Waves and Sound: Properties, Reflection, Superposition, Standing Waves, and Applications in Strings and Tubes

Waves and Sound

Wave Properties

Waves are disturbances that transfer energy through a medium or space. Understanding their properties is fundamental to analyzing sound and other wave phenomena.

• Amplitude: The maximum displacement from equilibrium. It measures the "strength" of the wave. Units: metres (m).

• 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. $f = 1/T$. Units: Hertz (Hz).

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

Example: A sound wave with a frequency of 440 Hz and a wavelength of 0.78 m travels at $v = 440 \times 0.78 = 343.2$ m/s.

Sound Waves in Air – Longitudinal Waves

Sound waves in air are longitudinal waves, meaning the oscillation of particles is parallel to the direction of wave propagation.

• Compression: Regions where air molecules are close together.

• Expansion (Rarefaction): Regions where air molecules are spread apart.

Example: The vibration of a drum membrane creates alternating compressions and expansions in the air, producing sound.

Graphical Representation of Sound Waves

Sound waves can be represented graphically in two ways:

• (a) As regions of compression and rarefaction in the medium.

• (b) As a sinusoidal graph showing the variation in air density or pressure with position.

Example: The density of air oscillates above and below normal as a sound wave passes through.

Reflection of Wave Pulses

When a wave pulse reaches a boundary, its behavior depends on the nature of the boundary.

• Fixed End: The pulse is reflected and inverted.

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

Example: A pulse sent down a rope attached to a wall (fixed end) returns inverted; if the end is free to move, it returns upright.

Superposition of Waves

When two or more waves overlap, their displacements add together at each point in space. This is called the superposition principle.

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

• The resulting displacement is the algebraic sum of the individual displacements.

Example: Two pulses traveling towards each other on a string will combine when they meet, then continue unaffected.

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere. This occurs at specific frequencies and wavelengths, leading to resonance.

• Standing waves only form when the length of the medium matches the frequency of the wave.

• Node: Points of zero displacement or pressure.

• Antinode: Points of maximum displacement or pressure.

• Mode: Allowed patterns of standing waves, denoted by integer m.

Example: A string fixed at both ends can support standing waves with nodes at the ends and antinodes in between.

Standing Waves – Nodes and Antinodes

Each mode of a standing wave has a specific number of nodes and antinodes:

• m = 1: Two nodes (fundamental mode)

• m = 2: Three nodes (first overtone)

• m = 3: Four nodes (second overtone)

Example: The fundamental mode of a string has one antinode in the center and nodes at the ends.

Modes, Overtones, and Harmonics

Modes, overtones, and harmonics are related but distinct concepts in wave physics.

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

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

Waves on a String

Strings fixed at both ends support standing waves at discrete frequencies. The wavelength and frequency depend on the length of the string and the speed of the wave.

• Fundamental mode (m = 1): $L = \frac{1}{2}\lambda_1$, $\lambda_1 = 2L$, $f_1 = \frac{v}{2L}$

• First overtone (m = 2): $L = \lambda_2$, $f_2 = \frac{v}{L} = 2f_1$

• Second overtone (m = 3): $L = \frac{3}{2}\lambda_3$, $f_3 = \frac{v}{2L/3} = 3f_1$

Example: A guitar string of length 0.65 m and wave speed 520 m/s has a fundamental frequency $f_1 = 520/(2 \times 0.65) \approx 400$ Hz.

Waves on a Guitar String

The frequency of a guitar string depends on its length, tension, and mass per unit length.

• Fingered strings are shorter, producing higher frequencies (pitches).

• The speed of the wave on the string is $v = \sqrt{F_T/\mu}$, where $F_T$ is tension and $\mu$ is linear mass density ($\mu = m/L$).

Formulas:

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

• $v = \sqrt{\frac{F_T}{\mu}}$

Example: Increasing the tension $F_T$ raises the wave speed and thus the pitch.

Speed of Sound in Air

Sound travels at a finite speed in air, which depends on temperature.

• At room temperature (20°C), $v \approx 343$ m/s.

• General formula: $v = 331 + \frac{3}{5}T_C$ m/s, where $T_C$ is temperature in Celsius.

• At STP ($T_C = 0$), $v = 331$ m/s.

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

Example: Sound takes about 3 seconds to travel 1 km.

Standing Waves in Tubes

Standing waves can also form in tubes, with different boundary conditions than strings.

• In tubes, standing waves are visualized by considering the motion of air molecules.

• Nodes and antinodes correspond to points of minimum and maximum air displacement.

Example: Organ pipes and wind instruments use standing waves in tubes to produce sound.

A Tube Open at Both Ends

For a tube open at both ends:

• Pressure is constant at both ends (equal to atmospheric pressure).

• Antinodes at both ends; nodes occur within the tube.

• Same standing wave patterns as a string fixed at both ends.

Example: Flutes and some organ pipes are open at both ends.

A Tube Closed at One End, Open at the Other

Tubes can also be closed at one end and open at the other, leading to different standing wave patterns.

• Node at the closed end, antinode at the open end.

• Only odd harmonics are present (partial harmonic series).

Example: Clarinets and some organ pipes are closed at one end.

Modes – Tube Closed at One End, Open at the Other

For tubes closed at one end:

• Fundamental mode: $\lambda_1 = 4L$, $f_1 = \frac{v}{4L}$

• Third mode: $\lambda_3 = 4L/3$, $f_3 = \frac{3v}{4L} = 3f_1$

• General: $f_m = mf_1$, where $m = 1, 3, 5, \ldots$

Additional info: Even harmonics (2nd, 4th, ...) are missing in this series.

Modes, Overtones, and Harmonics – Revisited

The relationship between modes, overtones, and harmonics differs for strings/open-open tubes and open-closed tubes.

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: Open-closed tubes lack even harmonics (odd overtones are missing).

Problem Hints

When solving problems, use the correct wave speed for the instrument:

• Wind instrument (tube): Use the speed of sound in air. For open-open tubes: $f_1 = \frac{v}{2L}$

• Stringed instrument: Use the speed of the wave on the string: $f_1 = \frac{v}{2L} = \frac{1}{2L}\sqrt{\frac{F_T}{m/L}} = \frac{1}{2L}\sqrt{\frac{F_T}{\mu}}$

Example: For a tube of length 0.33 m and $v = 340$ m/s, $f_1 = 340/(4 \times 0.33) \approx 258$ Hz.