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Physics of Sounds

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Physics of Sounds

Overview

Properties of vibrating systems

Free and forced vibrations

Resonance and frequency response

Sound waves in air

Frequency, wavelength, and velocity of a sound wave

Simple and complex sound waves

Periodic and aperiodic sound waves

Fourier analysis and sound spectra

Sound pressure and intensity

The decibel (dB) scale

The acoustics of speech production

Speech spectrograms

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)?

- As we have so far described them, the mass-spring system and the tuning fork representsystems 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.

- 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?

- 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.

- 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)?

- 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.

- 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.

- 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?

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

Frequency

Time

- 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 (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.