RF Cafe Software

RF Cascade Workbook 2005 - RF Cafe
RF Cascade Workbook

Calculator Workbook
RF Workbench
Smith Chart™ for Visio
Smith Chart™ for Excel
RF & EE Symbols Word
RF Stencils for Visio

About RF Cafe

Kirt Blattenberger - RF Cafe WebmasterCopyright
1996 - 2022
Kirt Blattenberger,

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The Internet was still largely an unknown entity at the time and not much was available in the form of WYSIWYG ...

All trademarks, copyrights, patents, and other rights of ownership to images and text used on the RF Cafe website are hereby acknowledged.

My Hobby Website:

Try Using SEARCH
to Find What You Need. 
There are 1,000s of Pages Indexed on RF Cafe !

Electronics World Cover,TOC,and list of posted Popular Electronics articles QST Radio & TV News Radio-Craft Radio-Electronics Short Wave Craft Wireless World About RF Cafe RF Cafe Homepage RF Cafe in Morse Code Google Search of RF Cafe website Sitemap Electronics Equations Mathematics Equations Equations physics Manufacturers & distributors Engineer Jobs Twitter LinkedIn Crosswords Engineering Humor Kirt's Cogitations Engineering Event Calendar RF Engineering Quizzes AN/MPN-14 Radar 5CCG Notable Quotes App Notes Calculators Education Magazines Software,T-Shirts,Coffee Mugs Articles - submitted by RF Cafe visitors Simulators Technical Writings RF Cafe Archives Test Notes RF Cascade Workbook RF Stencils for Visio Shapes for Word Thank you for visiting RF Cafe!

Electronic Warfare and Radar Systems Engineering Handbook
- Mach Number and Airspeed vs. Altitude Mach Number -

[Go to TOC]


MACH NUMBER is defined as a speed ratio, referenced to the speed of sound, i.e.

MACH NUMBER = Mach number equation - RF Cafe     [1]
(at the given atmospheric conditions)      

Since the temperature and density of air decreases with altitude, so does the speed of sound, hence a given true velocity results in a higher MACH number at higher altitudes.

AIRSPEED is a term that can be easily confused. The unqualified term airspeed can mean any of the following:

a. Indicated airspeed (IAS) - the airspeed shown by an airspeed indicator in an aircraft. Indicated airspeed is expressed in knots and is abbreviated KIAS.

b. Calibrated airspeed (CAS) - indicated airspeed corrected for static source error due to location of pickup sensor on aircraft. Calibrated airspeed is expressed in knots and is abbreviated KCAS. Normally it doesn't differ much from IAS.

c. True airspeed (TAS) - IAS corrected for instrument installation error, compressibility error, and errors due to variations from standard air density. TAS is expressed in knots and is abbreviated KTAS. TAS is approximately equal to CAS at sea level but increases relative to CAS as altitude increases. At 35,000 ft, 250 KIAS (or KCAS) is approximately 430 KTAS. 

IAS (or CAS) is important in that aircraft dynamics (such as stall speed) responds largely to this quantity. TAS is important for use in navigation (True airspeed ± windspeed = groundspeed).

Figures 1 and 2 depict relations between CAS and TAS for various altitudes and non-standard temperature conditions. The first graph depicts lower speed conditions, the second depicts higher speeds.

As an example of use, consider the chart on the next page. Assume we are in the cockpit, have read our IAS from the airspeed indicator, and have applied the aircraft specific airspeed correction to obtain 370 KCAS. We start at point "A" and go horizontally to our flight altitude at point "B" (25,000 ft in this case). To find our Mach, we go down vertically to point "C" to obtain 0.86 Mach. To get our TAS at our actual environmental conditions, we go from point "B" vertically until we hit the Sea Level (S.L.) reference line at point "D", then travel horizontally until we reach our actual outside air temperature (-20°C at altitude) at point "E", then go up vertically to read our actual TAS from the scale at point "F" (535 KTAS). If we wanted our TAS at "standard" temperature and pressure conditions, we would follow the dashed lines slanting upward from point "B" to point "G" and read 515 KTAS from the scale. Naturally, we could go into the graph at any point and go "backwards" to find CAS from true Mach or TAS.

Figure 3 shows a much wider range of Mach numbers. It contains only TAS and Mach, since aircraft generally do not fly above Mach 2, but missiles (which don't have airspeed indicators) do. The data on this graph can be obtained directly from the following formula for use at altitudes of 36,000 ft and below:

Speed of sound equation - RF Cafe      [2]

The speed of sound calculated from this formula can be used with the equation on the first page to obtain Mach number. This equation uses the standard sea level temperature of 59E F and a lapse rate of -3.57°/1000 ft altitude. Temperature stabilizes at -69.7° F at 36,000 ft so the speed of sound stabilizes there at 573 knots. See the last page of this section for a derivation of equation [2].

TAS and CAS Relationship with Varying Altitude and Temperature - RF Cafe

Figure 1. TAS and CAS Relationship with Varying Altitude and Temperature




TAS and CAS Relationship with Varying Altitude and Temperature (Continued) - RF Cafe

Figure 2. TAS and CAS Relationship with Varying Altitude and Temperature (Continued)

Mach Number vs TAS Variation with Altitude  - RF Cafe

Figure 3. Mach Number vs TAS Variation with Altitude

The following is a derivation of equation [2] for the speed of sound:

Given:   p = pressure (lb/ft2)                       T = absolute temperature (°Rankine) = °F + 459.7
             v = specific volume (ft3/lb)              w = specific weight (lb/ft3) = 1/ν
             R = a constant (for air: R = 53.3)    ρ = density = w/g = 1/gν ? ν = 1/gρ
From Boyle's law of gasses: pν = RT , therefore we have: p/ρ = gRT = (32.2)(53.3)T = 1718 T     [3]

It can also be shown that: p/ργ = constant; for air γ = 1.4       [4]

From the continuity equation applied to a sound wave: ρAVa = (ρ+dρ)A(Va + dVa)      [5]

Expanding and dropping insignificant terms gives: dVa = -Va dρ/ρ      [6]

Using Newton's second law (p + ρVa/2 = a constant) and taking derivatives: dp = -ρVadVa

substituting into [6] gives: Va2 = dp/dρ      [7]
Then taking derivatives of [4] and substituting in [7] gives: RF Cafe - formula    [8]

Then using [3] gives: RF Cafe - formula   [9]

Using a "Standard" atmosphere of 59° F @ Sea Level (S.L.) and a lapse rate of -3.57°/1000 ft altitude:

RF Cafe - equation        which is equation [2]

Table of Contents for Electronics Warfare and Radar Engineering Handbook
Introduction | Abbreviations | Decibel | Duty Cycle | Doppler Shift | Radar Horizon / Line of Sight | Propagation Time / Resolution | Modulation | Transforms / Wavelets | Antenna Introduction / Basics | Polarization | Radiation Patterns | Frequency / Phase Effects of Antennas | Antenna Near Field | Radiation Hazards | Power Density | One-Way Radar Equation / RF Propagation | Two-Way Radar Equation (Monostatic) | Alternate Two-Way Radar Equation | Two-Way Radar Equation (Bistatic) | Jamming to Signal (J/S) Ratio - Constant Power [Saturated] Jamming | Support Jamming | Radar Cross Section (RCS) | Emission Control (EMCON) | RF Atmospheric Absorption / Ducting | Receiver Sensitivity / Noise | Receiver Types and Characteristics | General Radar Display Types | IFF - Identification - Friend or Foe | Receiver Tests | Signal Sorting Methods and Direction Finding | Voltage Standing Wave Ratio (VSWR) / Reflection Coefficient / Return Loss / Mismatch Loss | Microwave Coaxial Connectors | Power Dividers/Combiner and Directional Couplers | Attenuators / Filters / DC Blocks | Terminations / Dummy Loads | Circulators and Diplexers | Mixers and Frequency Discriminators | Detectors | Microwave Measurements | Microwave Waveguides and Coaxial Cable | Electro-Optics | Laser Safety | Mach Number and Airspeed vs. Altitude Mach Number | EMP/  Aircraft Dimensions | Data Busses | RS-232 Interface | RS-422 Balanced Voltage Interface | RS-485 Interface | IEEE-488 Interface Bus (HP-IB/GP-IB) | MIL-STD-1553 & 1773 Data Bus |
  This HTML version may be printed but not reproduced on websites.