Resonance and Sound Waves

The Doppler effect is the apparent change in frequency or wavelength of a wave caused by relative motion between a source and an observer. It is commonly experienced with sound waves. For example, when an ambulance passes by with its siren on, the pitch sounds higher as it approaches and lower as it moves away. Although the siren produces a constant frequency, the motion changes how frequently the sound waves reach the observer.

When a source moves toward an observer, the wavefronts become compressed. This decreases the wavelength and increases the observed frequency. As a result, the sound appears higher in pitch. When the source moves away, the wavefronts spread out, increasing the wavelength and decreasing the observed frequency. This causes the sound to appear lower in pitch.

The relationship between wave speed, frequency, and wavelength is:

$v=f\lambda$v=fλ

where:

• $v$v = wave speed
• $f$f = frequency
• $\lambda$λ = wavelength

In the Doppler effect, the wave speed in the medium usually remains constant, but the observed frequency and wavelength change because of relative motion.

The Doppler effect can occur because of either a moving source or a moving observer. If the observer moves toward a stationary source, they encounter wavefronts more frequently, increasing the observed frequency. If the observer moves away, the observed frequency decreases. When the source itself moves, the spacing between wavefronts changes directly, creating compressed or stretched wavelengths in the medium.

For sound waves, the Doppler shift equation is:

$f' = f\left(\frac{v \pm v_o}{v \mp v_s}\right)$

f′=f(v∓vs​v±vo​​)

where:

• $f'$f′ = observed frequency
• $f$f = emitted frequency
• $v$v = wave speed
• $v_o$vo​ = observer speed
• $v_s$vs​ = source speed

The signs depend on whether the source and observer are moving toward or away from one another.

Example 1: Moving Source

An ambulance siren emits a frequency of

$800 \, \text{Hz}$

800Hz. If the ambulance moves toward a stationary observer, the observed frequency may increase to approximately

$850 \, \text{Hz}$

850Hz. The observer hears a higher pitch because the sound waves are compressed in front of the moving source.

Example 2: Moving Observer

A cyclist riding toward a stationary bell ringing at

$500 \, \text{Hz}$

500Hz will hear a slightly higher frequency because they encounter wavefronts more rapidly.

The Doppler effect is also extremely important for electromagnetic waves, including visible light. In astronomy, scientists observe changes in the wavelengths of light from stars and galaxies. When an object moves toward Earth, the wavelengths become shorter and shift toward the blue end of the spectrum. This is called blueshift. When an object moves away, the wavelengths become longer and shift toward the red end of the spectrum. This is called redshift.

The shift in wavelength can be expressed using:

$z=\frac{\Delta \lambda}{\lambda_0}$

z=λ0​Δλ​

where:

• $z$z = redshift parameter
• $\Delta \lambda$Δλ = change in wavelength
• $\lambda_0$λ0​ = original wavelength

Positive values of

$z$

z indicate redshift, while negative values indicate blueshift.

Example 3: Redshift

Light from a distant galaxy may normally have a wavelength of

$500 \, \text{nm}$

500nm, but astronomers observe it at

$550 \, \text{nm}$

550nm. The increase in wavelength indicates the galaxy is moving away from Earth.

Redshift observations have provided important evidence that the universe is expanding. Most distant galaxies show redshifted light, indicating they are moving away from Earth. This discovery contributed to the development of the Big Bang theory and modern cosmology. Astronomers also use Doppler shifts to study the rotation of galaxies, detect exoplanets, and measure the motion of stars.

The Doppler effect has many practical technological applications. Radar systems use reflected radio waves to measure the speed of vehicles, aircraft, and weather systems. Police radar guns determine vehicle speed by measuring frequency shifts in reflected microwaves. Weather radar uses Doppler measurements to track wind patterns and storm motion.

Example 4: Radar Speed Detection

A police radar gun sends out microwaves that reflect off a moving car. If the reflected frequency is slightly higher than the emitted frequency, the system calculates the vehicle’s speed from the Doppler shift.

Sonar systems use sound waves underwater to detect objects and measure motion. Submarines and ships use sonar to determine the location and speed of underwater objects. Medical ultrasound devices also use Doppler techniques to measure blood flow inside the human body by detecting frequency shifts in reflected sound waves.

Example 5: Medical Ultrasound

Doctors use Doppler ultrasound to measure blood flow in arteries. Faster-moving blood produces larger frequency shifts in the reflected sound waves.

The Doppler effect demonstrates how motion affects the observation of waves. From changing siren pitches to expanding galaxies and advanced radar systems, Doppler shifts provide valuable information about motion, speed, and the structure of the universe.