Visualize sound wave compression and stretching as the source moves. Real-time Mach number and observed frequency calculation. Experience shock waves at supersonic speeds.
Parameters
Source speed vs
m/s
Speed of sound vc
m/s
Source frequency fs
Hz
Presets
Controls
Shock wave! Mach ≥ 1. Wavefronts pile up forming a Mach cone (conical shock wave).
Front (source approaching): $(v_c - v_s)$ in denominator
Rear (source receding): $(v_c + v_s)$ in denominator
Mach number: $Ma = v_s / v_c$
What is the Doppler Effect?
🙋
What exactly is the Doppler Effect? I've heard it's why a siren sounds different as it passes by.
🎓
Basically, it's the change in frequency (and therefore pitch) of a wave when the source and observer are moving relative to each other. As a source moves toward you, the sound waves get compressed, raising the pitch. As it moves away, they stretch out, lowering the pitch. Try moving the "Source Speed" slider in the simulator above to see the wavefronts bunch up in front of the moving source.
🙋
Wait, really? So the formula in the simulator, $f_\text{obs}= f_s \cdot \frac{v_c \pm v_o}{v_c \mp v_s}$, looks complicated. What do all the plus/minus signs mean?
🎓
In practice, you choose the sign based on whether the motion causes the wave to be compressed or stretched. The top sign is for when the source and observer are moving toward each other. The bottom sign is for when they're moving apart. For instance, if a police car ($v_s$) is moving toward a stationary observer ($v_o=0$), the formula simplifies to $f_\text{obs}= f_s \cdot \frac{v_c}{v_c - v_s}$. Notice the denominator gets smaller, so the observed frequency $f_\text{obs}$ increases!
🙋
What happens if you crank the source speed up really high in the simulator? The waves look like they're piling up into a straight line.
🎓
Great observation! That's a shock wave, or Mach cone. When the source speed ($v_s$) equals or exceeds the speed of sound ($v_c$), it outruns its own sound waves. The overlapping wavefronts create that sharp, V-shaped cone you see. The ratio $v_s / v_c$ is called the Mach number. Try setting the source speed to be greater than the speed of sound parameter in the simulator to see the shockwave form clearly.
Physical Model & Key Equations
The core equation governing the Doppler Effect for sound relates the frequency heard by an observer ($f_\text{obs}$) to the source's original frequency ($f_s$), the speed of sound in the medium ($v_c$), and the speeds of the source ($v_s$) and observer ($v_o$). The signs depend on the direction of motion.
$f_\text{obs}$: Observed frequency (Hz) $f_s$: Source frequency (Hz) $v_c$: Speed of sound in the medium (m/s) $v_s$: Speed of the source (m/s) — positive if moving toward observer $v_o$: Speed of the observer (m/s) — positive if moving toward source Sign Rule: Use the top signs (+ in numerator, - in denominator) if source and observer are moving toward each other. Use the bottom signs if they are moving apart.
When the source moves at or faster than sound, the Mach number ($Ma$) becomes critical. It defines the regime where a shock wave forms. The angle of the resulting Mach cone ($\theta$) is determined by the inverse sine of the Mach number.
$Ma$: Mach number (dimensionless). $Ma < 1$ is subsonic, $Ma = 1$ is sonic, $Ma > 1$ is supersonic. $\theta$: Half-angle of the Mach cone. As speed increases ($Ma$ increases), this angle gets smaller, making the cone narrower and sharper.
Frequently Asked Questions
When the speed of the sound source exceeds the speed of sound (Mach number 1), the source overtakes the sound waves it has emitted, causing the wavefronts to overlap and generate a shock wave (Mach cone). At this point, the way sound reaches the observer fundamentally changes, resulting in a discontinuous change in frequency.
The half-angle θ of the Mach cone can be calculated using the speed of sound v_c and the source speed v_s with the formula sinθ = v_c / v_s. Since the Mach number Ma is displayed in the simulator, you can find θ = arcsin(1/Ma). The larger the Ma, the sharper the cone.
In the Doppler effect formula f_obs = f_s × v_c / (v_c ∓ v_s), when the source is approaching, the denominator is v_c - v_s (smaller), so the frequency increases; when moving away, it is v_c + v_s (larger), so the frequency decreases. This difference in sign produces the asymmetric change.
Changing the speed of sound v_c alters the Mach number for the same source speed, affecting the conditions for shock wave generation and the degree of the Doppler effect. The default value is set to approximately 340 m/s (at 20°C), which is the typical speed of sound in air. This allows experiments with different media (air temperatures).
Real-World Applications
Speed Radar Guns: Police radar guns and baseball pitch speed trackers use the Doppler Effect with radio or laser waves. The device sends out a wave that reflects off the moving car or ball. The change in frequency of the reflected wave is used to calculate the object's speed with high precision.
Medical Ultrasound (Doppler Echocardiography): This is a critical diagnostic tool. Sound waves are directed into the body, and their reflection off moving blood cells is analyzed. The frequency shift reveals the speed and direction of blood flow, helping diagnose heart valve problems, blood clots, and blocked arteries.
Astronomy & "Redshift": The Doppler Effect applies to light waves too. By analyzing the spectrum of light from stars and galaxies, astronomers can see if the characteristic lines are shifted toward the red (lower frequency, moving away) or blue (higher frequency, moving toward us) end of the spectrum. This "redshift" is key evidence for the expansion of the universe.
Aviation & Sonic Booms: When an aircraft like a fighter jet breaks the sound barrier ($Ma \ge 1$), it generates a Mach cone. The pressure wave from this cone reaches the ground as a loud sonic boom—a sudden, sharp thunderclap. This is not a one-time event at the moment of crossing Mach 1, but a continuous effect along the supersonic flight path.
Common Misconceptions and Points to Note
Here are some points where beginners often get tripped up when mastering this simulator. First, don't assume that "the speed of sound is always constant". The simulator uses a fixed speed of sound $v_c$, but in the real world, it varies significantly with air temperature, pressure, and medium (e.g., water or metal). For instance, it's about 331 m/s in air at 0°C, but about 343 m/s at 20°C. When working with numbers in practice, your first step should be to confirm the speed of sound for your specific environment.
Next, understand that the point of abrupt change in observed frequency is not necessarily "the instant the source passes directly beside the observer". It may appear that way in the simulator because the observer is fixed, but in reality, it's the relative velocity vector between the source and the observer that matters. For example, if the observer is also moving or if the source is not approaching the observer head-on, the formulas become more complex. Think of this as a tool for learning the basic form.
Finally, a crucial pitfall: when the Mach number exceeds 1, the standard Doppler effect formula no longer applies directly. When a source becomes supersonic, sound waves pile up in front of it, forming a shock wave (a sonic boom). In this region, an observer hears a single "bang" from the shockwave (an N-wave) before the source passes by. The moment the simulator's Mach cone crosses the observer corresponds to this. Remember, the continuous change in pitch of a sound like "wee-oo wee-oo" does not occur in the supersonic regime.