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Picture is a photograph of the blue glow in a reactor pool.
The blue glow in this research reactor pool is Cerenkov radiation caused by subatomic particles traveling faster than the speed of light in water. (credit: U.S. Nuclear Regulatory Commission)


  • The Mach number is the velocity of a source divided by the speed of sound, M = v s v .
  • When a sound source moves faster than the speed of sound, a shock wave is produced as the sound waves interfere.
  • A sonic boom is the intense sound that occurs as the shock wave moves along the ground.
  • The angle the shock wave produces can be found as sin θ = v v s = 1 M .
  • A bow wake is produced when an object moves faster than the speed of a mechanical wave in the medium, such as a boat moving through the water.

Key equations

Pressure of a sound wave Δ P = Δ P max sin ( k x ω t + ϕ )
Displacement of the oscillating molecules of a
sound wave
s ( x , t ) = s max cos ( k x ω t + ϕ )
Velocity of a wave v = f λ
Speed of sound in a fluid v = β ρ
Speed of sound in a solid v = Y ρ
Speed of sound in an ideal gas v = γ R T M
Speed of sound in air as a function of temperature v = 331 m s T K 273 K = 331 m s 1 + T C 273 ° C
Decrease in intensity as a spherical wave expands I 2 = I 1 ( r 1 r 2 ) 2
Intensity averaged over a period I = P A
Intensity of sound I = ( Δ p max ) 2 2 ρ v
Sound intensity level β ( d B ) = 10 log 10 ( I I 0 )
Resonant wavelengths of a tube closed at one end λ n = 4 n L , n = 1 , 3 , 5 ,…
Resonant frequencies of a tube closed at one end f n = n v 4 L , n = 1 , 3 , 5 ,…
Resonant wavelengths of a tube open at both ends λ n = 2 n L , n = 1 , 2 , 3 ,…
Resonant frequencies of a tube open at both ends f n = n v 2 L , n = 1 , 2 , 3 ,…
Beat frequency produced by two waves that
differ in frequency
f beat = | f 2 f 1 |
Observed frequency for a stationary observer
and a moving source
f o = f s ( v v v s )
Observed frequency for a moving observer
and a stationary source
f o = f s ( v ± v o v )
Doppler shift for the observed frequency f o = f s ( v ± v o v v s )
Mach number M = v s v
Sine of angle formed by shock wave sin θ = v v s = 1 M

Conceptual questions

What is the difference between a sonic boom and a shock wave?

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Due to efficiency considerations related to its bow wake, the supersonic transport aircraft must maintain a cruising speed that is a constant ratio to the speed of sound (a constant Mach number). If the aircraft flies from warm air into colder air, should it increase or decrease its speed? Explain your answer.

The speed of sound decreases as the temperature decreases. The Mach number is equal to M = v s v , so the plane should slow down.

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When you hear a sonic boom, you often cannot see the plane that made it. Why is that?

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An airplane is flying at Mach 1.50 at an altitude of 7500.00 meters, where the speed of sound is v = 343.00 m/s . How far away from a stationary observer will the plane be when the observer hears the sonic boom?

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A jet flying at an altitude of 8.50 km has a speed of Mach 2.00, where the speed of sound is v = 340.00 m/s . How long after the jet is directly overhead, will a stationary observer hear a sonic boom?

θ = 30.02 ° v s = 680.00 m/s tan θ = y v s t , t = 21.65 s

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The shock wave off the front of a fighter jet has an angle of θ = 70.00 ° . The jet is flying at 1200 km/h. What is the speed of sound?

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A plane is flying at Mach 1.2, and an observer on the ground hears the sonic boom 15.00 seconds after the plane is directly overhead. What is the altitude of the plane? Assume the speed of sound is v w = 343.00 m/s .

sin θ = 1 M , θ = 56.47 ° y = 9.31 km

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Questions & Answers

what is mass
ilance Reply
it is the quantity of matter in a substance....expressed in Kg as the SI unit.
what is the universe of conductivity
Agbeyangi Reply
what is si unit of force?
Jose Reply
Kg m / s^2 You can see that this is the case from dimensional analysis of F = M a
Newton which is Kgm/s2
What's power in physics
Power is the rate of doing work. P=W/T
A car moving with a velocity of 54km/hr accelerates uniformly at the rate of 2mper seconds. Calculate the distance traveled from the place where acceleration began to where the velocity reaches 72km/hr and the time taken to cover d distance
Betty Reply
t=2.5s and s=43.75m
how to find amplitude of interfering wave
Roha Reply
how do we slove a pendulum clock shaft question
Seray Reply
An object of mass m moves with acceleration a down a rough incline.
Phyu Reply
If the average velocity of an object is zero in some time interval, what can you say about the displacement of the object fr that interval?
what is force
Biggie Reply
you ever seen star wars?
What is angular velocity
Amara Reply
refer to how fast an object rotates ot revolve to another point
An object undergoes constant acceleration after starting from rest and then travels 5m in the first seconds .determine how far it will go in the next seconds
Sophy Reply
what is the deference between precision and accuracy
In measurement of a set, accuracy refers to closeness of the measurements to a specific value, while precision refers to the closeness of the measurements to each other. 
Thank you Mr..
Are there is a difference between Error and Relative Error...
An aircraft flies 300km due east and 600km due north. determine the magnitude of its displacement
Sophy Reply
which formula did you use
what is Fermat principle.
Madhu Reply
find the angle of projection at which the horizontal range is twice the maximum height of a projectile
Akaji Reply
impulse by height fomula
Tigst Reply
what is impulse?
Tigst Reply
impulse is the integral of a force (F),over the interval for which it act
Practice Key Terms 3

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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