The speed of sound is the velocity at which sound travels through a medium, which can be a solid, liquid or a gas.
The following calculations, however, only relate to the speed of sound through a gas - any gas.
The conventional calculation method for the speed of sound is as follows:
v = √[γ.Rᵢ.Ṯ / RAM]
Where: γ is normally rounded off to 1.4
However, this calculation method only considers temperature as a variable. It takes no account of gas density or pressure.
CalQlata has established a more accurate value for our atmosphere; γ = cₚ/cᵥ = 965.4/719.3 = 1.34213819
When both options for 'γ' are applied to the above formula and a column of Earth's atmosphere, the following velocities are achieved:
| γ = 1.4 | γ = 1.342 | |||||
|---|---|---|---|---|---|---|
| Altitude (km) | Ṯₑ (K) | Ṯₚ (K) | vₑ (m/s) | vₚ (m/s) | vₑ (m/s) | vₚ (m/s) |
| 0 | 353.15 | 293.15 | 374.41 | 341.13 | 366.59 | 334 |
| 5 | 321.9 | 261.9 | 357.46 | 322.43 | 350 | 315.7 |
| 10 | 290.65 | 230.65 | 339.67 | 302.58 | 332.58 | 296.27 |
| 15 | 278.15 | 218.15 | 332.28 | 294.27 | 325.34 | 288.13 |
| 20 | 278.15 | 218.15 | 332.28 | 294.27 | 325.34 | 288.13 |
| 25 | 285.65 | 225.65 | 336.73 | 299.29 | 329.7 | 293.04 |
| 30 | 293.15 | 233.15 | 341.13 | 304.22 | 334 | 297.87 |
| 35 | 303.15 | 243.15 | 346.9 | 310.68 | 339.65 | 304.19 |
| 40 | 313.15 | 253.15 | 352.57 | 317 | 345.21 | 310.38 |
| 45 | 323.15 | 263.15 | 358.16 | 323.2 | 350.68 | 316.45 |
| 50 | 333.15 | 273.15 | 363.66 | 329.28 | 356.06 | 322.41 |
| 55 | 323.51 | 263.51 | 358.36 | 323.42 | 350.87 | 316.67 |
| 60 | 307.44 | 247.44 | 349.34 | 313.4 | 342.05 | 306.86 |
| 65 | 291.36 | 231.36 | 340.08 | 303.05 | 332.98 | 296.72 |
| 70 | 275.29 | 215.29 | 330.57 | 292.34 | 323.67 | 286.23 |
| 75 | 259.22 | 199.22 | 320.78 | 281.21 | 314.08 | 275.34 |
| 80 | 243.15 | 183.15 | 310.68 | 269.63 | 304.19 | 264 |
| 85 | 243.15 | 183.15 | 310.68 | 269.63 | 304.19 | 264 |
| 90 | 243.15 | 183.15 | 310.68 | 269.63 | 304.19 | 264 |
| 95 | 248.15 | 188.15 | 313.85 | 273.29 | 307.3 | 267.58 |
| 100 | 253.15 | 193.15 | 317 | 276.9 | 310.38 | 271.11 |
| The velocity of sound using the formula; v = √[γ.Rᵢ.Ṯ / RAM] suffix 'ₚ' refers to the earth's polar atmosphere, and suffix 'ₑ' refers to the earth's equatorial atmosphere |
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If, on the other hand, we apply the following formula:
v = √[γ.p / ρ]
Where: p is the pressure and ρ is the density at altitude, both of which include the effects of temperature and centrifugal force, we achieve the following velocities (assuming γ = 1.34213819):
| γ = 1.342 | ||||||
|---|---|---|---|---|---|---|
| Altitude (km) | pₑ (N/m²) | pₚ (N/m²) | ρₑ (kg/m³) | ρₚ (kg/m³) | vₑ (m/s) | vₚ (m/s) |
| 0 | 101118.51 | 102439.98 | 1.02 | 1.24 | 364.77 | 333.46 |
| 5 | 66585.15 | 67454.57 | 7.35E-01 | 9.09E-01 | 348.68 | 315.57 |
| 10 | 39908.36 | 40429.00 | 4.87E-01 | 6.17E-01 | 331.68 | 296.46 |
| 15 | 22646.65 | 22941.84 | 2.88E-01 | 3.70E-01 | 324.71 | 288.53 |
| 20 | 13281.62 | 13454.59 | 1.69E-01 | 2.16E-01 | 325.05 | 288.83 |
| 25 | 8068.07 | 8173.04 | 9.96E-02 | 1.27E-01 | 329.78 | 294.09 |
| 30 | 4843.87 | 4906.85 | 5.82E-02 | 7.37E-02 | 334.16 | 299.00 |
| 35 | 2915.25 | 2953.12 | 3.39E-02 | 4.25E-02 | 339.66 | 305.21 |
| 40 | 1936.72 | 1961.86 | 2.18E-02 | 2.71E-02 | 345.46 | 311.64 |
| 45 | 1216.99 | 1232.77 | 1.33E-02 | 1.64E-02 | 350.94 | 317.74 |
| 50 | 799.09 | 809.45 | 8.45E-03 | 1.04E-02 | 356.23 | 323.64 |
| 55 | 517.81 | 524.51 | 5.64E-03 | 6.97E-03 | 350.99 | 317.83 |
| 60 | 312.99 | 317.03 | 3.59E-03 | 4.49E-03 | 342.05 | 307.89 |
| 65 | 190.69 | 193.16 | 2.31E-03 | 2.92E-03 | 333.02 | 297.74 |
| 70 | 113.20 | 114.66 | 1.45E-03 | 1.87E-03 | 323.68 | 287.19 |
| 75 | 63.36 | 64.18 | 8.63E-04 | 1.13E-03 | 313.99 | 276.18 |
| 80 | 33.96 | 34.39 | 4.93E-04 | 6.59E-04 | 304.04 | 264.75 |
| 85 | 17.92 | 18.15 | 2.60E-04 | 3.48E-04 | 303.94 | 264.67 |
| 90 | 9.44 | 9.56 | 1.37E-04 | 1.83E-04 | 303.84 | 264.57 |
| 95 | 5.16 | 5.22 | 7.35E-05 | 9.75E-05 | 306.84 | 268.07 |
| 100 | 2.98 | 3.02 | 4.16E-05 | 5.49E-05 | 309.85 | 271.55 |
| The velocity of sound using the formula; v = √[γ.p / ρ] suffix 'ₚ' refers to the earth's polar atmosphere, and suffix 'ₑ' refers to the earth's equatorial atmosphere |
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As you can see, the differences are marginal:
| Altitude (km) | vₑ/vₑ | vₚ/vₚ |
|---|---|---|
| 0 | 0.995045838 | 0.99839082 |
| 5 | 0.996235601 | 0.999584906 |
| 10 | 0.997294057 | 1.000642774 |
| 15 | 0.998061472 | 1.001380355 |
| 20 | 0.999095301 | 1.002415099 |
| 25 | 1.000250803 | 1.003584712 |
| 30 | 1.000473071 | 1.00380485 |
| 35 | 1.000017093 | 1.003350699 |
| 40 | 1.000724125 | 1.004074591 |
| 45 | 1.00072954 | 1.004084586 |
| 50 | 1.000475475 | 1.003803713 |
| 55 | 1.000341549 | 1.003662682 |
| 60 | 1.000013853 | 1.003353763 |
| 65 | 1.000112844 | 1.003446124 |
| 70 | 1.000031717 | 1.003369555 |
| 75 | 0.999713438 | 1.00304549 |
| 80 | 0.999506792 | 1.002843592 |
| 85 | 0.999192749 | 1.002525425 |
| 90 | 0.998835172 | 1.002163978 |
| 95 | 0.998513188 | 1.00183269 |
| 100 | 0.99828851 | 1.001609543 |
| A comparison of the two calculation methods suffix 'ₚ' refers to the earth's polar atmosphere, and suffix 'ₑ' refers to the earth's equatorial atmosphere |
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Given the closeness of the two calculation methods, either can be considered appropriate for general use.
However, it is CalQlata's opinion, that v = √[γ.p / ρ] is the most reliable formula as it takes into account the pressure and density of the gas, which if calculated correctly, will never give reason to doubt the accuracy of your calculation result, irrespective of gas composition and conditions. It may, however, require a little more work, dependent upon the information available.
You will find further reading on this subject in reference publications(17)