Physics of SoundsOverview

Properties of vibrating systemsFree and forced vibrationsResonance and frequency responseSound waves in airFrequency, wavelength, and velocity of a sound waveSimple and complex sound wavesPeriodic and aperiodic sound wavesFourier analysis and sound spectraSound pressure and intensityThe decibel (dB) scaleThe acoustics of speech productionSpeech spectrograms

Properties of Vibrating Systems

Some terms • displacement: momentary

distance from restpoint B• cycle: one complete

oscillation• amplitude: maximum

displacement, “average” displacement

• frequency: number of cycles per second (hertz or Hz)

• period: number of seconds per cycle

• phase: portion of a cycle through which a waveform has advanced relative to some arbitrary reference point

What is the relation between frequency (f) and period (T)?

How do these differ?

How do these differ?

How do these differ?

Another case of harmonic motion:tuning fork

Damping

Free vibration

• As we have so far described them, the mass-spring system and the tuning fork represent systems in free vibration. An initial external force is applied, and then the system is allowed to vibrate freely in the absence of any additional external force. It will vibrate at its natural or resonance frequency.

Forced vibration

• Now assume that the mass-spring system is coupled to a continuous sinusoidal driving force (rather than to a rigid wall).

How will it respond?

Resonance curve(aka: frequency response or

transfer function or filter function)

• In free vibration, the response amplitude depends only on the initial amplitude of displacement.

• In forced vibration, the response amplitude depends on both the amplitude and the frequency of the driving force.

Resonance

Sound waves

Sound waves (cont.)

Frequency, wavelength, and velocity of sound waves

• Wavelength: the spatial extent of one cycle of a simple waveform. (Compare this to period).

• If we know the frequency (f) and the wavelength (λ) of a simple waveform, what is its velocity (c)?

Simple vs. complex waves

• So far we’ve considered only sine waves (aka: sinusoidal waves, harmonic waves, simple waves, and, in the case of sound, pure tones).

• However, most waves are not sinusoidal. If they are not, they are referred to as complex waves.

Examples of complex waves:sawtooth waves

Examples of complex waves:square waves

Examples of complex waves:vowel sounds

Periodic vs. aperiodic waves

• So far all the waveforms we’ve considered (whether simple or complex) have been periodic—an interval of the waveform repeats itself endlessly.

• Many waveforms are nonrepetitive, i.e., they are aperiodic.

Some examples of aperiodic waves:

• A sine wave can be described exactly by specifying its amplitude, frequency, and phase.

• How can one describe a complex wave in a similarly exact way?

Fourier analysis

• Any waveform can be analyzed as the sum of a set of sine waves, each with a particular amplitude, frequency, and phase.

How to approximate a square wave

From time-domain to frequency-domain

Time Frequency

Periodic vs. aperiodic waves (cont.)

• Periodic waves consist of a set of sinusoids (harmonics, partials) spaced only at integer multiples of some lowest frequency (called the fundamental frequency, or f0).

• Aperiodic waves fail to meet this condition, typically having continuous spectra.

Sound pressure and intensity• Sound pressure (p) = force per square centimeter

(dynes/cm2)• Intensity (I) = power per square centimeter

(Watts/cm2)

• I = kp2

• Smallest audible sound = 2 x 10-4 dynes/cm2

= 10-16 Watts/cm2

• A problem: Between a just audible sound and a sound at the pain threshold, sound pressures vary by a ratio of 1:10,000,000, and intensities vary by a ratio of 1: 100,000,000,000,000! More convenient to use scales based on logarithms.

• Decibels (dBSPL,IL) = 20 log (p1/p0)= 10 log (I1/I0)

• where p1 is the sound pressure and I1 is the intensity of the sound of interest, and p0 and I0 are the sound pressure and intensity of a just audible sound.

Decibel scale

Acoustics of speech production

Spectrogram