College of Santa Fe Auditory Theory

Lecture 004-Sound III


  1. Superposition
  2. Sound refraction
  3. Refraction over ground
  4. Refraction over water
  5. Calculation of refraction
  6. Sound absorption
  7. Sound reflection from hard boundaries
  8. Sound reflection from bounded to unbounded boundaries
  9. Sound interference
  10. Standing waves at hard boundaries
  11. Standing waves at other boundaries
  12. Sound diffraction
  13. Sound scattering
  14. Brain Bullets

1.5 Sound Interactions

1.5.1 Superposition

Fig 1.14 Superposition of a sound wave in the golf ball spring model

When sounds destructively interfere with each other they do not disappear. Instead they travel through each other. Similarly, when they constructively interfere they do not grow but simply pass through each other. This is because, although the total pressure, or velocity component, may be either equal to zero or the sum of the amplitude of the individual waves, the energy flow of the sound wave is still preserved and so the wave continues to propagate. Thus the pressure or velocity at a given point in space is simply the sum, or superposition, of the individual waves that are propagating through that point, as shown in Figure 1.14. This characteristic of sound waves is called linear superposition and is very useful as it allows us to describe, and therefore analyse, the sound wave at a given point in space as the linear sum of individual components.

Two waves of different frequencies produce an output which is alternately louder and softer

1.5.2 Sound refraction

Sound Refraction (Video Demo)

Sound bends towards the slower medium

Refraction over ground

Refraction over water

Those of you who are students of the Bible will recognize that Jesus of Nazareth understood acoustics very well. There are many accounts of him getting into a boat and pushing off from the shore to teach. Over water there is a thermal inversion and the sound is bent back down and reinforces the direct sound like this:

Teaching from a boat meant that his voice would carry very much further than if he had been standing on land where his voice would have been carried up and away from the crowds. This also explains why sounds carry much farther at night. Under normal conditions the sun heats the earth which then radiates the heat back into the atmosphere creating the condition shown in the upper refraction diagram. At night the earth there is no longer a source of radiant energy and cool air sinks to the bottom of the atmosphere creating a thermal inversion.

Calculation of refraction

Change in Angle due to Refraction

Sine Theta 1/Sine Theta 2 = sqrt(Temperature1/Temperature2)
Temperature of Layer 1 Celsius
Temperature of Layer 2 Celsius
Angle of refraction:

1.5.3 Sound absorption

Sound is absorbed when it interacts with any physical object. One reason is the fact that when a sound wave hits an object then that object will vibrate, unless it is infinitely rigid. This means that energy is transferred from the sound wave to the object that has been hit as vibration. Some of this energy will be absorbed because of the internal frictional losses in the material that the object is made of. Another form of energy loss occurs when the sound wave hits, or travels through, some porous material. In this case there is a very large surface area of interaction in the material, due to all the fibres and holes. There are frictional losses at the surface of any material due to the interaction of the veloc ity component of the sound wave with the surface. A larger surface area will have a higher loss which is why porous materials such as cloth or rockwool absorb sound waves strongly.

1.5.4 Sound reflection from hard boundaries

Sound is also reflected when it strikes objects and we have all experienced the effect as an echo when we are near a large hard object such as a cliff, or large building. There are two main situations in which reflection can occur. In the first case the sound wave strikes an immovable object, or hard boundary, as shown in Figure 1.19. At the boundary between the object and the air the sound wave must have zero velocity, because it can't move the wall. This means that at that point all the energy in the sound is in the compression of the air, or pressure. As the energy stored in the pressure cannot transfer in the direction of the propagating wave, it bounces back in the reverse direction, which results in a change of phase in the veloc ity component of the wave. Figure 1.19 shows this effect using our golf ball and spring model. One interesting effect occurs due to the fact that the wave has to change direction and that is that the spring connected to the immovable boundary is compressed twice as much compared to a spring well away from the bound ary. This occurs because the velocity components associated with the reflected (bounced back) wave are moving in contrary motion to the velocity components of the incoming wave, due to the change of phase in the reflected velocity components. In acoustic terms this means that while the velocity component at the reflecting boundary is zero, the pressure component is twice as large.

Fig 1.19 Reflection of a soundwave due to a rigid barrier

Reflection from a concave surface (Video Demo)

1.5.5 Sound reflection from bounded to unbounded boundaries

In the second case the wave moves from a bounded region, for example a tube, into an unbounded region, for example free space, as shown in Figure 1.20. At the boundary between the bounded and unbounded regions the molecules in the unbounded region find it a lot easier to move than in the bounded region. The result is that, at the boundary, the sound wave has a pressure component which is close to zero and a large velocity component. Therefore at this point all the energy in the sound is in the kinetic energy of the moving air molecules, in other words, the velocity component. Because there is less resistance to movement in the unbounded region the energy stored in the velocity component cannot transfer in the direction of the propagating wave, due to there being less 'springiness' to act on. Therefore the momentum of the molecules is transferred back to the 'springs' in the bounded region which pushed them in the first place, by stretching them still further. This is equivalent to the reflection of a wave in the reverse direction in which the phase of the wave is reversed, because it has started as a stretching of the 'springs', or rarefaction, as opposed to a compression. Figure 1.20 shows this effect using the golf ball and spring model in which the unbounded region is modelled as having no springs at all. An interesting effect also occurs in this case, due to the fact that the wave has to change direction. That is, the mass that is connected to the unbounded boundary is moving with twice the velocity compared to masses well away from the boundary. This occurs because the pressure components associated with the reflected (bounced back) wave are moving in contrary motion to the pressure components of the incoming wave, due to the change of phase in the reflected pressure components. In acoustic terms this means that while the pressure component at the reflecting boundary is zero, the velocity component is twice as large. To summarise, reflection from a solid boundary results in a reflected pressure component that is in phase with the incoming wave whereas reflection from a bounded to unbounded boundary results in a reflected pressure component which is in antiphase with the incoming wave. This arises due to the difference in acoustic impedance at the boundary. In the first case the impedance of the boundary is greater than the propagating medium and in the second case it is smaller. For angles of incidence on the boundary, away from the normal, the usual laws of reflection apply.

Fig 1.20 Reflection of a sound wave due to bounded-unbounded transition



In a wind instrument (like a saxophone) this works as follows.

  1. Blowing in the mounthpiece causes a wave of pressure to travel down the horn.
  2. The Benouilli effect causes the reed to close the mouthpiece thus cutting off the air supply.
  3. When the wave reaches the bell or an open tone hole there is a sudden impedance drop and the wave is inverted and reflected back up the horn.
  4. When the reflected wave strikes the reed it forces the reed open thus allowing more air through which generates another pressure wave which starts down the horn again.

This looping cycle creates a vibrating column of air which then translates into a sound whose frequency is proportional to the frequency of the vibrating column.

1.5.6 Sound interference

We saw earlier that when sound waves come from correlated sources then their pressure and associated velocity components simply add. This meant that the pressure amplitude could vary between zero and the sum of the pressure amplitudes of the waves that are being added together, as shown in Example 1.9. Whether the waves add together constructively or destructively depends on their relative phases and this will depend on the distance each one has had to travel. Because waves vary in space over their wavelength then the phase will also spatially vary. This means that the constructive or destructive addition will also vary in space. Consider the situation shown in Figure 1.21, which shows two correlated sources feeding sound into a room. When the listening point is equidistant from the two sources (P1), the two sources add in constructively because they are in phase. If one moves to another point (P2) which is not equidistant, the waves no longer necessarily add constructively. In fact if the relative delays between the two paths is equal to half a wavelength then the two waves will add destructively and there will be no net pressure amplitude at that point. This effect is called interference, because correlated waves interfere with each other; note that this effect does not occur for uncorrelated sources.

Fig 1.23 Effect of frequency or wavelength on interference at a given position.

The relative phases of the waves depends on their relative delays, and this depends on the relative distances to the listening point from the sources. Because of this the pattern of constructive and destructive interferences depends strongly on position, as shown in Figure 1.22. Less obviously the interference is also strongly dependent on frequency. This is because the factor that determines whether or not the waves add constructively or destructively is the relative distances from the listening point to the sources measured in wavelengths. As the wavelength is inversely proportional to frequency one would expect to see the pattern of interference vary directly with frequency, and this is indeed the case. Figure 1.23 shows the amplitude that results when two sources of equal amplitude but different relative distances are combined. The amplitude is plotted as a function of the relative distance measured in wavelengths (A.). Figure 1.23 shows that the waves constructively interfere when the relative delay is equal to a multiple of a wavelength, and that they interfere destructively at multiples of an odd number of half wavelengths. As the number of wavelengths for a fixed distance increases with frequency, this figure shows that the interference at a particular point varies with frequency. If the two waves are not of equal amplitude then the interference effect is reduced, as shown in Figure 1.23. In fact once the amplitude interfering wave is less than one eighth of the other wave then the peak variation in sound pressure level is less than 1 dB.

There are several acoustical situations which can cause interference effects. The obvious ones are when two loudspeakers radiate the same sound into a room or when the same sound is coupled into a room via two openings which are separated.

Interference (Video Demo)

Java Applet demonstrating intereference

It is important to realize that playing monophonic data through stereo speakers can cause considerable interference. A pattern of phase cancelling is developed which looks like this:

The precise spacing of this interference pattern is different for every frequency .

Monophonic data stills has a strong place in the music library because it is easy to pan and locate in a stereo field and it often does not carry a strong ambient signature. Stereo data often carries the print of the room in which it was sampled which then forces the composer to join several dissimilar rooms in his mix. This can be problematic.

Fig 1.24 Interference arising from reflections from a boundary

Phase cancelling due to speakers

It is important when setting up a project studio to be able to predict the frequencies at which phase cancelling will occur.

Fig 1.25 Interference at a point due to two speakers

Which freqencies will cancel for a listener on axis to one speaker?

Distance between speakers: meters
Distance to listener: meters
frequency of wavelength/2:
frequency of wavelength/3:

There are many situations which can cause interference. Generally you should avoid recording instruments next to hard boundaries and be careful about how you position mono data in a mix. From experimenting with the above calculator you can soon see that the closer you are to the speakers the more likely tyou are to cancel bass frquencies. If you are trying to impress a client follow these rules.

  1. Sit the client exactly between the two speakers (most musicians position themselves in the sweet spot).
  2. If they must sit to one side position them at least two or three meters from the speakers

1.5.7 Standing waves at hard boundaries

The linear superposition of sound can also be used to explain a wave phenomenon known as standing waves, which is applicable to any form of sound wave. Standing waves occur when sound waves bounce between reflecting surfaces. The simplest system in which this can occur consists of two reflecting boundaries as shown in Figure 1.26. In this system the sound wave shuttles backwards and forwards between the two reflecting surfaces. At most frequencies the distance between the two boundaries will not be related to the wavelength and so the compression and rarefaction peaks and troughs will occupy all positions between the two boundaries, with equal probability, as shown in Figure 1.27. However, when the wavelength is related to the distance between the two boundaries then, as it travels between the two boundaries, the wave keeps tracing the same path. This means that the compressions and rarefactions always end up in the same position between the boundaries. Thus the sound wave will appear to be stationary between the reflecting boundaries, and so is called a standing wave. It is important to realise that the wave is still moving at its normal speed, it is merely that, like a toy train, the wave endlessly retraces the same positions between the boundaries with respect to the wavelength, as shown in Figure 1.28. Figures 1.28 and 1.29 show the pressure and velocity components respectively of a standing wave between two hard reflecting boundaries. In this situation the pressure component is a maximum and the velocity component is a minimum at the two boundaries. The largest wave that can fit these constraints is a half wavelength and this sets the lowest frequency at which a stand ing wave can exist for a given distance between reflectors, and can be calculated using the following equation:

flowest = L => λ/2 => flowest = ν/2/L


Any multiple of half wavelengths will also fit between the two reflectors as well and so there is, in theory, an infinite number of frequencies at which standing waves occur which are all multiples of flowest These can be calculated directly using:

fn = nν/2L


Fig 1.28 The pressure components of a standing wave between two hard boundaries

An examination of Figures 1.28 and 12.9 also shows that there are points of maximum and minimum amplitude of the pressure and velocity components. For example, in Figure 1.28 the pressure component's amplitude is a maximum at the two boundaries and the velocity component is zero at the midpoint. The point at which the pressure amplitude is zero is called a pressure node and the maximum points are called pressure antinodes. Note that as the number of half wavelengths in the standing waves increases then the number of nodes and antinodes increases, and for hard reflecting boundaries the number of pressure nodes is equal to, and the number of pressure antinodes is one more than, the number of half wavelengths. Velocity nodes and antinodes also exist, and they are always in the opposite sense to the pressure nodes, that is, a velocity antinode occurs at a pressure node and vice versa, as shown in Figure 1.30. This happens because the energy in the travelling wave must always exist at a pressure node carried in the velocity component and at a velocity node the energy is carried in the pressure component.

Fig 1.29 The velocity components of a standing wave between two hard surfaces

Fig 1.30 The pressure and velocity components of a standing wave between two hard boundaries

Java Applet demonstrating sound reflection

Standing Waves (Video Demo)

Standing waves form whenever hard boundaries face each other. Recording studios are usually built with non square floor plans to eliminate standing waves. However a rectangular floor plan is the normal situation in the thing we call a "room" and it will be a problem in a project studio. When the distance between two walls equals a particular wavelength then a sound of that wavelength will keep tracing and retracing the same path across the room. Thus the sound will appear to be stationary and therefore the name "standing wave". The largest wave that can fit between two walls is 1/2 a wavelength (because of the phase inversion of a wave at a hard boundary)

Standing wave between hard boundaries

frequency (f) lowest = Velocity of sound (v)/2*distance between boundaries (L)
Distance between boundaries: meters
Speed of sound: meters/sec
Lowest Standing Wave: Hz

It is important to realize that standing waves may form at an almost infinite number of multiples of this lowest frequency so there are really only two solutions: Avoid a square room or absorb/diffuse sound on the back wall.

Standing wave harmonics

frequency (f) lowest = (N*Velocity of sound v)/(2*distance between boundaries L)
Distance between boundaries: meters
Speed of sound: meters/sec
Harmonic Number:
Lowest Standing Wave: Hz

1.5.8 Standing waves at other boundaries

There are two other pairs of boundary arrangements which can generate standing waves

In the first case at each end is a boundary where where the pressure component is zero and the velocity component is at maximum.

Like hard boundaries the minimum frequency for standing waves occurs when there is half a wavelength between the two boundaries. This means that the calculator for hard boundaries can also be used to calculate the modal frequencies of an open pipe. (See above).

Standing waves in air colums (Video Demo)

When a pipe is closed at one end the situation is more interesting. Because there is a pressure node at the hard boundary and a pressure antinode at the unbounded end it allows a standing wave to exist when there is only a quarter wavelength between the boundaries.

However a standing wave cannot exist at twice this frequency as this would require a pressure node or antinode at both ends. Instead the next standing wave that can be supported occurs at a frequenct that is three times the lowest frequency


Standing wave in mixed boundary

frequency (f) lowest = ((2N+1)*Velocity of sound v)/(4*distance between boundaries L)
Distance between boundaries(L): meters
Speed of sound(V): meters/sec
Harmonic Number(N):
Lowest Standing Wave: Hz

Standing waves can occur for any type of wave propagation and can occur in one two or three dimensions.


Standing waves in acoustic systems are commonly referred to as MODES. The lowest frequency is refferred to as the FIRST ORDER MODE and the multiples of this are higher order modes. Thus the third order mode is the third lowest frequency standing that can be supported by a particular system.

Standing waves also exists in flat objects like sound boards or indeed in any object where a cyclic path in which a wave may trave in phase with the previous round may exist.

1.5.9 Sound diffraction

Sound diffraction (Video Demo)

The diffraction of sound can be fairly well expressed with a simple rule: Low frequencies turn corners while high frequencies don't. This is very important for our ability to perceive the direction of sound since our head creates an acoustic shadow between our ears. The ear which is faurther from the sound woill hear more low frequency than the ear which is towards the sound threby giving us at least one directional cue.

The degree of diffraction depends upon the wavelength. Shorter wavelengths are unable to turn the corner


The effect is similar for sound passing though an opening:

Low frequencies:

High frequencies

1.5.10 Sound Scattering

If the size of the object struck by a sound wave is significantly larger than the wavelengtgh of the sound then reflection will occur, however if the size of the object is smaller than the wavelength of the sound then some scattering will occur from the smaller object.

Reading Assignment

Before next class please read Section

Pages 51 to 63 of Acoustics and Psychoacoustics. We may have a brief quiz on these sections at the beginning of the next class.

You Need to Know