Waves, Beats, and the Doppler Effect: Study Notes

Final Exam Overview

• Date & Time: Tuesday, Dec. 16, 9–12, in the assigned room.

• Duration: Three-hour exam, covering approximately 50% more material than midterms.

• Scope: Comprehensive, with greater emphasis on material since the last exam: Momentum, Simple Harmonic Motion (SHM), and Waves.

• Resources: Equation sheet provided; calculators allowed.

• Preparation: Practice problems available on Mastering Physics; study guide on Moodle.

Beats in Sound Waves

Superposition and Interference of Waves

When two sound waves of slightly different frequencies interfere, they produce a phenomenon known as beats. The resulting sound alternates between loud and soft, corresponding to constructive and destructive interference.

• Constructive Interference: Occurs when waves are in phase, resulting in increased amplitude (louder sound).

• Destructive Interference: Occurs when waves are half a cycle out of phase, resulting in decreased amplitude (softer sound).

Beat Frequency is the frequency at which the amplitude of the combined wave fluctuates:

• Definition: The number of beats per second is equal to the absolute difference between the two frequencies.

$f_{\text{beat}} = |f_1 - f_2|$

• Beat Period: The time for one complete cycle of loudness variation is the reciprocal of the beat frequency.

$T_{\text{beat}} = \frac{1}{f_{\text{beat}}}$

• Example: If you hear a sound with a frequency of 256 Hz and the amplitude varies with a period of 2 seconds, the beat frequency is $f_{\text{beat}} = \frac{1}{2} = 0.5$ Hz. The two component frequencies are 255.75 Hz and 256.25 Hz.

Doppler Effect

Definition and Physical Basis

The Doppler Effect is the change in observed frequency of a wave when the source and/or observer are moving relative to the medium in which the wave propagates. It applies to all types of waves, including sound and light.

• Wave velocity (v): Speed of the wave in the medium (e.g., speed of sound in air).

• Observer velocity ($v_o$): Speed of the observer relative to the medium.

• Source velocity ($v_s$): Speed of the source relative to the medium.

• Source frequency ($f_s$): Frequency emitted by the source.

• Observed frequency ($f_o$): Frequency detected by the observer.

Doppler Effect Formulas

• Moving Source (observer at rest):

$f_o = \frac{v}{v \mp v_s} f_s$

• Use minus sign if the source moves toward the observer (frequency increases).

• Use plus sign if the source moves away from the observer (frequency decreases).

• Moving Observer (source at rest):

$f_o = \frac{v \pm v_o}{v} f_s$

• Use plus sign if the observer moves toward the source (frequency increases).

• Use minus sign if the observer moves away from the source (frequency decreases).

• Both Source and Observer Moving:

$f_o = \frac{v \pm v_o}{v \mp v_s} f_s$

• Signs depend on the direction of motion (toward or away).

Conceptual Examples

• Ambulance Example: If you are standing still and an ambulance approaches at constant speed, the frequency you hear does not change as it gets closer (assuming constant speed and direction).

• Moving Source: If a sound source moves toward you (no wind, you at rest), you hear a higher frequency and a shorter wavelength compared to when the source is stationary.

Doppler Effect with Light

The Doppler Effect also applies to electromagnetic waves such as light. In astronomy, it is used to determine the motion of stars and galaxies (redshift and blueshift).

• Redshift: Light from an object moving away from the observer is shifted to longer wavelengths (lower frequency).

• Blueshift: Light from an object moving toward the observer is shifted to shorter wavelengths (higher frequency).

Applications

• Medical Imaging: Doppler ultrasound is used to measure blood flow in the body.

• Astronomy: Measuring the speed and direction of stars and galaxies.

• Everyday Life: The change in pitch of a passing siren is a classic example of the Doppler Effect.

Summary Table: Doppler Effect Formulas

Situation	Observed Frequency ($f_o$)
Source moving toward observer	$f_o = \frac{v}{v - v_s} f_s$
Source moving away from observer	$f_o = \frac{v}{v + v_s} f_s$
Observer moving toward source	$f_o = \frac{v + v_o}{v} f_s$
Observer moving away from source	$f_o = \frac{v - v_o}{v} f_s$
Both moving (general case)	$f_o = \frac{v \pm v_o}{v \mp v_s} f_s$

Additional info: The notes also reference pop culture (Sheldon Cooper) and medical imaging (Doppler ultrasound), which help illustrate the real-world relevance of the Doppler Effect.