 2 exists to compare, and its last clauses are the comparisons. The **insertion loss** of an elastic element (Annex B, Formula (B.1)) is the drop in each third-octave level at one point when the element is built in, and the annex adds a warning worth repeating: it belongs to the point it was measured at. And the results of a class of trains are **averaged energetically** (Clause 9): r.m.s. values as the root of the mean square, levels as the level of the mean energy. ```python from phonometry import vibration before = [70.4, 62.4] # 40 Hz and 63 Hz, without the mat after = [61.0, 50.9] print(vibration.elastic_insertion_loss(before, after)) # [ 9.4 11.5] average = vibration.passage_average_level([70.4, 72.1, 69.0]) print(f"{average:.1f} dB") # 70.7 dB rms = vibration.passage_average_velocity([0.18, 0.21, 0.16]) print(f"{rms:.4f} mm/s") # 0.1845 mm/s ``` ## 7. The ground the vibration travels through Part 1 is procedure, with one exception. Clause 4.5 describes the ground by the speeds of its compression and shear waves, and derives from them the constants a prediction needs. An unbounded elastic continuum carries two kinds of wave, and their speeds are fixed by two elastic constants and the density, so measuring both speeds fixes everything: the shear modulus from the shear wave, Poisson's ratio from their ratio (Formula (3)), and the elastic modulus from the two together. Two of the clause's formulas are printed wrong, and the [errata page](https://jmrplens.github.io/phonometry/reference/errata/) has the reading. Formula (1) writes the compression speed first as `√(E/ρ)`, the speed of a wave in a thin rod, and then as `√(G(1−ν)/(ρ(1−2ν)))`, which is short of a factor 2. Formula (5) builds on the first and writes `E = v_p² ρ`, which from a measured `v_p` returns the P-wave modulus instead of `E`. Formula (3), printed between them, is right, and the library uses the relations it is derived from: ```python from phonometry import vibration v_p, v_s, rho = 400.0, 180.0, 1900.0 # a medium-dense sand, m/s and kg/m3 nu = vibration.poisson_ratio_from_wave_speeds(v_p, v_s) g = vibration.shear_modulus_from_wave_speed(v_s, density_kg_m3=rho) e = vibration.youngs_modulus_from_wave_speeds(v_p, v_s, density_kg_m3=rho) print(f"nu = {nu:.3f}, G = {g / 1e6:.1f} MPa, E = {e / 1e6:.0f} MPa") # nu = 0.373, G = 61.6 MPa, E = 169 MPa print(f"E as Formula (5) prints it: {v_p**2 * rho / 1e6:.0f} MPa") # 304 MPa strain = vibration.shear_strain_amplitude(0.2e-3, shear_wave_speed_m_s=v_s) print(f"shear strain of 0.2 mm/s: {strain:.1e}") # 1.1e-06 print(strain < vibration.SHEAR_STRAIN_LINEAR_LIMIT) # True ``` Poisson's ratio from 0 to 0.49 on the horizontal axis and the ratio of compression to shear wave speed on the vertical, from 1 to 8. A blue curve, the unbounded continuum that Formula (3) inverts, starts at the square root of 2 and climbs steeply towards the incompressible limit, passing 2.2 at 0.37. A grey dashed curve is the second radical of Formula (1) as printed, the same shape a factor square root of two lower. A red dotted line is the thin-rod speed of the first radical, rising gently from 1.41 to 1.73. The two printed expressions cross at a Poisson's ratio of 0.39, marked with a dot and a label below it, the only place where Formula (1) agrees with itself
Figure code ```python import matplotlib.pyplot as plt import numpy as np from phonometry import vibration nu = np.linspace(0.0, 0.49, 300) continuum = [ vibration.compression_wave_speed(1.0, poisson_ratio=n, density_kg_m3=1.0) for n in nu ] fig, ax = plt.subplots(figsize=(10, 6)) ax.plot(nu, continuum, label="continuum, the inverse of Formula (3)") ax.plot(nu, np.sqrt((1 - nu) / (1 - 2 * nu)), "--", label="Formula (1), second radical") ax.plot(nu, np.sqrt(2 * (1 + nu)), ":", label="Formula (1), sqrt(E / rho)") ax.set_xlabel("Poisson's ratio") ax.set_ylabel("v_p / v_s") ax.legend() ```
The strain matters because a soil's shear modulus falls as the strain grows: Figure 1 of the clause has it constant up to a shear strain of about `10⁻⁴` and dropping below 20 % of that value further on. Rail traffic, and the measurements that read the speeds, strain the ground around `10⁻⁶`, so the modulus measured is the one that applies. ## 8. Where the record comes from A cross-section at right angles to a railway line. On the left a train stands end-on on its track at ground level, the track axis drawn through it, and a coordinate cross beside it gives x along the track out of the page, y across the track and z vertical. On the ground to the right, transducers stand 8, 16, 32, 64 and 128 metres from the track axis, each with an arrow for the vertical direction: the first, preferably in undisturbed ground, is the emission point and the other four are transmission points, the distance doubling at each step. Further right a two-storey building is the immission area, with one transducer on the floor slab measuring vertically along z prime and one at the foot of the wall facing the track, at ground level, measuring at right angles to it along y prime, these immission points on foundations and floors chosen after DIN 45669-2, DIN 4150-2 and DIN 4150-3. Three insets follow. In plan, three cross-sections run at right angles to the track, in the middle of a straight test section 100 to 200 metres long where a track form is what is being studied, and a building turned by the angle alpha carries x prime along its wall. An embankment and a cutting carry their points at projected distances of 8, 16 and 32 metres. A rectangular tunnel with two tracks has points on the invert at a track centre and between the tracks, on the wall 1.5 metres above the rail, at the middle of the roof, on the ground above it with the cover between, and on the ground beside it, about two thirds of the cover out from the track axis but at least 8 metres. Under the insets, a thirty-second record of one passage carries its three stretches: T3 over the whole event, T2 from where the amplitude reaches about a quarter of the most frequent maxima to where it falls back to it, and T1 of about four seconds with the largest values near its middle. Two boxes at the foot give the minimum test-track length, l MG equals l Z plus r times v Z over v R, and the event value, v E equals v tilde 3 times the square root of T3 over 3600 seconds. Two notes close the plate: a meter to DIN 45669-1 from 4 hertz to 315 hertz, with every point recorded at once where possible, and a record without a train, at the same points and through the same chain, to show the background Part 1 says where the transducers go and what a record has to be before Part 2 may reduce it: a cross-section at right angles to the track (6.1), the emission point on the ground 8 m from the track axis (6.2.1.2), the transmission points at twice the distance each time out to 128 m (6.2.2), and the immission points that DIN 45669-2, DIN 4150-2 and DIN 4150-3 place (6.2.3). **How the measurement goes.** Use a meter to DIN 45669-1 working from 4 Hz to 315 Hz and check it after every set-up. Put the emission point on the ground 8 m from the track axis, or on the invert in a tunnel, and the transmission points at 16 m, 32 m, 64 m and 128 m, with the distances projected on an embankment or in a cutting; the building points go where DIN 45669-2, DIN 4150-2 and DIN 4150-3 put them. Record several passages of each kind of train, every point at once where possible, then the same points with no train for the background, and cut each record into `T₁`, `T₂` and `T₃`. ## 9. What this page is not DIN 45672-1 is a measuring method: measurement points on the way from the track to the building, directions, coupling, the trains to record and the report to write. None of that is arithmetic and the library computes none of it, but that procedure is what decides what a record is worth, so it is set out above; the coupling it points to is DIN 45669-2. Nor is any assessment: the numbers this page produces are compared with something only in DIN 4150-2 and DIN 4150-3. ## What this guide covers The **three stretches** of Clause 5, the **running r.m.s.** and the velocity and acceleration levels of Formulae (1) to (3), the **interval r.m.s.** of Formulae (4) and (5), and the **event value**, its level and its sum over the passages of an hour, Formulae (8) to (10). The **third-octave levels** of Formulae (6) and (7) from 4 Hz to 315 Hz, or to 80 Hz as DIN 45672-1 allows, and the **amplitude distribution** of Formula (11). The **narrow-band densities** of Formulae (12) to (14) at the resolution of 7.3.2, the conversion to third octaves of Formula (23) with every count of **Table 1**, the density level of Figure 7, the dimension conversion of **Annex A**, the quantities of **Annex B** and the energy averaging of **Clause 9**. The **ground constants** of DIN 45672-1 Clause 4.5 from two wave speeds and a density, with Formulae (1) and (5) in the form the continuum gives and the printed forms registered as errata. **No analyser.** The calibration checks of 7.3.3 test the amplitude display of a narrow-band analyser, with and without a Hanning window; the library's density is correct by construction and has nothing to calibrate. **No measuring procedure is checked, and no verdict.** The measurement points, directions, coupling and choice of trains of DIN 45672-1 and DIN 45669-2 are summarised above, but nothing here checks them; what the numbers mean for people and buildings is DIN 4150-2 and DIN 4150-3. ## See also - [Measuring vibration immission (DIN 45669-1)](https://jmrplens.github.io/phonometry/vibration/immission/vibration-meter/): the meter every quantity here is read with, and the `KB_F` it produces. - [Vibration damage to structures (DIN 4150-3)](https://jmrplens.github.io/phonometry/vibration/structural/structural-damage/): the guideline values a peak velocity from a railway is compared with. - [Spectral analysis](https://jmrplens.github.io/phonometry/signals/spectra/spectral-analysis/): the Welch estimate behind the narrow-band density and the confidence it carries. - API reference: [`vibration.immission.railway`](https://jmrplens.github.io/phonometry/reference/api/vibration/railway/) and [`vibration.immission.ground`](https://jmrplens.github.io/phonometry/reference/api/vibration/ground/). ## References - Deutsches Institut für Normung. (1995). *Schwingungsmessungen in der Umgebung von Schienenverkehrswegen — Teil 2: Auswerteverfahren* (DIN 45672-2:1995-07). The three stretches of Clause 5, the running r.m.s. and levels of Formulae (1) to (3), the interval r.m.s. of Formulae (4) and (5), the third-octave levels of Formulae (6) and (7), the event value and level of Formulae (8) to (10), the amplitude distribution of Formula (11), the power and energy spectral densities of Formulae (12) to (14) with the parameters of 7.3.2, the narrow-band to third-octave conversion of Formula (23) with Table 1, the dimension conversion of Annex A, the quantities of Annex B and the averaging over passages of Clause 9. The analyser calibration checks of 7.3.3 are not implemented. - Deutsches Institut für Normung. (2009). *Schwingungsmessungen in der Umgebung von Schienenverkehrswegen — Teil 1: Messverfahren* (DIN 45672-1:2009-12). Clause 4.5: the wave speeds of Formulae (1) and (2), Poisson's ratio of Formula (3), the shear strain of Formula (4) and the moduli of Formula (5), with Formulae (1) and (5) in the corrected form registered in the errata. The measuring procedure of the rest of the standard is not arithmetic and is not implemented; the measuring conditions of Clause 6, the operating states of Clause 7, the execution of Clause 8 and the report of Clause 10 are set out in the measurement section. - Deutsches Institut für Normung. (2010). *Messung von Schwingungsimmissionen — Teil 1: Schwingungsmesser — Anforderungen und Prüfungen* (DIN 45669-1:2010-09). The meter a railway measurement is made with, in its 4 Hz to 315 Hz working range, whose band limitation every quantity here is read after.