Waves: Properties, Behavior, and Sound (Physics 2025 Study Guide)

Waves

Wave Properties

Waves are fundamental to physics, describing the transfer of energy without the permanent transfer of matter. Understanding wave properties is essential for analyzing phenomena in sound, light, and other physical systems.

• Energy Transfer: Waves transfer energy from one location to another. The medium itself does not move with the wave; only energy and information are transmitted.

• Mechanical Waves: These require a physical medium (such as air, water, or a solid) to propagate. Examples include sound waves, seismic waves, and vibrations in strings.

• Transverse Waves: The oscillations are perpendicular to the direction of wave travel. Examples: vibrations of stringed instruments, electromagnetic waves.

• Longitudinal Waves: The oscillations are parallel to the direction of wave travel. Examples: sound waves, seismic P-waves.

• Key Terms:

  • Compression: Region where particles are closest together (longitudinal waves).

  • Rarefaction: Region where particles are furthest apart (longitudinal waves).

  • Crest: Highest point of a transverse wave.

  • Trough: Lowest point of a transverse wave.

  • Displacement: Distance a point in the medium moves from its equilibrium position.

  • Amplitude: Maximum displacement from equilibrium; relates to energy carried by the wave.

  • Period (T): Time for one complete cycle of the wave.

  • Frequency (f): Number of cycles per second (measured in Hertz, Hz).

  • Wavelength (\(\lambda\)): Distance between two consecutive crests or compressions.

  • Velocity (v): Speed at which the wave propagates through the medium.

• Graphical Representation: These properties can be identified and measured on wave graphs (e.g., displacement vs. time or displacement vs. position).

Analyzing Waves

Wave graphs allow for the determination of amplitude, period, frequency, and wavelength for both transverse and longitudinal waves.

• Amplitude: Measured from equilibrium to crest or trough.

• Period: Time between successive crests (or compressions).

• Frequency: Inverse of period: $f = \frac{1}{T}$

• Wavelength: Distance between successive crests or compressions.

• Velocity: Related to frequency and wavelength: $v = f \lambda$

• Example: For sound in air, $v = 346\ \text{m}\ \text{s}^{-1}$ (approximate value at room temperature).

Wave Behavior at Boundaries

Waves exhibit various behaviors when encountering boundaries or obstacles.

• Reflection: Wave bounces off a boundary; angle of incidence equals angle of reflection.

• Refraction: Wave changes direction and speed when passing into a different medium.

• Diffraction: Wave spreads out after passing through a narrow opening or around an obstacle.

• Superposition: When two or more waves overlap, their displacements add algebraically.

Interference and Superposition

Interference occurs when waves overlap, resulting in constructive or destructive patterns.

• Constructive Interference: Waves add to produce a larger amplitude.

• Destructive Interference: Waves add to produce a smaller (or zero) amplitude.

• Resultant Amplitude: Determined by the principle of superposition: $A_{\text{resultant}} = A_1 + A_2$ (for simple cases).

• Standing Waves: Formed by the superposition of two waves traveling in opposite directions; characterized by nodes (points of zero amplitude) and antinodes (points of maximum amplitude).

Sound Waves

Sound is a longitudinal mechanical wave, and its behavior in pipes and strings is governed by harmonics and resonance.

• Fundamental (First) Harmonic: The lowest frequency at which a system naturally vibrates.

• Natural Frequency: Frequency at which a system oscillates when not driven by an external force.

• Standing Waves in Pipes:

  • Pipes Open at Both Ends: $L = n \frac{\lambda}{2}$, where $n = 1, 2, 3, ...$

  • Pipes Closed at One End: $L = (2n - 1) \frac{\lambda}{4}$, where $n = 1, 2, 3, ...$

• Standing Waves on Strings: $L = n \frac{\lambda}{2}$, where $n$ is the harmonic number.

• Resonance: Occurs when a system is driven at its natural frequency, resulting in efficient energy transfer and large amplitude oscillations.

Table: Comparison of Transverse and Longitudinal Waves

Property	Transverse Wave	Longitudinal Wave
Oscillation Direction	Perpendicular to wave travel	Parallel to wave travel
Examples	String vibrations, electromagnetic waves	Sound waves, seismic P-waves
Key Features	Crest, trough	Compression, rarefaction

Table: Standing Wave Formulas

System	Formula	Harmonic Number
Pipe open at both ends	$L = n \frac{\lambda}{2}$	$n = 1, 2, 3, ...$
Pipe closed at one end	$L = (2n - 1) \frac{\lambda}{4}$	$n = 1, 2, 3, ...$
Stretched string	$L = n \frac{\lambda}{2}$	$n = 1, 2, 3, ...$

Example: For a pipe open at both ends with length $L = 1.0\ \text{m}$, the fundamental wavelength is $\lambda = 2L = 2.0\ \text{m}$.

Additional info: The velocity of sound in air is temperature-dependent; $v = 346\ \text{m}\ \text{s}^{-1}$ is typical at 25°C.