Waves and Sound: Properties, Superposition, Standing Waves, and Applications

Waves and Sound

Introduction

This study guide covers the fundamental concepts of waves and sound, including wave properties, superposition, standing waves, and their applications to strings and tubes. These topics are essential for understanding oscillatory motion and wave phenomena in physics.

Wave Properties

Basic Definitions

• Amplitude (A): The maximum displacement from equilibrium. Units: metres (m). Represents the "strength" or intensity of the wave.

• Wavelength (\(\lambda\)): The distance over which the wave repeats itself. Units: metres (m).

• Period (T): The time it takes for the wave to repeat itself. Units: seconds (s).

• Frequency (f): The number of waves that pass a point in one second. Units: Hertz (Hz). $f = \frac{1}{T}$

• Velocity (v): The speed of propagation of the wave. Units: metres per second (m/s). $v = f \lambda$

Sound Waves

Longitudinal Waves in Air

• Sound waves in air are longitudinal waves, where the oscillation of air molecules is parallel to the direction of wave propagation.

• Regions of compression (high pressure) and expansion (low pressure) move through the medium.

• Graphically, sound waves can be represented as variations in air density or pressure, often shown as sinusoidal curves.

Reflection of Waves

Wave Pulse Reflection

• When a wave pulse reaches a boundary, it can be reflected in different ways depending on the boundary condition:

• Fixed end: The pulse is inverted upon reflection.

• Free end: The pulse is reflected upright (not inverted).

• These principles apply to both mechanical and sound waves.

Superposition of Waves

Principle of Superposition

• The superposition principle states that when two or more waves are present at a single point in space, the total displacement is the sum of the displacements due to each individual wave.

• Unlike particles, waves can pass through each other without being destroyed or altered permanently.

• This principle leads to phenomena such as constructive and destructive interference.

Standing Waves

Formation and Properties

• Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere.

• This occurs at certain frequencies (resonant frequencies) where the length of the medium matches the wave's wavelength appropriately.

• Node: A point of zero displacement or pressure.

• Antinode: A point of maximum displacement or pressure.

• Mode (m): The allowed standing wave patterns, labeled by an integer m (mode number).

Nodes and Antinodes

• For a string fixed at both ends, the number of nodes increases with the mode number:

• m = 1: Two nodes (fundamental mode)

• m = 2: Three nodes (first overtone)

• m = 3: Four nodes (second overtone)

Waves on a String

Modes and Harmonics

• For a string fixed at both ends:

• Fundamental mode (m = 1): $L = \frac{1}{2} \lambda_1$

• First overtone (m = 2): $L = \lambda_2$

• Second overtone (m = 3): $L = \frac{3}{2} \lambda_3$

• Frequency for the fundamental: $f_1 = \frac{v}{2L}$

• For higher modes: $f_m = m f_1$ where $m = 1, 2, 3, \ldots$

Modes, Overtones, and Harmonics

Mode (m)	Overtone	Harmonic
1	Fundamental	1st
2	1st	2nd
3	2nd	3rd
...	m-1	m

Additional info: The correspondence between mode, overtone, and harmonic is more complex for partial harmonic series, such as in tubes closed at one end.

Waves on a Guitar String

• The wavelength of a fingered string is shorter than that of an unfingered string, resulting in a higher frequency (pitch).

• Frequency: $f = \frac{v}{\lambda} = \frac{v}{2L}$

• The speed of a wave on a string depends on the tension ($F_T$) and the linear mass density ($\mu$): $v = \sqrt{\frac{F_T}{\mu}}$ where $\mu = \frac{m}{L}$

Speed of Sound in Air

Properties and Formula

• At room temperature (20°C), the speed of sound in air is about 343 m/s.

• General formula for speed of sound in air: $v = (331 + \frac{3}{5} T_C)$ m/s, where $T_C$ is the temperature in degrees Celsius.

• At standard temperature and pressure (STP, $T_C = 0$), $v = 331$ m/s.

• Air is a non-dispersive medium: all sound frequencies travel at the same speed.

Standing Waves in Tubes

General Concepts

• Standing waves can form in tubes, but visualization is more challenging than with strings.

• Sound waves in tubes are created by pressure variations, but are often analyzed in terms of air molecule motion.

A Tube Open at Both Ends

• Both ends are open to the atmosphere, so both ends are antinodes (maximum displacement).

• The simplest standing wave has an antinode at each end and a node in the middle.

• Higher-order modes require an antinode at each end, with additional nodes and antinodes in between.

• This is analogous to a string fixed at both ends.

A Tube Closed at One End, Open at the Other

• One end is closed (node), the other is open (antinode).

• The simplest standing wave has a node at the closed end and an antinode at the open end.

• Only odd harmonics are present (partial harmonic series): $f_m = f_1, 3f_1, 5f_1, \ldots = m f_1$ where $m = 1, 3, 5, \ldots$

• Even harmonics (2nd, 4th, etc.) are missing.

Example: Organ Pipe

• For a tube closed at one end, the fundamental frequency is $f_1 = \frac{v}{4L}$.

• Given $f_1 = 256$ Hz and $v = 340$ m/s, the required length is $L = \frac{v}{4f_1} = 0.33$ m.

Modes, Overtones, and Harmonics – Comparison Table

String or Open-Open Tube			Open-Closed Tube
Mode (m)	Overtone	Harmonic	Mode (m)	Overtone	Harmonic
1	Fundamental	1st	1	Fundamental	1st
2	1st	2nd	3	2nd	3rd
3	2nd	3rd	5	4th	5th
m	m-1	m	odd m	(odd m)-1	odd m

Additional info: Both the string and open-open tube have a full harmonic series, but the open-closed tube has a partial harmonic series (even harmonics are missing).

Problem Hints

• For wind instruments (tubes), use the speed of sound: $f_1 = \frac{v}{2L}$ (for open-open tubes).

• For stringed instruments, use the speed of the wave on the string: $f_1 = \frac{v}{2L} = \frac{1}{2L} \sqrt{\frac{F_T}{m/L}} = \frac{1}{2L} \sqrt{\frac{F_T}{\mu}}$