Physics 4830 Course Notes                                                               Thursday 9/4/01

 

Last time:  Vibrations and complex periodic waveforms, frequency and period

 

frequency = 1 / period

 

Outline

 

Next Tues:  Wave reflection and begin architectural acoustics

 

Last time we discussed what a complex periodic waveform was and what was special about it and that it produced the sensation of distinct pitch.  Often we will drop the

adjectives "complex" - meaning that it is not sinusoidal, and "periodic" - meaning that

the waveform repeats itself at regular intervals, and just call waveforms "waveforms" or "waves" or sometimes "wave files" if they are samples stored on the computer.

 

As I said last time, I'd like to make a distinction between vibrations or oscillations and

waves.  This distinction is important, and unfortunately the English language uses one word to describe two things that are conceptually different.  When we look at a vibration we are looking at the motion as a function of time only.  When we look at waves, we are looking at the motion as both a function of time and space.  We will talk more about waves later today.  Of course, we call vibrations complex periodic waveforms, so keep in mind that this is a conceptual distinction that is often not made when using the word "wave" to describe something.

 

Digital Sampling

 

The best way to understand these complex waveforms is to really use them and see them using digital sampling.  Let's go through the process of recording a sample starting from

the microphone and understand each step.

 

1) Microphone - There are a variety of microphones, dynamic, condenser, electret condenser.  Let's take a look at a dynamic mic.  These mics are nice because they don't need an external power feed (either battery or phantom power), they are solid (pretty indestructible) and usually pretty inexpensive.  Sound waves cause the pressure to increase/decrease at the mic diaphragm in the audible range (20-20,000 Hz).  These increases/decreases in the background pressure push/pull on the diaphragm causing it to move.  The diaphragm is connected to a coil of wire, which moves in an external magnetic field supplied by a permanent magnet.

 

There is a law of physics called "Faraday's Law" that says that if the magnetic field passing through a loop of wire changes, it induces a voltage drop around that loop.  In the case of the dynamic microphone, as the coil moves, causing the magnetic field passing through the coil to change.  This, in turn, induces a voltage.  An electric generator works by using the same principle.   The microphone converts a pressure fluctuation into a time varying voltage.

 

2) The sound card - Now, this voltage from the mic, then goes into the sound card.  The sound card has a "preamp" which boosts the time varying voltage (or signal).  Next, the signal is converted from a smooth analog signal to a discrete digital signal (represented as integer numbers).  You can see visually this sampling process by using Soundprobe to resample at a very low resolution.  The soundcard takes discrete samples in time and takes the voltage at each time and converts it to an integer number.  The electrical component that converts the analog voltage to a digital voltage (or integer) is called an analog-to-digital converter.   The whole process is called digital sampling.

 

3)  The digital sound editing software then takes this digital signal, stores it, displays it and plays it back.

 

Sample Properties

 

Computers use binary integer numbers to represent data.  So, we need to know a little (very little) about the conventions used when digital samples are stored on the computer.

 

As an example, let's take CD quality recording.  There are 44,100 samples taken each second. Or, one every 0.0000227 seconds.  Two signals are stored, a left and a right channel.  The signal is represented as a "16 bit" binary number.  This means in base-ten

(the number system we are familiar with working in) the signal runs from:

 

0 to 216             or from            0 to 65,536      or from            -32,768 to +32,768

 

This is a pretty good quality sample, and it will sound good.

 

We can calculate how much data is involved in storing a CD quality sample for say one second.

 

44,100 samples x 2 channels x 16 bits = 1,411,200 bits/sec

 

There are 8 bits per "byte", so

 

1,411,200 bits / 8 = 176,400 bytes/sec

 

CDs hold about 74 minutes or

 

176,400 bytes/sec x 74 minutes x 60 sec/min = 783 Mbytes

 

Typically, for music, we want a good quality sample (CD resolution or better). You don't always want a CD quality sample rate and resolution because it uses up a lot of memory.

 

In Soundprobe, you have the option of 8, 16 or 32-bit resolution and 8,000-192,000 Hz sampling rates.  I do not know how high a sampling rate the soundcards we have in the lab are capable of.  My experience is that CD quality is pretty good, and it is hard to hear the difference between 16 bit, 44,100 Hz and a higher quality sample.

 

Nyquist Frequency

 

Nyquist frequency = sampling rate / 2

 

The Nyquist frequency is an upper bounds on the frequency range of a sample.  There will be no sounds present with frequencies higher than the sampling rate.  For, CD quality samples the Nyquist frequency is 22,050 Hz, which is just above the audible range.  For a 8,000 Hz sample, the Nyquist frequency would be 4,000 Hz and this would be where the frequency spectrum would be cut off.  4,000 Hz is well within the audible range, so this is not a good quality sample rate.  Maybe it would be ok for speaking voice sound.

 

Sound Waves

 

The plan is to discuss the physics of waves some, then launch into architectural acoustics.  Concert hall acoustics gives a solid example of sound wave propagation and its importance.

 

Air is compressible and springy.  If you compress it, it springs back.  Try clogging the hole of a bicycle pump or squeezing an empty 1-liter plastic bottle with the cap on it.

It is this springiness of air, which supports sound waves.

 

If we compress air in one place and let it then spring back, it expands and compresses

air in a region near to it.  This compression/expansion moves out from the source at the

sound speed, which is

 

v = 344 m/s  or

 

v = 331.3 m/s + 0.6 T (in Celsius) m/s

 

Sound waves are a little bit hard to develop an intuitive feel for, since we can't see them.

There are fairly exact partial differential equations, which treat the air as a fluid and describe the forces on a small volume element of air.  These equations can be used to quantitatively predict sound waves.  But, what we want is an intuitive feel for what these waves are and some ways to predict their behavior.   It is important to realize that the air molecules themselves are not moving from the source to your ear.  The air molecules are actually moving at a speed comparable to the sound speed, but they are continually colliding with one-and-other and do not move very far.  There are 1015 air molicules per m3.  What is moving from the source to your ear is the compressions/expansions.  This is true of waves in general:  The medium moves a short distance back and forth, but the wave can go great distances.

 

Wave Motion

 

A good way to understand sound waves is to examine wave phenomena in other mediums. We know that wave motion is a generic physical phenomenon and we can take what we learn from observing waves we can actually see and apply it to sound.  Of course, you should be skeptical and ask "Why should we expect waves on the ocean to behave like sound waves?"  Well, physicists have learned that waves of various types have common properties.  But, more importantly, we can do various experiments with sound waves, and gradually begin to trust that "Yes, sound does behave like a water wave in this context."  When using wave analogs you need to keep in mind both the similarities AND the differences between various types of waves.  For example, there are two types of waves:

 

Transverse Waves   waves on a string, water waves, electromagnetic waves

 

Motion of the medium is perpendicular to the direction the wave propagates.

 

Longitudinal Waves  Sound waves, some seismic waves, waves on a spring

 

Motion of the medium is in the direction of wave propagation.

 

For all mechanical waves, we have a "medium" which moves back and forth slightly, and a wave, which carries information, and energy, which moves great distances (relative to the motion of the medium).

 

 

The wavelength is the distance it takes for the wave to repeat itself.  Look at a snapshot in time of the wave (keep time fixed).

 

The period is the time it takes the wave to repeat itself. Look at a fixed location and plot the wave motion versus time (keep position fixed).

 

frequency = 1 / period (as before and always)

 

The wave speed is how fast the wave is propagating (or moving).

 

frequency = speed / wavelength

 

Why?  Take a fixed wave shape and move past a fixed point with a fixed speed, you will then see an oscillation that has a frequency corresponding with this formula.