Wave Properties & Doppler Effect

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Visualization of a sound wave transforming from a physical longitudinal compression wave into a mathematical transverse sine wave graph.

Waves transport energy, not matter. Understanding their properties is the key to unlocking acoustics, optics, and quantum mechanics.

Whether it’s a ripple in a pond, the sound of a guitar string, or the light from a distant galaxy, all waves share fundamental mathematical properties. In this section, we will define those properties and explore how waves behave when they move, hit barriers, or are emitted by a moving object.

1. The Wave Equation (Topics 14.1 & 14.2)

A wave is a disturbance that travels through a medium (or a vacuum, in the case of light). The most important equation you will use to describe periodic waves relates speed ( $v$ ), wavelength ( $\lambda$ ), and frequency ( $f$ ).

$v = \lambda f$

Where $v$ is wave speed (m/s), $\lambda$ is wavelength (m), and $f$ is frequency (Hz).

Concept First: The speed of a wave ( $v$ ) is determined exclusively by the properties of the medium it is traveling through. If you increase the frequency of a sound wave, the sound doesn’t travel faster; the wavelength just gets shorter to compensate!

Comparison diagram of a transverse wave on a string vs. a longitudinal wave in a spring, labeling crests, troughs, compressions, and rarefactions.

Transverse vs. Longitudinal: Transverse waves (like light) vibrate perpendicular to the direction of travel. Longitudinal waves (like sound) vibrate parallel to the direction of travel.

2. Boundary Behavior (Topic 14.3)

What happens when a wave reaches the end of its medium or tries to enter a new one?

Fixed Boundary: The reflected wave pulse is inverted (flipped upside down).
Free Boundary: The reflected wave pulse remains upright.
Entering a New Medium: The frequency ( $f$ ) never changes when a wave enters a new medium. Because the speed ( $v$ ) changes, the wavelength ( $\lambda$ ) must change to compensate.

3. The Doppler Effect (Topic 14.5)

Have you ever noticed how a police siren sounds higher pitched as it drives toward you, and lower pitched as it drives away? This apparent shift in frequency is the Doppler Effect.

When a source is moving toward an observer, it “catches up” to its own sound waves, compressing the wavefronts together. This creates a shorter wavelength and a higher perceived frequency. When moving away, the wavefronts stretch out, creating a lower perceived frequency.

⚙️ Interactive Doppler Simulator

Adjust the speed of the source. Notice how the wavefronts bunch up in the direction of motion, creating a higher frequency for an observer standing in front of the dot.

Source Velocity ( $v_s$ ): 0 m/s

State: Stationary Source (f_observed = f_source)

4. Quick AP Practice

📚 Unit 14.1 – 14.5 Mastery Challenge

1. A student shakes a spring to create a wave. If the student shakes the spring twice as fast (doubling the frequency), what happens to the wave speed?

Check Answer

It remains exactly the same. Wave speed is determined entirely by the medium (the tension and mass of the spring). If frequency doubles, the wavelength simply halves.

2. Light from a distant galaxy is observed on Earth. The spectral lines are shifted toward the red end of the spectrum (lower frequency). What does this tell us about the galaxy?