Doppler Effect and Mach 1
This activity belongs to the GeoGebra book The Domain of the Time.
Even something as straightforward as uniform motion can provide fascinating scenarios. Here, we combine the uniform motion of sound waves with the constant motion of an F18 aircraft, where the user can choose the speed at which the plane travels.
This setup allows us to observe the compression of sound waves, which results in the variation of frequency perceived by the observer (the Doppler Effect). Additionally, it demonstrates the breaking of the sound barrier, where a shock wave appears in the form of a conical surface. This shock wave is a pressure wave that travels faster than the speed of sound in that medium—i.e., it exceeds Mach 1. When this wave reaches an observer, they will hear a sonic boom.
- Note 1: In the construction, the aircraft may not appear to be to scale. However, we can imagine that the aircraft is on a different plane, much closer to the observer (who is always in the foreground) than the x-axis.
- Nota 2: Since sound is essentially a variation in air pressure, the cone represents a region of intense sound (the sonic boom). If the aircraft were very small, the effect would be a singular burst. However, with a larger aircraft, there may be two or more cones of pressure, resulting in multiple sonic booms—at least one from the nose and another from the tail.
SCRIPT FOR SLIDER anima
# Calculate the elapsed seconds dt; add one second if t1(1) < tt
SetValue(tt, t1(1))
SetValue(t1, First(GetTime(), 3))
SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)
# Move M while keeping focus on it
SetValue(M, M + dt v)
SetActiveView(1)
CentreView(M + (-100, 500))
# Control automatic changes in velocity
SetValue(v0, If(desp ∧ v0 < 140 ∨ mach1 ∧ v0 < 343 ∨ mach2, v0 + 1, v0))
SetValue(v, (v0, 0))
SetValue(M, If(80 < v0 < 140, (x(M), 17v0 - 1210), M))
Author of the activity and GeoGebra construction: Rafael Losada.