Superposition and Standing Waves: Physics for Life Sciences I (Lecture 27, Sections 16.1–16.4)

Chapter 16: Superposition and Standing Waves

Introduction

This chapter explores the principle of superposition, interference, and the formation of standing waves in various physical systems. These concepts are fundamental to understanding wave phenomena in physics, including sound, light, and mechanical waves.

The Wave Model

Traveling Waves

• Traveling wave: An organized disturbance that moves through a medium at a well-defined speed v.

• Transverse waves: Particles of the medium move perpendicular to the direction of wave travel (e.g., waves on a string).

• Longitudinal waves: Particles move parallel to the direction of wave travel (e.g., sound waves in air).

Mechanical and Electromagnetic Waves

• Mechanical waves require a material medium. The speed of the wave depends on the properties of the medium, not the wave itself.

• For a wave on a string: $v_{string} = \sqrt{\frac{T}{\mu}}$ where T is tension and μ is linear mass density.

• For a sound wave in a gas: $v_{sound} = \sqrt{\frac{\gamma RT}{M}}$ where γ is the adiabatic index, R is the gas constant, T is temperature, and M is molar mass.

• Electromagnetic waves do not require a medium and travel at $c = 3.00 \times 10^8$ m/s in vacuum.

Mathematical Representation of Waves

• Sinusoidal waves are produced by sources in simple harmonic motion.

• General equation: $y(x, t) = A \cos \left[2\pi \left(\frac{x}{\lambda} + \frac{t}{T}\right)\right]$

• Period: $T = \frac{1}{f}$

• Wave speed: $v = \lambda f$

Wave Intensity and Sound Intensity Level

Intensity of a Wave

• Intensity (I): Ratio of power to area: $I = \frac{P}{a}$

• For spherical waves: $I = \frac{P_{source}}{4\pi r^2}$

Sound Intensity Level

• Sound intensity level (β): Logarithmic measure in decibels (dB): $\beta = (10 \text{ dB}) \log_{10} \left(\frac{I}{I_0}\right)$

• I0 = $1.0 \times 10^{-12}$ W/m2 is the threshold of hearing.

Doppler Effect and Shock Waves

Doppler Effect

• Doppler effect: Shift in frequency due to relative motion between source and observer.

• Moving source, stationary observer: Receding: $f_o = \frac{f_s}{1 + v_s / v}$ Approaching: $f_o = \frac{f_s}{1 - v_s / v}$

• Moving observer, stationary source: Approaching: $f_o = \left(1 + \frac{v_o}{v}\right) f_s$ Moving away: $f_o = \left(1 - \frac{v_o}{v}\right) f_s$

• Reflection from a moving object (ultrasound): For $v_o \ll v$, $\Delta f = 2 f \frac{v_o}{v}$

• Shock wave: Formed when an object moves faster than the wave speed in a medium, causing overlapping waves and a large amplitude disturbance.

The Principle of Superposition

Definition and Application

• Principle of superposition: When two or more waves are present at a point, the displacement is the sum of the individual displacements.

• To apply: Calculate the displacement each wave would cause alone, then add them point by point.

Interference

• Interference: The result of superposition of two or more waves.

• Constructive interference: Occurs when waves are in phase, resulting in larger amplitude.

• Destructive interference: Occurs when waves are out of phase, resulting in reduced or zero amplitude.

Standing Waves

Formation and Properties

• Standing wave: Formed by two identical waves traveling in opposite directions, resulting in a wave that oscillates in place.

• Individual points oscillate, but the wave does not travel.

Nodes and Antinodes

• Nodes: Points that never move; spaced $\lambda/2$ apart.

• Antinodes: Points of maximum displacement, halfway between nodes.

• Wavelength of standing wave: Twice the distance between successive nodes or antinodes.

• At nodes: Destructive interference, zero intensity.

• At antinodes: Constructive interference, maximum intensity.

Standing Waves on a String

Reflection and Boundary Conditions

• Waves reflect at boundaries (fixed ends or discontinuities).

• At a fixed end, the reflected pulse is inverted but amplitude is unchanged.

• At a discontinuity, part of the wave is transmitted, part is reflected.

Standing Wave Modes

• Standing waves form when the string is fixed at both ends, requiring nodes at each end.

• Mode number m quantifies the number of antinodes.

• Allowed wavelengths: $\lambda_m = \frac{2L}{m}$, where L is string length and m = 1, 2, 3, ...

• Allowed frequencies: $f_m = m f_1$, where $f_1$ is the fundamental frequency.

• Fundamental frequency: $f_1 = \frac{v}{2L}$

• Wave speed: $v = f \lambda$

Stringed Musical Instruments

• Fundamental frequency depends on tension and linear density: $f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$

• Resonance ensures only allowed standing wave frequencies persist.

Standing Sound Waves

Standing Waves in Tubes

• Sound waves are longitudinal pressure waves, creating compressions and rarefactions.

• At open ends, pressure is fixed at atmospheric value (node of pressure).

• At closed ends, pressure varies maximally (antinode of pressure).

• Open-open and closed-closed tubes have the same allowed wavelengths and frequencies as a string fixed at both ends.

• Open-closed tube fundamental frequency is half that of open-open or closed-closed tubes of the same length.

Standing Sound Wave Modes in Tubes

Tube Type	Allowed Wavelengths	Allowed Frequencies	Mode Numbers
Open-Open / Closed-Closed	$\lambda_m = \frac{2L}{m}$	$f_m = m \frac{v}{2L}$	$m = 1, 2, 3, ...$
Open-Closed	$\lambda_m = \frac{4L}{m}$	$f_m = m \frac{v}{4L}$	$m = 1, 3, 5, ...$

Wind Instruments

• Changing the effective length of the tube (e.g., opening holes) alters the resonant frequencies.

• Musical notes are produced by frequencies matching the tube's resonances.

Summary of Key Principles

• Principle of Superposition: The displacement at any point is the sum of the displacements from all waves present.

• Interference: Superposition leads to constructive (in-phase) or destructive (out-of-phase) interference.

• Standing Waves: Formed by two identical waves traveling in opposite directions; boundary conditions determine allowed modes.

• Nodes and Antinodes: Nodes are points of zero displacement; antinodes are points of maximum displacement.

• Standing Sound Waves: Boundary conditions (open/closed ends) determine the pattern and frequencies of standing waves in tubes.

Additional info: These notes are based on lecture slides for Physics for Life Sciences I, Lecture 27, covering Chapter 16 (Sections 16.1–16.4) on superposition, interference, and standing waves in strings and tubes.