Wave Properties

Frequency, wavelength, amplitude, phase

Lorenz Schwarz
Karlsruhe University of Arts and Design (HfG)

Winter Semester 2024/25
Course info

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Audio signals (electrical representation)

A transducer (e.g., microphone) converts the varying air pressure of a sound wave into a continuous electrical signal, which is proportional to the sound pressure variations.

→ Electrical sound signals in this form are analogous to the sound pressure levels.

Electronic sound and audio signals

Electronic audio signals are variations in electrical energy (changes in voltage or current), that correspond to sound pressure variations.

→ Audio signals can be converted into audible sound through a speaker or other transducers.

Wave properties

The shape of the graph of a periodic function can be described using the following terms:

1. Amplitude - The maximum instantaneous value of the wave.
2. Period - The time interval after which the wave repeats.
3. Frequency - The reciprocal of the period (oscillations per second).
4. Phase angle - Describes the offset or difference between two sine waves.
5. Wavelength - The distance over which the wave repeats (spatial period).

Related:
Zero crossing - Point where the wave crosses zero.

Amplitude

Amplitude describes the maximum variation of a periodic signal (such as air pressure, displacement, or voltage) within a single period:

• Maximum instantaneous value of the variable.
• Maximum distance between the resting position (equilibrium) and the point of maximum displacement.
• Perceived as the loudness of sound in the context of audio signals.

→ Amplitude has no influence on frequency, wavelength, phase, period of time, and speed of sound.

Amplitude and energy transfer in waves

Amplitude relates to the wave's energy:

• Higher amplitude corresponds to greater energy transfer in the wave

This energy can be converted into work, heat, or other forms depending on the medium and interaction.

Amplitude

Root mean square (RMS)

The average value of a sine wave over a full cycle is zero (positive and negative halves cancel). RMS solves this by squaring values first, making them all positive:

RMS represents the effective value: the equivalent DC level that delivers the same power.

→ Electrical and acoustic power are proportional to the square of the RMS value.

Calculating RMS

RMS and power

Electrical and acoustic power are proportional to the square of RMS values:

• Amplifier power ratings use RMS voltage and current
• Sound intensity is proportional to RMS sound pressure squared

→ RMS enables meaningful comparison of signal strength and power delivery across different waveforms.

Period

Period

Example:

Frequency

Variations or alterations between 0,05 ms (20 kHz) and 50 ms (20 Hz) are perceived as sound.

• Ultrasound: higher than 20 000 Hz.
• Infrasound: lower than 20 Hz.

→ Hearing range for humans is 20 Hz to 20 000 Hz.

Frequency

Angular frequency ()

Whereas Hertz [Hz] counts cycles per second, radians per second [rad/s] measure the angle swept per second by a rotating pointer.

• Hz: cycles per second (cps)
• rad/s: angle swept per second (rotating motion)

Hertz and radian can be expressed as reciprocal seconds:

Radian

Example: Relating rad/s to Hz

One radian per second:

Frequency and period

The period is the reciprocal of the frequency and vice versa.

Phase

Phase angle

Zero crossing

The wave equation

All wave properties are interconnected through a single relationship:

• c is the propagation speed of the wave (e.g. 343 m/s in air)
• Amplitude determines energy, independent of the other properties
• Frequency and period are reciprocals:
• Wavelength depends on both frequency and medium:
• Phase describes position within a cycle

→ Changing one property (except amplitude) necessarily affects others.

Wavelength

Wavelength

Spatial period:

• distance over which the wave's shape repeats (related to frequency)
• parallel to the direction of propagation

Wavelength
Frequency
Speed of sound

Calculating wavelength from frequency

Example:

What is the wavelength of a 440 Hz tone in air, where sound speed is 343 m/s?

Wavelength

Wavelength and frequency

Frequency and wavelength are inversely proportional to each other:

Speed of sound

The speed of sound is the distance a sound wave travels per unit of time through a medium.

Speed of sound:

Factors affecting the speed of sound

• Elasticity: The more elastic (less compressible) the medium, the faster sound travels.
• Density: Higher density generally slows sound in gases but may increase speed in solids or liquids if accompanied by high elasticity.
• Temperature: In gases, higher temperatures increase the speed of sound by reducing the medium's density and increasing molecular energy.

Speed of sound in gases

heat capacity ratio
density
pressure
molar gas constant
molar mass
thermodynamic temperature in kelvin
Boltzmann constant
molecular mass

Example: Time–distance relationship of sound

• 34 cm/ms
  • sound travels about 34 cm per millisecond
• 3 ms/m
  • Sound takes roughly 3 ms to travel 1 meter

(Assuming speed of sound = 343 m/s)

Applying the speed of sound formula

Example:
Determining the distance of a lightning bolt (and thunderstorm cell):

• Every 3 seconds of delay ≈ 1 kilometre distance from the lightning bolt.

Question:
How long does sound need to travel 2 m?

Wave propagation

Sound speed in an elastic medium depends on temperature.

• Lower temperature → lower speed
• 1°C change ≈ 60 cm/s change in sound speed

Practical use: delay lines in PA systems

In large venues with multiple speaker arrays, a listener may hear sound from nearby speakers arrive before sound from distant speakers.

Electronic delay compensates for physical distance, time-aligning all sources at the listener position.

Speed of sound and related terms

Generally, sound travels faster in denser and less compressible media.

• Subsonic: Motion or speed less than the speed of sound in a given medium.
• Infrasonic: Sound waves with frequencies lower than ~20 Hz (below the range of human hearing).

Doppler effect

The change in frequency or pitch of sound waves perceived by an observer due to the relative motion between the sound source and the observer.

• The pitch is higher than the stationary pitch as the source approaches.
• The pitch decreases as the source passes the observer.
• The pitch becomes lower than the stationary pitch as the source moves away.

Used in rotary (Leslie) speakers and film sound design plug-ins.

▶ Doppler effect applied to a moving sound source (plugin simulation)

Doppler effect formula

Stationary receiver ():

• = observed frequency
• = source frequency
• = speed of sound
• = receiver velocity
• = source velocity

   Stopped      Subsonic    Speed of sound   Supersonic

Polarity inversion

Polarity inversion

Applications:

• Differential signalling for transmitting analog audio.
• "Phase" button on mixing desks to avoid phase cancellation

Superposition and interference

Interference, a consequence of superposition, describes the interaction between sound waves. The resultant amplitude is the sum of the individual amplitudes:

• amplification (constructive, even multiple of )
• attenuation (destructive)
• cancellation (destructive, odd multiple of )

view in graphing calculator

Constructive and destructive interference

Applications:

• Chorus (multiple copies of the same signal, slightly delayed and out of tune)

• Phaser (copied signal runs through an all-pass filter and is then mixed with its original)

• Active noise control

▶ Phasing effect applied to white noise

Wave properties and sound design

The following slides on envelope and amplitude modulation connect amplitude concepts to time-varying behavior.

Listening examples: Amplitude modulation

Varying the amplitude of a 400 Hz sound with a lower-frequency modulation signal:

1. Slow rates ( < 4 Hz ) → Pulsation.
2. Moderate rates ( 4 - 30 Hz ) → Tremolo.
3. Faster rates ( 30 - 70 Hz ) → Roughness.
4. Very fast rates ( > 70 Hz ) → Spectral coloration.

Copyright and Licensing

Original content: © 2025 Lorenz Schwarz
Licensed under CC BY 4.0. Attribution required for all reuse.

Includes: text, diagrams, illustrations, photos, videos, and audio.

Third-party materials: Copyright respective owners, educational use.

Contact: lschwarz@hfg-karlsruhe.de

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